Theoretical DFT study of phosphorescence from porphyrins

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Chemical Physics 315 (2005) 215–239

Theoretical DFT study of phosphorescence from porphyrins

Boris Minaev *, Hans Agren

Laboratory of Theoretical Chemistry, Department of Biotechnology, SCFAB, The Royal Institute of Technology, SE-10691, Sweden

Received 16 March 2005; accepted 13 April 2005Available online 7 July 2005

Abstract

Geometrical structure of free-base porphin (H2P) and Mg- and Zn-porphyrins together with their vibrational frequencies andvibronic intensities in phosphorescence are investigated by density functions theory (DFT) with the standard B3LYP functional.These molecules have a closed-shell singlet ground state (S0) and low-lying triplet (T1) excited states of pp* type. The S0–T1 tran-sition probability and radiative lifetime of phosphorescence (sp) of these molecules are calculated by time-dependent DFT utilizingquadratic response functions for account of spin–orbit coupling (SOC) and electric-dipole transition moments including displace-ments along active vibrational modes. The infrared and Raman spectra in the ground singlet and first excited triplet states are alsostudied for proper assignment of vibronic patterns. The long radiative lifetime of free-base porphin phosphorescence (sp � 360 s atlow temperature limit, 4.2 K) gets considerably shorter for the metalloporphyrins. An order of magnitude reduction of sp is pre-dicted for Mg-porphyrin but no change of phosphorescence polarization is found. A forty times enhancement of the radiative phos-phorescence rate constant is obtained for Zn-porphyrin in comparison with the H2P molecule which is accompanied by a strongchange of polarization and spin-sublevel radiative activity. A strong vibronic activity of free-base porphin phosphorescence is foundfor the b2g mode at 430 cm�1, while the 679 and 715 cm�1 vibronic bands of b3g symmetry are less active. These and other out-of-plane vibrations produce considerable changes in the radiative constants of different spin sublevels of the triplet state; they also pro-mote the S1 ! T1 intersystem crossing. Among the in-plane vibrations the ag mode at 1614 cm�1 is found very active; it produces along progression in the phosphorescence spectrum. The time-dependent DFT calculations explain the effects of the transition metalatom on phosphorescence of porphyrins and reproduce differences in their phosphorescence and EPR spectra.� 2005 Elsevier B.V. All rights reserved.

Keywords: Free-base porphin; Mg-porphyrin; Zn-porphyrin; Phosphorescence; Radiative lifetime; Vibronic bands

1. Introduction

Porphyrins constitute an important class of p-conjugated organic chromophores, which can be dopedby metal ions. These derivatives are involved in a numberof biological processes (photosynthesis, dioxygen trans-port and activation) which are of crucial importance forthe Earth biosphere [1–3]. In recent time the porphyrinshave been employed for applications in dye industry,

0301-0104/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2005.04.017

* Corresponding author. Present address: State University of Tech-nology, Department of Chemistry, bvd. Shevchenko 460, 18006Cherkassy, Ukraine. Tel.: +46 8 5537 8417; fax: +46 8 5537 8590.

E-mail address: [email protected] (B. Minaev).

solar energy conversion, artificial photosynthesis, pho-todynamic therapy, electrooptics and nonlinear optics[4–6], to mention a few out of many examples. To eluci-date the mechanisms of this biophotonics and bioma-chinery, the electronic structure and spectra ofporphyrins have been widely investigated [2–20].

Studies of triplet state spectroscopy present mostinteresting aspects of porphyrins [4,8–14]. The intensetriplet–triplet (T–T) absorption in the visible region isnow utilized in nonlinear optical devices and opticalstorage applications [4,8,9]. Lifetimes of triplet-excitedporphyrins and triplet quantum yields are importantquantities also in photobiology since they determinethe singlet oxygen O2(

1Dg) generation [1,6,21] under

mailto:[email protected]

216 B. Minaev, H. Agren / Chemical Physics 315 (2005) 215–239

irradiation of tissues with visible light. High concentra-tion of porphyrin pigments in the skin and in the bloodcan cause damage by the O2(

1Dg) generation upon light,since the singlet oxygen is a hazardous molecule thatoxidizes biopolymers [1,6]. In cancer treatment withphotodynamic therapy the O2(

1Dg) generation by por-phyrins is used in a controlled manner to oxidize a tu-mor [1,22].

Porphyrin derivatives which contain zinc, palladiumand platinum ions are known to exhibit phosphores-cence even at room temperature (zinc - to a less extent)[2,4,11–14]. The phosphorescence (radiative transitionfrom the first excited triplet state, T1, to the ground sin-glet state, S0), depends on spin–orbit coupling (SOC),which increases with atomic number. This notion ex-plains the so-called heavy atom effect [23], which is ob-served also in phosphorescence of porphyrins [2,12,24].At the same time the closed shell porphyrins withoutheavy atoms provide some exception from commonrules of luminescence of organic p-conjugated com-pounds [10,12,13]. Such porphyrins have extremelyweak phosphorescence though they have a very efficientnonradiative S1 [ T1 intersystem crossing [12,25]. Forexample, free-base porphin (H2P), which is the basicbuilding block and the electronic ‘‘heart’’ of porphyrins,has a quantum yield of the intersystem crossing equal to0.9 [25], which is typical for many other tetrapyrrolecompounds [10,14]. The weak phosphorescence meansthat sufficient triplet state population can be createdby high-intensity laser sources [4,8] and intense T–Tabsorption can make the transparent porphyrins be-come opaque in some part of the visible spectrum. Thus,the efficient intersystem crossing of porphyrins, whichmaintains a high concentration of triplet-excited mole-cules, is used in a wide variety of applications from pho-todynamic therapy to nonlinear optics and opticallimiting devices [8].

The photophysical properties of porphyrins are notcompletely understood in spite of their fundamental sig-nificance for numerous applications [10,12,26,27]. Themain puzzle of the luminescence of porphyrin free-basesand their complexes with light metals (Mg, Al) is con-nected with the question why their T1 ! S0 decay is al-most completely dark in solid organic solvents [13],though these molecules have nitrogen atoms with lonepairs of electrons. Usually such molecules with n-elec-trons exhibit strong phosphorescence with a relativelyshort lifetime because of the effective SOC-induced mix-ing between 1np* and 3pp* states [23].

The study of luminescence of porphyrins and theirrelative fluorescence and phosphorescence intensitieswas pioneered by Becker et al. [28–31]. They establishedthat only porphyrins with heavy metal ions exhibitappreciable phosphorescence. In porphyrins with a dia-magnetic ground state, the phosphorescence grows withthe weight of the ion. van der Waals et al. [13,32] studied

the fine structure and kinetics of populating and depop-ulating of the lowest triplet state of free-base porphinfrom a pulsed microwave technique. They analyzedtransient changes in the fluorescence intensity inducedby microwaves, which depend on the S1–T1 coupling,but did not detect phosphorescence itself in n-octaneat 1.3 K [13]. Tsvirko et al. [12,25] detected phosphores-cence of many porphyrin free bases in EPA and otherorganic solids. They found the phosphorescence quan-tum yield to be equal to about /p � 10�4. Since the mea-sured phosphorescence lifetime (s) is of the order 0.1 sthis leads to a very small radiative rate constant of phos-phorescence kp = /p/s � 0.001 s�1. Even pure hydrocar-bons (benzene, naphthalene) have higher kp values(0.1–0.01 s�1) [23,33].

The time-dependent DFT calculations with quadraticresponse functions [33–37] can be useful in solving thispuzzle in order to understand the very weak phospho-rescence of free-base porphin and its strong sensitivityto the external heavy atom effect. For more systematicstudies of the photophysics of porphyrins one needs toconsider the vibrational movements and their interac-tion with electrons and spins.

Infrared (IR) and Raman spectra of porphyrins arenow well documented and analyzed [20,24,38–41,43–45]. The knowledge of the force field, vibrationalfrequencies and their intensities are important for corre-lating IR and Raman spectroscopy measurements withstructural information and with luminescence vibronicactivity. It is difficult to obtain the empirical force fieldfor such large molecules like porphyrins, making quan-tum chemical calculations of the force constants the onlyeffective way to solve the problem [20]. Pulay et al.[20,44,45] calculated the force field of free-base porphin(H2P) and metalloporphyrins by the DFT B3LYPmethod [46,47] with the 6-31G* basis set [48]. For theZn ion [20] they used the TZV basis set of Ahlrichsgroup [49]. Using scaling factors optimized on the H2Pisotopomers Pulay et al. [44,45] interpreted the IR activefundamental frequencies (a2u,eu) and Raman spectra(a1g,b1g,b2g,eg) in agreement with empirical force fields[40,43]. Time-resolved resonance Raman spectra re-cently applied to excited states of porphyrins [50–52]provide very important information about changes ofvibrational modes on excitation to the triplet state.The simplest molecule studied so far is free-base octa-ethylporphyrin [50]. Though the time-resolved reso-nance Raman spectra are not available for themolecules studied in the present work, we will make aqualitative comparison of our triplet state data withthose known for more complicated derivatives [50–52].

In this paper, we are going to address the issue oftriplet state radiative properties and calculate theT1 ! S0 transition dipole moments of a number of por-phyrins in the framework of time-dependent DFT usingquadratic response (QR) functions. This method has

B. Minaev, H. Agren / Chemical Physics 315 (2005) 215–239 217

been recently developed in our laboratory [53] andproved to be very useful for calculations of phosphores-cence lifetimes [35,37]. We also present vibronic patternsof the T1 ! S0 transitions together with vibrationalanalyses of the ground and excited states.

2. Method of calculations

Calculations of phosphorescence, excitation energiesand oscillator strengths for porphyrins are carried outby time-dependent density functional theory (TD-DFT) implemented in a recent version of the Daltonprogram [53,54]. TD-DFT is actually an application ofthe response technique of Kohn–Sham DFT for calcula-tions of excitation energies and many related propertieswithin the density functional context [15,17]. The advan-tages of response theory have been documented manytimes in connection with applications to triplet statesproperties which depend on spin–orbit coupling [33].

The ground state density is determined by solving theKohn–Sham equations [55,56]:

dE ¼h0j½dj; H �j0i ¼ 0; ð1Þ

H ¼Xr

ZdsWrð~rÞy

dEdqrð~rÞ

Wrð~rÞ; ð2Þ

Wrð~rÞ ¼Xk

/kð~rÞakr. ð3Þ

Here, E[qa,qb] is a chosen functional with usual nota-tions for the Kohn–Sham determinant

j~0i ¼ e�jj0i; jrs ¼Xrsr

jrsayrrasr. ð4Þ

The Kohn–Sham orbitals /k form a representation ofthe ground state density

qr ¼ h0jWð~rÞyrWrð~rÞj0i � h0jqð~rÞj0i. ð5ÞThe influence of SOC perturbation on the singlet–triplet(S–T) transitions can be formulated in terms of qua-dratic response functions. The first order perturbationis an interaction between the photon electric field andthe electric dipole moment of the molecular electroniccloud. In this case the S–T transition is spin-forbidden.The transition is realized by the spin–orbit coupling per-turbing these two states and producing non-pure spinstates of mixed multiplicity. The expansion describingthese spin contaminations can then be written in termsof the ‘‘sum-over-states’’ perturbation approach [23,87]

MaS;T ¼ hS0jqjTa

1i

¼Xn

hS0jqjSnihSnjHSOjTa1i

ET 1� ESn

þXn

Xb

hS0jHSOjT bnihT b

njqjTa1i

ES0 � ETn

. ð6Þ

Here, q ¼P

iqi denotes all electronic coordinates; Ta

means zero-field spin sublevel of the triplet state withzero spin projection on the a-axis, connected withmolecular frame [13,14,87]. This can be used for calcula-tion of phosphorescence radiative lifetimes for each spinsublevel

1

sa¼ const:ðDEÞ3ðMa

S;TÞ2. ð7Þ

Application of the ‘‘sum-over-states’’ method, Eq. (6),can be efficient when only a few states contribute. Forpolyatomic molecules it has been shown [33,87] that thissum converges very slowly. Truncation of the row in Eq.(6) is very risky in many cases. At the same time, MRCIab initio calculations with large basis sets include toomany terms to be feasible. In response theory the‘‘sum-over-states’’ expression is implicitly obtained bysolving systems of linear equations without explicitaddressing the excited states [33]. Response theory isespecially effective in the DFT context when the ‘‘sum-over-states’’ value is readily obtained for big moleculeswhere the explicit summation is absolutely impossible.

The SOC perturbation destroys the spin symmetryand a non-vanishing S–T transition dipole momentcan be obtained as a residue of a quadratic responsefunction involving the combined perturbations of thespin–orbit coupling and the electric dipole–photon inter-actions [33,57]. The perturbation is denoted by the sym-bol V and the operator H þ V defines the Hamiltonianof the non-interacting system, which governs the time-evolution of the Kohn–Sham orbitals. The ground-statedensity can be expanded in a power series in the pertur-bation strength

qr ¼ qð0Þr þ qð1Þ

r þ qð2Þr þ � � � ð8Þ

One can assume that the zero-order ground-state a- andb-densities are equal.

Following Salek, Vahtras et al. [53], interpreting Eq.(4) as a parametrized time-evolution operator acting onthe converged Kohn–Sham determinant, one can applya time-dependent variational principle, which is basedon the Ehrenfest theorem for an operator Q, to solve re-sponse equations for the nth-order parameters jðnÞ

dn 0 Q; ej H ½qr� þ V � i�ho

ot

� �e�j

� ��0

� �¼ 0. ð9Þ

The nth order response functions, i.e., the nth-orderchanges of a property, may then be identified from theexpansion of a property A

A ¼ hAi þ h0j½jð1Þ; A�j0i þ h0j½jð2Þ; A�j0i

þ 1

2h0j½jð1Þ; ½jð1Þ; A��j0i þ � � � ð10Þ

The quadratic response function in the frequency do-main is given by


hhA; V ; V iix1;x2¼ h0j½jx1;x2 ; A�j0i

þ P 12h0j½jx1 ; ½jx2 ; A��j0i; ð11Þand the singlet–triplet transition amplitude is evaluatedas the residue at a triplet state excitation energy withthe dipole operator as the primary property A ¼~q andthe spin–orbit operator as a perturber V �x1 ¼ HSO, let-ting x1 = 0 (X is an arbitrary triplet operator) [57]

hSj~qjT i ¼ limx2!xT

ðx2 � xT Þhh~q;H so;X ii0;x2

hSjX jT i . ð12Þ

The SOC matrix elements hS0|HSO|T1i can be evaluatedas residues of linear response functions and SOC be-tween excited states are effectively obtained as residuesof quadratic response functions [33]. This formalism is

Fig. 1. Numeration of atoms and choice of a

implemented in the DALTON program [54] and hasbeen utilized in a number of recent works [35,53,37].For more details we refer to the original paper [53].

The geometrical structures of free-base porphin andMg, Zn-porphyrins are optimized for the ground stateand for the first excited triplet state; vibrational frequen-cies, IR and Raman intensities are found with Gaussiancode [47]. Becke�s three-parameter hybrid functional(B3LYP) method is used for all calculations [46].Numeration of atoms and choice of axes are shown inFig. 1. Mass-weighted displacements along the activenormal modes of gerade symmetry are used for vibroniccalculations.

Phosphorescence of porphyrins in low temperaturesolids [10,12,24,30] results from vibronic transitions

xes for the free-base porphin molecule.


where an excitation of vibrational modes occurs simulta-neously with the electronic T1–S0 transition. The phos-phorescence vibronic fine structure containsinformation not only about the frequencies of the vibra-tional modes but also about the weak magnetic pertur-bations of three spin-sublevels of the TMs

1 state andabout their dependence on vibrational movement. Therevelation of this dependence is an important step inunderstanding the phosphorescence of porphyrins.

For analysis of phosphorescence vibronic patternsone needs to consider vibrational frequencies and modesof the ground state; the initial state for this emission isthe lowest vibrational level of the T1 electronic excitedterm and the final states are vibrational excitations ofthe ground electronic S0 term. The general expansionof the total wave function of the ground state in the adi-abatic approximation WS0;nðq;QÞ ¼ S0ðq;QÞUnðQÞ canbe used, where S0 is the electronic ground state wavefunction, and q and Q denotes electronic and nuclearcoordinates [58]. In the harmonic approximation the nu-clear wave function Un(Q) is a product of wave func-tions of all vibrational modes Qi; we consider thatonly one of them is excited in the state with quantumnumber n; thus we neglect combinational lines.

The 0–n vibronic Ta1–S0 transition intensity is deter-

mined by the transition dipole moment [59,33]

hWS0;njqþQjWaT 1;0

i ¼ hS0jqjTa1i0hnj0i

þoMa

S;TðQÞoQi

hUnjQijU0i; ð13Þ

where

MaS;TðQÞ ¼ hS0ðq;QÞjqjTa

1ðq;QÞi. ð14Þ

The first term corresponds to the vertical transitionweighted by a vibrational overlap; in the intensityexpression it includes the Franck–Condon factorhn|0i2. The second term is known as the Herzberg–Tellercontribution to vibronic 1–0 band in the case of the sin-glet–singlet transitions [61,72]. For the singlet–triplettransition we use in Eq. (14) the value from Eq. (6)determined by quadratic response TD-DFT methodand provide numerical differentiation in the second termof Eq. (13) along the DFT-calculated normal mode.Quantization of the spin projection is implemented herefor the so-called ‘‘zero-field splitting’’ (ZFS) approach[23]. ZFS for the free-base porphin lowest triplet statehas been calculated before [37]; the spin-quantizationaxes coincide with the molecular symmetry axes andTa means a spin-sublevel with the total spin projectionon the a-axis being equal to zero. It is easy to see thatselection rules for vibronic activity in the singlet–singletand singlet–triplet transitions are quite different, since inthe former case the electric dipole operator is important,while for the S–T transition the SOC integrals and theirdependence on normal modes get into play.

The TD-DFT quadratic response calculations of theS–T transition moments in all porphyrins and theirvibrational mode dependences are performed with thesplitted 3-21 G basis set [60]. More sophisticated basissets are used for geometry optimization and vibrationalfrequency analysis and for comparative studies of theinternal heavy atom effect. The FC factors are calculatedthrough the excited state gradients with the code devel-oped by Macak et al. [61]. Calculations of the g-factor inthe the lowest triplet state are done by the restrictedDFT linear response method [62]. In this case SOC inte-gral calculations are performed with two approxima-tions: by means of the scaled atomic charges proposedby Koseki et al. [63] and with the atomic mean field inte-grals (AMFI) for the two-electron part [64]. In the phos-phorescence calculations only the AMFI approximationis used.

3. Results and discussion

We shall start with an analysis of the structure of themolecules in the ground and the first excited states. Thenwe shall discuss the singlet–singlet (S–S) absorptionspectra. Though this aspect of porphyrins photonics iswell studied including spectral interpretation at theDFT level, one needs to make some comparisons withtransition energies and intensities of singlet excitedstates in order to be assured of the triplet–singlet spec-trum. We shall compare the published data on S–Sabsorption and our results obtained with different basissets and with different geometries and we shall also pro-vide some additional arguments in favor of assignmentof new absorption bands.

3.1. Geometry optimization

The optimized geometries are presented in Table 1.All basis sets including the effective core potential(ECP) basis Lanl2DZ [47] and the 3-21G basis predictplanar structure for all studied molecules in the groundsinglet and first excited triplet states. Account of polar-ization function on hydrogen atoms in the 6-31G** ba-sis set provides some essential differences with previousstudies [8,18] performed with the 6-31G* basis. The dis-tance between nitrogen atoms and the center of the por-phyrin ring is an important parameter in calculations ofphosphorescence. For the H2P molecule in the 6-31G**basis set we get 2.028 and 2.117 A for deprotonated andprotonated nitrogens, respectively (the distances 2.046and 2.128 A optimized for the T1 state with the cc-pVTZbasis set have been used earlier [36,37]). The averageddistance of all four nitrogens is 2.0725 A. This indicatesthat the porphyrin cavity radius is larger than that esti-mated before [65]. For ZnP our result (2.047 A) is closerto the experimental value than the distance optimized by

Table 1Bond lengths (A) calculated by DFT B3LYP methods for the ground state of the H2P, MgP and ZnP molecules

Bond H2P MgP ZnP

6-31G**a 6-31G*b Exp.c 6-31G**a 6-31G**a 6-31G*b 6-311G**a Exp.d

b 0-b 0 C5–C7 1.3714 1.372 – 1.3651 1.362 1.364 1.361 1.349b-b C17–C18 1.3568 1.356 1.365 1.3651 1.362 1.364 1.361 1.349a 0-b 0 C5–C1 1.4353 1.435 1.438 1.4467 1.442 1.446 1.444 –a-b C13–C17 1.4592 1.460 – 1.4467 1.442 1.446 1.444 –a 0-m C1–C9 1.3932 1.394 1.387 1.4005 1.396 1.397 1.395 1.400a-m C13–C9 1.4004 1.400 – 1.4005 1.396 1.397 1.395 1.400a-NC14–N 1.3638 1.364 – 1.3724 1.371 1.373 1.370 –a 0-NC1–N 1.3721 1.372 1.380 1.3724 1.371 1.373 1.370 –H34–N35 1.0145 1.015 – – –b 0-H C5-H31 1.0813 1.082 1.0823 1.083 1.083 1.082b-H C17-H24 1.0826 1.083 1.0823 1.083 1.083 1.082m-H C9-H27 1.0857 1.086 1.0860 1.086 1.086 1.086M–N – – – 2.0642 2.047 2.056 2.048 2.037

a This work.b Ref. [8].c Ref. [91].d Ref. [92].


Nguyen et al. [18] with the 6-31G* basis (2.056 A). Devi-ations in vibrational frequencies are more serious inthese two basis sets as one can see from comparison ofdata in Table 5 and rescaled IR active frequencies in[18]. We note that the larger basis set 6-311G** doesnot alter our 6-31G** results (Table 1).

It is interesting to observe that for MgP the metal-nitrogen distance is much larger than for ZnP (2.064A) in spite of the smaller ionic radius of Mg2+ (0.65A) in comparison with Zn2+ (0.74 A) [20]. This is inagreement with Jarzecki et al. [20] and their contentionthat the size of the porphyrin cavity is larger than for-merly assumed [65] and that light metals can fit the cav-ity without ruffling and other alternations of thestereochemistry of the porphyrin core.

3.2. The singlet–singlet absorption spectra of porphyrins

All porphyrins have two major absorption bands,namely the weak Q band in the visible region (500–650nm) and the B band (or Soret band) in the near UV re-gion (350–400 nm). Excitation of free-base porphin, Mg-and Zn-porphyrins in the Soret band leads to internalconversion to the lowest singlet excited state S1; afterthat about 5% of the molecules decay back to theground state under fluorescence, while the S1 [ T1

intersystem crossing pushes about 90% of them to thelowest triplet state [12]. Thus, the proper analysis ofphosphorescence includes knowledge of the singlet ex-cited states. Results of the present calculations of thelow-lying excited states and the S–S absorption spec-trum of free-base porphin and metal porphyrins are pre-sented in Table 2.

The first interpretation of the S–S absorption spectraof porphyrins was achieved by Gouterman on theground of semi-empirical calculations and the four orbi-

tal model [7,66]; later the ZFS in triplet states was stud-ied with a similar approximation [67]. In recent time theold semi-empirical results [7,66] have been confirmed onthe ground of ab initio [16,68,69] and DFT calculations[15,17,19,27,70,71]. The TD-DFT extension for the sin-glet–triplet transition intensity calculation has beenfound quite successful for small two- and three-atomicmolecules [35,53] as well for free-base porphin [36,37].Relatively large basis sets (cc-pVTZ) have been used inthe H2P calculation [37]. As follows from Table 2 thesmaller basis sets also provide quite reasonable interpre-tations of the singlet–singlet absorption spectrum of theH2P molecule.

The excited states calculated with B3LYP by the TD-DFT code [37] are in a good agreement with the elec-tronic absorption spectra of all three porphyrinmolecules [7,93]. The first Qx absorption band, namelyits 0–0 branch in the visible region (k = 625 nm,DE = 1.984 eV in anthracene matrix [73]) is attributedto the 11B3u state (with the present choice of axes whichcoincides with Gouterman�s choice used by manyauthors [16,19,27,68,70,71]). The Qx (0–0) line is polar-ized along the N–H� � �H–N axis (x direction, Fig. 1),while the Qy(0–0) band (k = 531 nm, DE = 2.33 eV)has y-polarization along the N� � �N direction (transitionto the 11B2u state) [73]. Both the Qx and Qy bands havestrong vibronic components (0–1) for a number of vibra-tions in the range 1600–1400 cm�1 in the anthracene ma-trix [73]; but the Qy 0–1 band is much more intense. OurDFT calculations predict low intensity for both verticaltransitions (6-31G** basis set). Calculation with the 3-21G basis set which are used here for numerous studiesof vibronic bands slightly overestimates the verticaltransition probability of the Qx band since the observedintensity is determined mostly by vibronic 0–1 compo-nent [73,74]. In Mg- and Zn-porphyrins with the D4h

Table 2Electronic absorption spectrum of free-base porphin, Mg-porphyrin and Zn-porphyrin calculated by different methods

State H2P MgP ZnP

6-31G** 3-21G Exp.a 6-31G** Exp.a 6-31G** Exp.a

13B2u 1.46 1.46 1.56 1.69 1.7 1.74 1.8213B3u 1.81 1.76 1.87 1.69 1.7 1.74 1.8211B3u 2.27(0.0001) 2.15(0.0034) 1.98(0.01) 2.38(0.001) 2.14(0.08) 2.45(0.001) 2.18(0.09)11B2u 2.44(0.0003) 2.28(0.0008) 2.42(0.06) 2.38(0.001) 2.14(0.08) 2.45(0.001) 2.18(0.09)11B2g 2.85(0) 2.82(0) – 3.46(0) 3.20(0)11B3g 2.91(0) 2.88(0) – 3.46(0) 3.20(0)21B3u 3.33(0.411) 3.44(0.575) 3.33(1.15) 3.40(0.855) 3.22(1.39) 3.40(0.90) 3.22(1.39)21B2u 3.41(0.603) 3.56(0.156) 3.40(0.855) 3.22(1.39) 3.40(0.90) 3.22(1.39)31B2u 3.56(0.449) 3.78(0.808) 3.50(0.052) 3.82(0.102) 3.50(0.06) 3.82(0.10)31B3u 3.61(0.864) 4.10(0.637) 3.50(0.052) 3.82(0.10) 3.50(0.06) 3.81(0.10)11B1u 3.85(0.0014) 3.69(0.0027) 3.65(<0.1)41B3u 3.89(0.565) 4.10(0.637) 4.12(0.241) 4.96(0.10)41B2u 3.89(0.565) 4.10(0.637) 5.37(0.092) 5.97(0.10)21B1u 5.89(0.003) 7.13(0.0001)

DE is excitation energy (eV), oscillator strengths (f) are given in parentheses.a Ref. [93].


symmetry point group the Qx,Qy lines are merged sincethe two states are degenerate in the 1Eu term.

The vibronic 0–1 transitions of the Q bands are moreintense than the 0–0 transitions in absorption spectra ofall porphyrins [17]. From our calculations it follows thatthe 0–1 transitions of the Qx band are connected withthe m76, m87, m91, and m94 vibrations of ag symmetry inH2P. Their experimental frequencies determined fromthe resonance Raman spectra [40] are in the range1360–1614 cm�1 which correlate pretty well with the fre-quencies of the 0–1 bands of H2P excitation and fluores-cence spectra in different matrices [24,73,74]. The m94vibration is the most important one in this respect. Itsfrequency is not changed much upon excitation. In thelowest triplet state 3B2u of H2P the m94 vibrational fre-quency is 1659 cm�1; thus a 5 cm�1 shift is predicted.Though we have not optimized the singlet excited stategeometry we can anticipate a qualitatively similarbehavior for the S1 potential energy surface. The m94vibration is also well resolved in fluorescence spectra[24,73,74] and in phosphorescence of H2P in Xe matrix[24]; the late will be discussed in the next section.

We have to note that the in-plane vibrations of b1gsymmetry are also very active as 0–1 lines in absorption(Qx and Qy bands) and in fluorescence spectra of H2P[24,42,73,74,89]. Our calculations predict that m92 vibra-tional mode is the most intense in the Qx band in agree-ment with polarization measurements [24]. Thisvibration and other b1g modes, m75, m77 and m86, are veryactive in Qy (S0 ! S2) absorption band; this assignmnetagrees well with fluoresecnce excitation spectrum insupersonic jet [89] and Fourier-transform absorptionin noble gas matrices [42]. The 0–1 transitions with exci-tation of the m75 and m79 modes are relatively intense inthe Qx band, while the other two b1g modes (m77 andm86) are rather weak in the Qx absorption and fluores-

cence. Such selectivity of the b1g vibrations in respectto vibronic structure of the Qx and Qy absorption bands,predicted by our DFT calculations within the frame-work of the Herzberg–Teller mechanism agrees wellwith all available experimental information[24,42,73,89]. Together with the above-mentioned vibra-tions of ag symmetry all these modes are identified in Q

bands of H2P by polarization measurements [42]. Inmore details we shall adress this issue elsewhere [75].

The problem of identification of the np* state in por-phyrins has been unsolved for a long time [7,66,73]. Inthe present work we have predicted the lowest tripletand singlet B1u(np*) states in H2P at 3.6 and 3.85 eV,respectively. The first S0–S(np*) transition is very week(Table 2) and is buried by strong p–p* absorption withan onset at lower excitation energy. Other n–p* transi-tions are also weak and cannot be detected.

The very strong B band (or Soret band) in the nearUV region (k = 372 nm, DE = 3.33 eV) can be attributedto the 21B3u and 21B2u states (Table 2). Two other in-tense transitions to the 31B3u and 31B2u states are prob-ably also connected with the Soret band. Comparison ofdifferent methods indicates that even the small 3-21Gbasis set provides a good interpretation of the absorp-tion spectrum in the framework of the TD-DFT methodwith the B3LYP functional. It therefore seems reason-able to apply this approximation for the phosphores-cence study.

3.3. Calculations of phosphorescence radiative lifetime

In the present work, the ground state optimizedgeometry and vibrational modes are used for the phos-phorescence radiative lifetime calculations. Geometrieswere optimized with the B3LYP functional and 6-31G** basis set. The geometry is found to retain D2h


symmetry for H2P and D4h symmetry for ZnP, MgPmolecules with positive definite Hessians and withgeometries that are quite similar to those found in otherDFT studies [8,37]. The phosphorescence rate constantsfor the H2P molecule have been calculated recently bythe same quadratic response TD-DFT method withthe cc-pVTZ basis at the optimized T1 state geometry[37]. This optimized triplet state geometry has also beenused for calculations of phosphorescence in complexesof H2P with noble gas atoms using the smaller basisset (3-21G) [36]. The calculated T1 ! S0 transition en-ergy at the optimized T1 state geometry [36,37] is natu-rally lower than the 0–0 transition energy. The latevalue determined from phosphorescence spectra[12,10,24] is used for comparison with theoretical predic-tions [36,37,83]. That is why this comparison is not in fa-vour of simple 3-21G method, where the calculatedDET–S energy gap is underestimated [36,83]. In the pres-ent work, the calculated T1 state energy is in good agree-ment with experiment for all basis sets (Table 2). Theorbital symmetry of the lowest triplet state of the H2Pmolecule (3B2u) is firmly established from comparisonof experimental [13,14,32,79] and theoretical [37] analy-sis of EPR spectra, including zero-field splitting, hyper-fine interactions, ENDOR and microwave-inducedtransient optical signals.

In Table 3 the T1 ! S0 transition moments and radi-ative rate constants (Einstein coefficient for spontaneousemission) calculated by the quadratic response TD-DFT/B3LYP method with different basis sets are pre-sented (most results are obtained with the 3-21G basisset). An extremely weak phosphorescence of free-baseporphin is predicted (Table 3) with unusually low rateconstants for spontaneous emission. This is in generalagreement with all available experimental data [10–13],

Table 3Quadratic response DFT/B3LYP calculations of phosphorescence intensity

Molecule DE h3Wx1jzj1W0i h3Wz

1jxj1W0iH2P(

3B2u) 1.46 2.4 · 10�5 6.0 · 10�7

H2P(3B2u)

a 1.45 0.8 · 10�5 6 · 10�8

H2P, Exp.b

H2P, Exp.c 1.58

H2P, Exp.d 1.56

MgP(3Eu) 1.72 6.6 · 10�5 1.2 · 10�6

MgP(3Eu)a 1.80 3.9 · 10�5 0.3 · 10�6

MgP, Exp.d 1.70ZnP(3Eu) 1.81 7.1 · 10�5 1.80 · 10�4

ZnP(3Eu)a 1.80 6.5 · 10�5 1.64 · 10�4

ZnP, Exp.e 1.82

DE is the transition energy (eV), electric dipole S0–T1 transition moments areTa1 spin sublevel, sp(L) is the phosphorescent radiative lifetime (s) at low-tem

high-temperature limit (77 K); z-axis is perpendicular to the plane, x-axis ina Calculations are done with the 6-31G** basis set. Otherwise the 3-21G bb Ref. [13]c Ref. [10]d Ref. [12]; averaged k value is measured at 77 K.e Ref. [77,80]; relative ka values are presented.

though direct measurements of the quantum yield andradiative phosphorescence lifetime of free-base porphinin the absence of the external heavy atom (EHA) effectare very scarce. Gouterman and Khalil [10] have ob-served phosphorescence of a number of porphyrins ina mixture ‘‘EPA–ethyl iodide’’: it ranges from 773 nmfor mesoporphyrin IX to 785 nm for H2P. The presenceof iodides in solvents is essential for the phosphores-cence detection; this is manifestation of the EHA effect[10,11].

In Table 3, we present radiative lifetime of phospho-rescence which could be detected at the low-temperaturelimit (<4.2 K), i.e., at cooled liquid helium, when spin–lattice relaxation in the triplet state is frozen [76]. In thiscase the radiative phosphorescence lifetime is deter-mined by the most active spin sublevel. For porphyrinsystems doped with heavy ions such measurements havebeen carried out by van der Waals et al. [13,32,77–81].Direct detection of H2P and MgP phosphorescence hasbeen performed in glassy solvents at liquid nitrogen tem-perature [10,11,25,30]. In this condition the spin–latticerelaxation is very fast and smears out the spin alignmentin the triplet state before the molecule emits phosphores-cence, which means that the three spin sublevels haveequilibrium population and that the radiative lifetimeis an averaged value of all three sublevels of the tripletstate. This is the so-called ‘‘high-temperature limit’’,though 77 K is not a high-temperature in a normal sense[77].

From calculated results (Table 3) one comes to con-clusion that the observed phosphorescence lifetime ofporphyrin free bases has very little in common withthe natural (radiative) lifetime since the observed life-time [12] of H2P at 77 K is equal to 11 s, i.e., about hun-dred times shorter than the calculated sp(H) value. It

at the ground state geometry optimized with 6-31G** basis set

kx kz sp(L) sp(H)

0.0018 6.0 · 10�7 540 16190.0002 5.2 · 10�9 5128 153810.002 500

>700.00780.024 1.0 · 10�5 42.4 1270.008 1.0 · 10�6 126 3770.021 470.031 0.203 4.9 12.80.014 0.071 14 35.30.16c 1.00c <10

in atomic units (ea0), ka is the rate constants (s�1) for emission from the

perature limit (4 K), sp(H) is the phosphorescent radiative lifetime atH2P coincides with the NH� � �HN direction.asis set is used.


means that the main decay of the triplet state is nonra-diative. This also follows from the low phosphorescencequantum yield [12,13]. It is very difficult to measure thequantum yield of week phosphorescence for porphyrinfree bases [10,11,25,30]. The first quantitative measure-ment was carried out by the Minsk group [2] which re-ported /p = 0.002 for etioporphyrin. Later [11] thisresult has been changed to a smaller value/p = 0.00027. Tsvirko et al. [12] have determined thephosphorescence quantum yield of free-base porphinin solid hydrocarbon glass to be equal to 1.4 · 10�4.For mesoporphyrin IX they reported /p = 3.2 · 10�4

in the same octane + benzene mixture. This value ishigher than /p < 6 · 10�5 determined independently byGouterman and Khalil in solid EPA [10]. Finally, onehas to mention that van der Waals et al. [13] have pre-dicted the phosphorescence quantum yield to be/p � 10 · 10�5 from indirect measurements of micro-wave-induced transients in free-base porphin fluores-cence. Thus, our quadratic-response prediction of verysmall transition moments and radiative rate constants(Table 3) in the free-base porphin phosphorescenceseems to be reasonable if we keep in mind the experi-mental uncertainty. Nevertheless, the results of the 6-31G** basis set calculation definitely underestimatesthe phosphorescence rate constant.

Our calculations predict that the most active spinsublevel is Tx; this is the state with zero projection ofthe total electron spin on the N–H� � �H–N direction.In the double point group the spin symmetry of tripletsublevels coincides with rotation around axes [23]. Thus,the Tx spin symmetry is B3g and the total (spin–orbit)symmetry of this sublevel is equal B2u · B3g = B1u whichprovides z polarization of the 0–0 band in phosphores-cence emission from this spin sublevel; such polarizationreally dominate [82] in free-base porphin phosphores-cence (Table 3). For the out-of-plane zero spin projec-tion (sublevel Tz of the B1g spin symmetry) thecalculated radiative decay rate constant is practicallynegligible in free-base porphin as one can see from Table3. The total symmetry of this sublevel is equal toB2u · B1g = B3u; this spin sublevel can emit 0–0 line withx polarization, which was not detected in H2P phospho-rescence from all matrices [24,82]. In this molecule the0–0 transition from the Ty spin sublevel (N� � �N direc-tion) is forbidden even with SOC account by the symme-try selection rule: B2u · B2g = Au. A long radiativelifetime of H2P phosphorescence is quite unusual for amolecule with lone pairs of electrons at nitrogen atoms.The calculations indicate that intensity borrowing fromthe S–S lowest n ! p* transition provides much fasterH2P phosphorescence than what is finally obtained. Thispartial contribution is equal to 0.1 s�1 for the rate con-stant in the 3-21G basis set which is about two order ofmagnitudes higher than the final result of the QR calcu-lation (Table 3). It can be explained by that a slow H2P

phosphorescence results from cancellation of many con-tributions in Eq. (6) which come with opposite signsfrom different sources of intensity borrowing.

Substitution of two protons in H2P by a metal ionprovides considerable change in the phosphorescence,which gets larger with increase of the atomic number.It is here relevant to mention the following trends:

(1) The S–T transition energy increases and our calcu-lations reproduce this trend quit well even with thesmall basis set (Tables 2 and 3).

(2) The rate constant of the nonradiative T1 ! S0relaxation (qTS) decreases with increase of theatomic number of the central ion [12,13]. The larg-est qTS value is obtained for H2P: Tsvirko et al.[12] provided 52.6 s�1 and van der Waals et al.[13] – 103 s�1 (they have also measuredqTS = 230 s�1 for the Tx spin sublevel [13]). InMgP, Tsvirko et al. [12] determined qTS = 10 s�1;in ZnP van der Waals et al. [13] have found analmost isotropic T1 ! S0 relaxation rate with anaveraged constant qTS = 6.6 s�1. We can explainthis trend by the increase of the T–S energy gapin the row ZnP > MgP > H2P. We shall addressthis problem in more detail later.

(3) The rate constant of the radiative T1 ! S0 transi-tion (k) increases with increase of the atomic num-ber [12,13] which indicates a normal internal heavyatom effect. Our quadratic-response method repro-duces this trend reasonably well (Table 3). Withthe light metal ion (MgP) calculations predict anincrease of the emission rate for the same Tx spinsublevel like in free-base porphin; the same polar-ization of phosphorescence (perpendicular to themolecular plane) is found. Table 4 illustrates thedependence of the calculated MgP phosphores-cence parameters on the type of the functionaland on the basis set. One can see that the bestagreement for the phosphorescence lifetime is pro-vided by the LDA functional. The 6-31G** basisset gives longer lifetime; this trend is obtained forall molecules (Table 3). It is interesting to note thatan account of d-orbitals in the 6-31G** basis setdoes not lead to an increase of spin–orbit couplingin MgP in comparison with the 3-21G basis set cal-culation, where d-orbitals are not included. Theradiative T1 ! S0 transition probability gets evensmaller with account of d-orbitals (Table 3). Thisput little credit to numerous speculations on therole of d-orbitals in SOC calculations for mole-cules with non-transition elements [23].

A great change of spin-sublevel selectivity and phos-phorescence polarization is calculated for Zn-porphyrinin comparison with H2P and MgP molecules. The moreactive spin sublevel is Tz in ZnP which provides the

Table 4Calculation of Mg-porphyrin phosphorescence by different methods

Method DE h3Wx1jzj1W0i h3Wz

1jxj1W0i kx kz spa sp

b

3-21G basis setB3LYP 1.72 6.6 · 10�5 1.2 · 10�6 0.024 1.0 · 10�5 42.3 126.9BLYP 1.80 9.2 · 10�5 0.8 · 10�6 0.052 2.0 · 10�6 19.2 57.7LDA 1.86 9.9 · 10�5 1.0 · 10�6 0.067 7.0 · 10�6 14.9 44.6BP86 1.79 8.6 · 10�5 0.4 · 10�6 0.045 2.1 · 10�7 22.2 66.4B3P86 1.71 6.2 · 10�5 1.3 · 10�6 0.021 1.1 · 10�5 48.5 145.2

6–31G** basis setBLYP 1.77 6.4 · 10�5 0.3 · 10�7 0.024 0.5 · 10�7 40.9 122.7LDA 1.83 7.3 · 10�5 0.4 · 10�6 0.035 0.2 · 10�6 28.5 85.4BP86 1.77 6.1 · 10�5 0.3 · 10�7 0.022 0.4 · 10�7 46.5 139.5B3P86 1.68 3.2 · 10�5 0.8 · 10�6 0.005 0.2 · 10�5 191.3 573.9Exp.c 1.70 47

Electric dipole S0–T1 transition moments (ea0), transition energy (DE, eV), rate constants ka (s�1) for emission from the Ta1 spin sublevel, sp (s).

a sp – the phosphorescent radiative lifetime at low-temperature limit (4 K).b sp – the phosphorescent radiative lifetime at high-temperature limit (77 K).c Ref. [80]


predominant in-plane polarization of the T1 ! S0 tran-sition. There are strong experimental evidences in favorof this prediction [13,82]. If our results for the perpen-dicular polarization of the H2P and MgP phosphores-cence are correct then we get full agreement with theexperimental findings on the spin-sublevel radiativeactivity in ZnP [13] and polarization measurements [82].

Only one spacial component of the 3Eu degeneratestate is presented in the Table 3 for metal porphyrins.The same results are obtained for the other spatial com-ponent; in this case the Tz spin sublevel should emit lightwith the in-plane y polarization. At the ground stategeometry (D4h) of ZnP the x and y axes are equivalentwith respect to polarization of the emitted light. Geom-etry optimization of the lowest triplet state of ZnP bythe B3LYP method with 6-31G** and 6-311G** basissets leads to Jahn–Teller distortion in agreement with re-sults of [9]. The lowest T1 state has 3B1u symmetry andthe second triplet state, 3B2u, lies 0.25 eV higher in en-ergy at the T1 state-optimized geometry. The distortionoccurs along the b1g mode in agreement with EPR anal-ysis [80]. The D4h square structure of the ground stateZnP is distorted into a D2h rectangular structure similarto that of the H2P molecule. A simple comparison ofphosphorescence for all three molecules calculated atthe same T1 state-optimized geometry of the free-baseporphin [83] gives results which are similar to those pre-sented in Table 3. The Tz spin sublevel in ZnP is still themost active one and provides phosphorescence polariza-tion along the x-axis in which the molecule is elongated.Only the lowest triplet state of 3B1u symmetry emitsphosphorescence and it correlates with one spatial com-ponent of the 3Eu degenerate state presented in Table 3.Only 17% of the ZnP phosphorescence is out-of-planepolarized. This is in a qualitative agreement with polar-ization measurements of Zn-tetraphenylporphyrin phos-phorescence by the dynamic photoelectric method [82].

So far we use the common approximation of traditionalquantum chemistry to interpret molecular spectra on thebasis of calculated vertical transition energies which, inprincipal, does not correspond to any observable andgives a rough estimate for the total intensity at the max-imum of the phosphorescence band. In low-resolutionspectra of solid solvents of porphyrins [10,82] the maxi-mum of phosphorescence band is interpreted as due tothe 0–0 transition. The results of Table 3 can be referredto this band. Better resolution has been obtained in [12]where some vibronic bands with relatively high intensityare distinguished. Highly resolved phosphorescencespectra of free-base porphin and its deutereated deriva-tives in noble gas matrices have been detected at 4–10 K[24]. However, the vibrational analysis of phosphores-cence has been presented only for a xenon matrix, wherethe EHA effect is dominating. As follows from ourcalculations the vibronic structure of the porphyrinphosphorescence is completely different from theEHA-induced spectrum where only totally symmetricvibrations are active [24].

4. Spin selectivity and intensity of vibronic lines inphosphorescence spectra

We shall start the vibronic activity analysis in por-phyrin phosphorescence by presenting calculations andassignments of the IR and Raman spectra of groundstate porphyrins. We present results obtained with theB3LYP/6-31G** approach. Though the comprehensiveassignment of all vibrational modes has already beendone [20,44,45] we need to compare our results obtainedwith the larger basis set: inclusion of the p functions onhydrogen atoms provides some changes in vibrationalfrequencies in comparison with 6-31G* results of Pulayet al. [20,44,45]. We shall consider also a comparison of


the ground state IR spectra with the triplet state vibra-tional absorption, which can be measured by time-re-solved IR spectroscopy. The triplet state vibrationalfrequencies of gerade symmetry will be compared withthe time-resolved resonance Raman experimental datawhere available.

4.1. Vibrational analysis

Vibrational IR and Raman spectra of the groundstate porphyrins have been studied by numerous investi-gators (for a review, see [43]). Gladkov et al. [38–40,84]and Li and Zgierski [43] have developed the most accu-rate empirical force field for H2P, which reproducesquite well the IR and Raman resonance spectra of anumber of isotopomers including metalloporphyrins.Pulay et al. [20,44,45] have measured the nonresonanceRaman spectra of H2P, ZnP and MgP molecules. Theyhave also calculated the scaled quantum mechanical(SQM) force fields and assigned all vibrational frequen-cies with the B3LYP/6-31G* approach. A number ofother groups [18,52,85] have calculated the active fre-quencies of porphins by the B3LYP method with vari-ous scaling factors. In Tables 5, 6, 8, and 7, we presentour results of vibrational analysis obtained by B3LYP/6-31G** calculations for the ground states of H2P,ZnP and MgP molecules without scaling factors. The

Table 5Infrared active spectra of Mg-porphyrin and Zn-porphyrin calculated by B3

Symmetry MgP

i mi Int. mi, Ex

eu 14,13 327.7 0.6eu 16,15 350.6 6.5eu 23,24 467.2 7.2 431eu 37,36 755.5 26.3 741eu 43,44 808.6 3.6 799eu 56,57 1024.6 96.8 992eu 59,60 1040.3 2.16 1014eu 62,63 1083.8 59.8 1046eu 67,68 1183.9 9.0 1149eu 71,72 1276.7 0.53 1254eu 73,74 1333.7 9.6 1290eu 80,81 1427.6 9.7 1385eu 84,83 1473.7 2.0 1423eu 87,88 1559.7 4.2 1509eu 90,89 1587.4 13.3 1538eu 95,96 3192.5 14.6eu 99,100 3235.6 5.0eu 103,104 3255.1 31.1a2u 3 82.3 2.04a2u 11 241.8 46.8a2u 17 351.3 15.4a2u 33 726.8 11.8a2u 38 780.5 63.3a2u 49 876.7 128.0

IR intensity is in km/mol; frequency is in cm�1.a Ref. [20].

simulated IR spectra of the first two molecules are pre-sented in Figs. 2–4.

We have also calculated the H2P, MgP and ZnPground state IR spectra with 6-311G** and 3-21G ba-sis sets. Both basis sets give reasonable results, the onlyexclusion is that the 3-21G basis fails in calculation ofthe lowest-frequency a2u vibration; instead of 77 cm�1

obtained by Pulay et al. [20] it gives 12 cm�1 forZn-porphyrin. This mode includes a large amplitudeout-of-plane vibration of the central ion with smalldeformation of the pyrrole rings in the opposite direc-tion (tilt along Ca–Ca axes). The analogous mode inH2P is the first b1u vibrational mode which, in contrast,does not include any movement of the central protons.These low-frequency vibrations are almost forbidden inIR spectra of the two molecules, but they could beimportant in nonradiative relaxation of the tripletstate. In the excited T1 state their frequencies areslightly smaller (by 5 cm�1). The low frequency out-of-plane vibrations are quite different in H2P andmetalloporphyrins since there is no central metal ion.The first b1u band is extremely weak and occurs al-ready at 55 cm�1 (I = 0.004 km/mol; it is similar tothe first a2u vibration in ZnP, which is forbidden inthe IR spectrum by chance). The second b1u band inH2P is less intense than the analogous m6 (a2u) bandin ZnP and has a lower frequency, 95 cm�1 (compare

LYP DFT method with the 6-31G** basis set

ZnP

p.a i mi Int. mi, Exp.a

12,13 224 1.016,17 359 6.919,20 391 7.036,37 755 24.0 73943,42 808 5.0 79954,55 1016 95.0 99359,60 1044 0.1 101962,63 1086 56.2 105267,68 1186 8.1 115171,72 1280 0.273,74 1342 15.3 129980,81 1427 6.3 138482,83 1482 3.0 143887,88 1572 9.2 151790,89 1603 7.5 155895,96 3195 13.5

99,100 3236 4.8103,104 3257 28.8

2 64 0.16 148 29.315 343 7.034 714 8.8 69938 779 69 76549 879 122 849

Table 6Infrared and Raman spectra of free-base porphin

i Symmetry 6-31G**a Scaled 6-31G*b Exp.c Exp.b,d

mi IR Raman Dep(P) Dep(U) mi IR Raman Dep(P) mi mi

1 b1u 55.5 0.004 0 54.8 0.00 02 au 67.2 0 0 64.5 0 03 b1u 95.2 7.73 0 90.5 7.7 04 b1g 98.9 0 17.16 0.750 0.857 86.8 0 16.9 0.750 109d

5 b2g 124.7 0 2.64 0.750 0.857 122.6 0 2.6 0.7506 b3g 133.2 0 0.83 0.750 0.857 130.2 0 0.9 0.7507 ag 156.6 0 27.93 0.748 0.856 152.9 0.00 27.4 0.748 155b

8 b2g 187.7 0 5.61 0.750 0.857 183.7 0 5.5 0.7509 b3g 207.8 0 7.11 0.750 0.857 204.5 0.00 7.1 0.75010 b1u 209.7 2.08 0 0.750 0 204.5 1.8 0 21911 b2u 292.2 0.08 0 0 283.3 0.1 012 au 299.5 0 0 0 293.7 0 013 ag 310.4 0 74.82 0.114 0.205 303.8 0 77.0 0.115 309b

14 b3u 316.6 2.77 0 0 312.4 2.9 0 31015 b1u 336.7 5.64 0 0 331.3 5.6 0 0 33516 b2u 358.0 9.02 0 0 350.8 8.4 0 0 35717 b3u 358.2 6.96 0 0 0 350.3 6.5 0 0 35718 b1g 395.9 0 0.10 0.750 0.857 388.4 0 0.2 0.750 389d

19 b1g 420.1 0 1.29 0.750 0.857 410.2 0 1.0 0.750 418d

20 b2g 430.0 0 0.005 0.750 0.857 421.5 0 0 0.75021 b3g 443.2 0 0.08 0.750 0.857 435.5 0 0.1 0.75022 au 482.0 0 0 0 472.7 0 023 b2g 630.5 0 0.23 0.750 0.857 614.1 0 0.4 0.75024 b1u 655.5 0.85 0 0.750 0.857 639.4 2.2 0 63925 b3g 679.2 0 0.17 0.750 0.857 666.1 0 0.1 0.75026 b2g 680.2 0 0.59 0.750 0.857 667.2 0 0.7 0.75027 au 691.8 0 0 0 679.5 0 0 028 au 710.3 0 0 0 0.0 698.5 0 0 029 b1u 712.2 12.9 0 0.750 0.857 696.7 14.2 0 0 69130 b2g 712.9 0 9.89 0.750 0.857 700.3 0 9.9 0.75031 b3g 714.6 0 11.3 0.750 0.857 700.3 0 11.5 0.75032 ag 734.0 0 18.97 0.119 0.213 720.6 0 25.5 0.119 723b,d,e

33 ag 738.5 0 12.46 0.191 0.321 727.8 0 9.4 0.279 736d

34 b3u 740.6 30.94 0 0.0 0.0 726.2 31.1 0 0 72335 b1u 748.3 6.54 0.0 0.750 0.857 727.1 11.6 0 0 73136 b2u 758.2 26.88 0 00 0.0 745.2 25.5 0 0 74537 b3g 789.2 0 4.84 0.750 0.857 770.3 0 4.4 0.75038 b2u 791.8 0.30 0 00 0 781.6 0.30 0 039 b2g 792.8 0 3.06 0.750 0.857 772.2 0 2.7 0.75040 b3u 794.7 1.49 0 0.0 0.0 785.1 1.9 0 0 78041 b1u 795.8 60.54 0.0 0.750 0.857 776.1 53.2 0 0 77342 b1g 797.1 0 0.001 0.750 0.857 783.7 0 0.0 0.750 786d,e

43 b1u 808.1 120.13 0.0 0.750 0.857 786.0 140.1 0 0 78544 b1g 817.4 0 1.17 0.750 0.857 809.4 0 1.5 0.750 805d

45 au 850.9 0 0 0 0.0 831.7 0 0 046 b3g 860.6 0 2.42 0.750 0.857 841.0 0 1.9 0.75047 b2g 861.9 0 1.38 0.750 0.857 841.7 0 0.9 0.75048 b1u 872.0 143.45 0.0 0.0 0.0 853.3 145.4 0 0 85249 au 909.1 0 0 0 0.0 879.7 0 0 050 b2g 912.9 0 0.70 0.750 0.857 884.7 0 0.6 0.75051 b3g 917.9 0 0.93 0.750 0.857 889.3 0 1.0 0.75052 au 919.9 0 0 0 0.0 891.7 0 0 053 b2u 973.3 82.79 0 00 0.0 945.2 89.30 0 0 95154 ag 976.8 0 88.32 0.109 0.196 950.8 0 66.5 0.108 952b,d,e

55 b3u 994.3 54.61 0 0.0 0.0 968.0 56.4 0 0 97156 b1g 999.0 0 5.06 0.750 0.857 966.5 0 4.4 0.750 976d, 972e

57 b2u 1011.7 10.41 0.0 0.0 0.0 981.5 5.7 0 0 97758 ag 1013.1 0 43.56 0.174 0.296 984.7 0 51.4 0.149 987d, 988b,e

59 b3u 1023.4 0.08 0 0.0 0.0 996.3 0.4 0 0 99460 b1g 1028.5 0 8.59 0.750 0.857 1002.1 0 7.1 0.750 1005d,e

61 b1u 1079.3 43.58 0.0 0.0 0.0 1048.6 43.0 0 0 104362 b2u 1083.8 35.90 0.0 0.0 0.0 1053.0 35.7 0 0 1054


Table 6 (continued)

i Symmetry 6-31G**a Scaled 6-31G*b Exp.c Exp.b,d

mi IR Raman Dep(P) Dep(U) mi IR Raman Dep(P) mi mi

63 ag 1086.9 0 0.71 0.190 0.319 1055.8 0 0.6 0.180 1063d,e

64 ag 1093.7 0 6.13 0.194 0.325 1061.9 0 6.2 0.177 1064b

65 b1g 1169.6 0 0.14 0.750 0.857 1133.2 0 0.1 0.750 1138d

66 b3u 1172.9 20.62 0.0 0.0 0.0 1137.8 20.5 0 0 113467 b2u 1188.6 0.02 0.0 0.0 0.0 1155.8 0 0 0 116568 ag 1210.1 0 37.60 0.644 0.784 1179.3 0 36.2 0.644 1177b,e

69 b1g 1222.8 0 1.62 0.750 0.857 1186.3 0 1.6 0.750 1182d

70 b3u 1231.7 2.82 0.0 0.0 0.0 1206.4 3.4 0 0 117771 b1g 1261.9 0 2.46 0.750 0.857 1219.5 0 1.9 0.750 1226d, 1221e

72 b2u 1269.2 57.07 0.0 0.0 0.0 1225.4 57.8 0 0 122873 b2u 1282.9 0.78 0.0 0.0 0.0 1251.7 0.3 0 0 125574 b3u 1321.9 1.64 0.0 0.0 0.0 1286.3 1.8 0 0 128775 b1g 1357.3 0 55.74 0.750 0.857 1320.5 0 52.3 0.750 1313b, 1316e

76 ag 1392.1 0 42.21 0.373 0.544 1358.1 0 45.5 0.444 1353b,d, 1360e

77 b1g 1393.3 0 13.44 0.750 0.857 1354.7 0 17.4 0.750 1352b,e, 1374d

78 b2u 1393.7 4.41 0.0 0.0 0.0 1354.8 4.5 0 0 135779 b1g 1422.4 0 28.02 0.750 0.857 1381.1 0 31.3 0.750 1388d,e

80 ag 1442.2 0 224.43 0.692 0.818 1402.0 0 224.3 0.665 1384b

81 b3u 1444.7 28.9 0.0 0.0 0.0 1407.6 26.0 0 0 139682 b3u 1448.7 4.00 0.0 0.0 0.0 1400.0 6.4 0 0 141283 b2u 1451.8 10.81 0.0 0.0 0.0 1409.3 10.3 0 0 140684 ag 1477.7 0 109.15 0.169 0.289 1430.4 0 105.0 0.148 1425b,d

85 b2u 1538.5 2.19 0.0 0.0 0.0 1490.9 2.1 0 0 149086 b1g 1538.6 0 13.55 0.750 0.857 1491.2 0 15.3 0.750 1493d, 1497e

87 ag 1551.1 0 297.25 0.692 0.818 1504.2 0 279.3 0.750 1492b,d, 1502e

88 b3u 1557.7 0.70 0.0 0.0 0.0 1512.2 1.1 0 0 150789 b3u 1570.4 7.03 0.0 0.0 0.0 1522.3 5.0 0 0 152290 b2u 1591.8 24.76 0.0 0.0 0.0 1546.8 22.6 0 0 154091 ag 1604.3 0 291.85 0.144 0.252 1558.7 0 280.4 0.139 1544b,d, 1575e

92 b1g 1640.7 0 0.26 0.750 0.857 1590.3 0 0.4 0.750 1600d, 1578e

93 b2u 1644.8 17.23 0.0 0.0 0.0 1594.6 16.4 0 0 –94 ag 1654.8 0 203.44 0.684 0.813 1605.9 0 213.6 0.676 1609b,d, 1614e

95 b2u 3196.6 10.41 0.0 0.0 0.0 3058.8 13.0 0 0 304596 b1g 3196.7 0 227.7 0.750 0.857 3058.8 0 225.0 0.750 –97 b3u 3196.7 12.02 0.0 0.0 0.0 3058.8 14.9 0 0 304298 ag 3196.9 0 318.55 0.111 0.200 3058.9 0 314.5 0.113 –99 b1g 3230.4 0 263.37 0.750 0.857 3088.8 0 263.4 0.750 –100 b3u 3230.4 6.72 0.0 0.0 0.0 3088.8 8.3 0 0 3088101 b2u 3249.1 1.80 0.0 0.0 0.0 3107.3 2.0 0 0 3112102 b1g 3249.1 0 236.70 0.750 0.857 3107.3 0 239.5 0.750 3109e

103 b2u 3251.5 39.84 0.0 0.0 0.0 3109.0 46.7 0 0 3124104 ag 3251.6 0 652.11 0.215 0.354 3109.1 0 657.7 0.219 –105 b3u 3266.5 14.56 0.0 0.0 0.0 3123.7 18.2 0 0 3128106 ag 3266.6 0 605.31 0.205 0.340 3123.8 0 631.3 0.210 –107 b3u 3558.1 66.48 0.0 0.0 0.0 3329.8 63.7 0 0 3324108 ag 3601.1 0 4.54 0.716 0.834 3366.2 0 4.2 0.709 NH

‘‘IR’’ means IR absorption intensity (km/mol); ‘‘Raman’’ means scattering activities (A4/amu), Dep(P) – depolarization ratios for plane polarizedincident light, Dep(U) – for unpolarized light, mi – frequency (cm�1).a This work.b Ref. [45].c Ref. [41].d Ref. [38].e Ref. [24].


Tables 5 and 6.) This important out-of-plane vibrationof a2u symmetry of the Zn-ion (m6 = 148 cm�1) is pre-dicted in the far IR region and is quite intense(I = 29 km/mol; this is the first strongly allowed IRband in the spectrum, see Fig. 4). Unfortunately, the

far IR region is not available in KBr pellets used sofar for IR measurements [20,38]. In MgP this band isshifted to 242 cm�1 and there is no such intense IRband for the H2P molecule in this region (Table 6).A weak IR signal at 224 cm�1 corresponds to the first

Table 7Raman spectra of Mg-porphyrin calculated by B3LYP DFT method with two basis sets

i Symmetry 6-31G** 3-21G

mi Raman Dep(P) Dep(U) mi Raman Dep(P) Dep(U)

4,5 eg 143.9 1.11 0.75 0.857 150.1 0.62 0.75 0.8577 b2g 208.9 19.21 0.75 0.857 216.7 18.51 0.75 0.8578,9 eg 212.1 7.55 0.75 0.857 221.7 9.74 0.75 0.85710 b1g 220.8 26.40 0.75 0.857 221.7 23.72 0.75 0.85718 a1g 363.8 89.62 0.106 0.192 369.4 89.58 0.11 0.20120 b2g 420.3 1.35 0.75 0.857 426.5 0.84 0.75 0.85721,22 eg 443.2 0.0002 0.75 0.857 460.4 0.0008 0.75 0.85727,28 eg 685.9 0.03 0.75 0.857 701.1 0.94 0.75 0.85731.32 eg 722.4 10.15 0.75 0.857 752.4 13.41 0.75 0.85734 a1g 737.0 26.81 0.09 0.169 743.7 31.34 0.10 0.17635 b1g 748.7 4.80 0.75 0.857 767.1 5.23 0.75 0.85739,40 eg 791.6 3.61 0.75 0.857 835.2 3.55 0.75 0.85742 a2g 807.8 0 0 0 830.6 0 0 045 b2g 833.7 0 0.75 0.857 859.3 0.001 0.75 0.85747,48 eg 866.4 2.30 0.75 0.857 932.4 0.80 0.75 0.85751,52 eg 916.4 1.13 0.75 0.857 970.1 0.25 0.75 0.85754 a1g 1020.3 115.02 0.109 0.197 1006.3 88.23 0.11 0.20255 a2g 1023.5 0.00 0.0 0.0 994.6 0.00 0.0 0.058 b1g 1034.2 1.80 0.75 0.857 1003.8 0.18 0.75 0.85761 b2g 1068.7 16.70 0.75 0.857 1053.5 10.05 0.75 0.85764 b1g 1087.6 1.84 0.75 0.857 1111.3 0.47 0.75 0.85765 a1g 1093.0 1.51 0.14 0.247 1114.9 7.50 0.15 0.26669 b1g 1210.3 46.41 0.75 0.857 1170.0 83.03 0.75 0.85770 b2g 1212.9 0.80 0.75 0.857 1208.5 5.87 0.75 0.85777 b2g 1389.6 87.05 0.75 0.857 1349.1 96.01 0.75 0.85778 a1g 1396.9 31.30 0.12 0.211 1339.9 64.68 0.11 0.19679 b1g 1422.9 215.00 0.75 0.857 1414.2 214.63 0.75 0.85782 a1g 1472.7 107.11 0.10 0.184 1463.6 113.71 0.11 0.19185 b2g 1506.7 4.02 0.75 0.857 1482.3 7.63 0.75 0.85786 b1g 1543.6 410.11 0.75 0.857 1518.5 318.78 0.75 0.85791 a1g 1592.3 283.42 0.13 0.224 1573.3 268.93 0.13 0.23693 b1g 1643.9 215.30 0.75 0.857 1647.0 164.50 0.75 0.85794 b2g 3192.5 235.81 0.75 0.857 3272.9 223.23 0.75 0.85797 a1g 3192.8 317.90 0.11 0.203 3172.9 297.06 0.11 0.200101 b2g 3235.8 503.22 0.75 0.857 3336.0 424.65 0.75 0.857102 b1g 3255.0 296.83 0.75 0.857 3356.3 258.61 0.75 0.857105 a1g 3255.4 989.30 0.11 0.198 3356.6 834.30 0.11 0.196

Raman – scattering activities (A4/amu), Dep(P) – depolarization ratios for plane polarized incident light, Dep(U) – depolarization ratios forunpolarized light, mi – frequency (cm�1).


eu modes (m12,13 in Table 5) in ZnP; these are in-planeZn–N bending vibrations with weak IR intensity I = 1km/mol. In the H2P molecule these vibrations (b2u andb3u types) are absent which is important for the analy-sis of nonradiative degradation of the excited tripletstates.

In the region 1016–1086 cm�1 of the calculated ZnPspectrum there are two intense peaks of eu modes. Themost intense band in the IR spectrum corresponds tom54,55 = 1016 cm�1 vibrations (Fig. 4) which involvethe Zn–N-(pyrrole) breathing. It corresponds to the993 cm�1 band in the observed IR spectrum of ZnP[20]. Similar vibrations are easily identified in MgP(Table 5) and in free-base porphin (Table 6, b2u,b3umodes m53, m55) IR spectra. The big difference in theIR spectra of the ZnP and H2P molecules is connected

with an intense absorption in the range 750–1000 cm�1

(Figs. 2 and 4); in ZnP the IR absorption intensity in-creases to the higher frequency edge whereas for theH2P molecule it decreases. In MgP the intense out-of-plane a2u vibrations in the 700–850 cm�1 region arevery similar to those of the ZnP analoge. (Li andZgierski [43] did not consider the out-of-plane vibra-tions when they introduced their numeration of vibra-tional modes, thus we need to use our own numerationthroughout in Table 5.)

The out-of-plane C–H vibration of the methynebridges could be important in promoting the T1–S0intersystem crossing in porphyrins. The out-of-planeCm–H vibration in ZnP (m49,a2u) is calculated at 878cm�1; it has high IR intensity (I = 123 km/mol) and pro-vides the second intense peak in the IR spectrum of the

Table 8Raman spectra of Zn-porphyrin calculated by B3LYP DFT method with two basis sets

i Symmetry 6-31G** 3-21G

mi Raman Dep(P) Dep(U) mi Raman Dep(P) Dep(U)

4,5 eg 145.6 1.09 0.75 0.857 150.9 0.59 0.75 0.8577 b1g 177.6 23.20 0.75 0.857 177.2 22.51 0.75 0.8579,10 eg 208.5 7.50 0.75 0.857 227.7 8.79 0.75 0.85711 b2g 224.4 19.4 0.75 0.857 229.7 20.7 0.75 0.85718 a1g 372.3 91.4 0.11 0.192 390.8 89.96 0.11 0.20121 a2g 417.8 0 0 0 425.2 0 0 022 b2g 424.3 1.05 0.75 0.857 435.8 0.96 0.75 0.85723,24 eg 439.1 0.00 0.75 0.857 455.2 0.0005 0.75 0.85727,28 eg 683.7 0.03 0.75 0.857 702.4 0.09 0.75 0.85733,32 eg 715.4 10.77 0.75 0.857 754.3 12.80 0.75 0.85734 a1g 738.6 6.25 0.09 0.171 746.2 4.56 0.09 0.17635 b1g 749.3 4.82 0.75 0.857 771.1 5.23 0.75 0.85739,40 eg 792.6 3.61 0.75 0.857 831.0 3.55 0.75 0.85744 a2g 809.1 0 0 0 840.1 0 0 045 b2g 833.7 0 0.75 0.857 861.2 0.001 0.75 0.85747,48 eg 867.9 1.30 0.75 0.857 924.6 0.80 0.75 0.85751,52 eg 918.0 1.18 0.75 0.857 963.8 0.95 0.75 0.85756 a1g 1021.7 110.51 0.109 0.197 1019.5 98.23 0.11 0.20057 b1g 1024.7 1.22 0.75 0.857 996.9 0.44 0.75 0.85758 a2g 1025.9 0 0 0 991.6 0 0 061 b2g 1076.8 15.70 0.75 0.857 1053.0 11.25 0.75 0.85764 b1g 1088.8 1.99 0.75 0.857 1124.3 1.73 0.75 0.85765 a1g 1093.8 1.51 0.141 0.247 1128.1 2.50 0.15 0.26669 b1g 1208.5 48.54 0.75 0.857 1171.0 76.23 0.75 0.85770 b2g 1216.8 1.00 0.75 0.857 1220.2 3.97 0.75 0.85777 b2g 1392.8 87.05 0.75 0.857 1360.8 96.01 0.75 0.85778 a1g 1399.9 39.34 0.12 0.211 1408.0 55.68 0.109 0.19679 b1g 1419.8 215.00 0.75 0.857 1425.9 214.63 0.75 0.85784 a1g 1485.1 111.41 0.10 0.184 1492.2 113.0 0.11 0.19185 b2g 1519.1 7.82 0.75 0.857 1509.1 7.99 0.75 0.85786 b1g 1551.1 400.43 0.75 0.857 1539.0 398.78 0.75 0.85791 a1g 1603.2 289.01 0.13 0.224 1600.0 277.91 0.13 0.23693 b1g 1663.7 209.30 0.75 0.857 1688.1 188.53 0.75 0.85794 b2g 3195.6 235.82 0.75 0.857 3206.6 193.73 0.75 0.85797 a1g 3195.8 310.90 0.11 0.201 3206.8 299.44 0.11 0.200101 b2g 3236.6 523.81 0.75 0.857 3260.3 486.65 0.75 0.857102 b1g 3256.0 300.83 0.75 0.857 3282.7 281.60 0.75 0.857105 a1g 3256.5 986.34 0.10 0.193 3283.1 898.35 0.10 0.190

Raman – scattering activities (A4/amu), Dep(P) – depolarization ratios for plane polarized incident light, Dep(U) – depolarization ratios forunpolarized light, mi – frequency (cm�1).

Table 9Correlation of vibrational symmetry between ZnP (D4h) and H2P (D2h)molecules

In-plane Out-of-plane

ZnP H2P ZnP H2P

eu b2u eg b2geu b3u eg b3ga1g ag a1u aub1g ag b1u aua2g b1g a2u b1ub2g b1g b2u b1u


ZnP molecule (Fig. 4; observed at 849 cm�1). Instead ofZnP a2u modes in the H2P molecule we have b1u modes(Table 9). The H2P analoge is m48 = 872 cm�1 which pro-vides the most intense peak in the IR spectrum (Fig. 2).

In the range of high-frequency C–H stretchingvibrations there are two close-lying IR bands at 3195and 3257 cm�1 in the ZnP spectrum (Fig. 4; eu modesm95–104 in Table 6). This group of IR bands is almostidentical to the 3197 and 3252–3266 cm�1 close-lyingbands in the IR spectrum of the H2P molecule; the split-ting of the b2u and b3u modes in the latter case is verysmall. The occurrence of a new IR line with the highestfrequency m107 = 3600 cm�1 (Fig. 2) which correspondsto N–H stretching vibrations of b3u symmetry (3324cm�1 in experiment; Table 6) explains the big differencebetween the IR spectra of free-base porphin and metal-

loporphyrins. This natural difference of vibrationalmodes can be used for explanation of the sharp distinc-tion in the nonradiative quenching of the triplet states in

0 500 1000 1500 2000 2500 3000 3500 4000

Fig. 2. Infrared absorption spectrum of free-base porphin molecule inthe ground state 1Ag calculated by B3LYP/6-31G** method. (Thespectrum is simulated with the assumed half-width equal to 20 cm�1,which is close to typical IR data [38]. Maximum intensity is 191 km/mol.)

0 500 1000 1500 2000 2500 3000 3500 4000

Fig. 3. Infrared absorption spectrum of free-base porphin molecule inthe first triplet excited state 3B2u calculated by B3LYP/6-31G**method. Intensity scale differs from that in Fig. 2. (Maximum intensityis 325 km/mol; half-width is equal to 20 cm�1.)

0 500 1000 1500 2000 2500 3000 3500

Fig. 4. Infrared absorption spectrum of zinc porphyrin molecule in theground state 1A1g calculated by B3LYP/6-31G** method. (Maximumintensity is 2 · 94 km/mol; half-width is equal to 20 cm�1.)


the two molecules (the T1 state of H2P is quenched fasterthan that of ZnP in contrast to the expectation of theheavy atom effect); the reason is that b3u mode can in-duce SOC between the Tz(B1g) spin sublevel of the low-est 3B2u triplet and the ground singlet 1Ag state of theH2P molecule. We shall discuss this issue latter.

For the purpose of vibronic analysis of radiative T1–S0 transitions the most interesting modes are a1g, b1g, b2gand eg modes, since they are active not only in the Ra-man spectrum, but also in phosphorescence of ZnPand MgP molecules. In the H2P molecule those modesare ag, b1g, and b2g, b3g, respectively. The latter twoout-of-plane vibrations are expected to be the most ac-tive ones by SOC enhancement. They are discussed inthe next section.

Under the S–T excitation the largest structural defor-mations occur in protonated rings: the Cb–Cb bondlength increases and the the Ca–Cb bond length de-creases (by about 0.025 A). A large bond length increase(0.035 A) is obtained for the Ca–Cm methyne bridgeshoulder connected to the protonated rings. Thus, somevibrational frequencies are shifted by a few wavenum-bers upon the S–T excitation. It is interesting to notethat the IR spectrum of H2P in the ground singlet state(Fig. 2) and in the first excited triplet state (Fig. 3) arevery different though their frequencies are quite similar.This is because of a strong intensity redistribution in theIR absorption of the triplet state in comparison with thetraditional ground state IR spectrum. Deviation in fre-quencies for most vibrations are in the range 1–15cm�1. For example, a very weak band m17 = 358 (b3u)in the IR spectrum of H2P (Table 6) is shifted to the348 cm�1 IR band in the T1 state and its intensity is in-creased to 20.2 km/mol. Another band m61 = 1079 (b3u)in the IR spectrum (Table 6) is shifted to the 1069 cm�1

IR band in the T1 state and its intensity is increasedfrom 43 to 156 km/mol.

The most prominent change in IR spectra is con-nected with the m72 (b2u) vibration, the intensity of whichincreases from 57 km/mol to 325 km/mol upon S–Texcitation; this provides a very intense peak in the IRabsorption spectrum of the triplet state (the frequencydecreases by 34.5 cm�1 in comparison with the resultof Table 6). The character of this mode is only slightlychanged upon S–T excitation; this is mainly the C–N–H bending vibration. In the ground state it includes alsothe Ca–Cm–H deformations near the protonated rings.In the T state the addition of pyrrole vibrations is muchmore prominent.

A large IR intensity increase is predicted for other b2umodes, m83 and m90 (Figs. 2 and 3; Table 6), by 12 and 6times, respectively. Their frequencies are decreased onlyby about 10 cm�1 upon the S–T excitation. At the sametime the characters of the vibrational mode m83 is chan-ged significantly. It includes more the C–N–H bendingand protonated pyrrole ring deformations in the triplet


state. In the ground state the nonprotonated ringbreathing prevails. Since the time-resolved IR spectrumof H2P in the triplet state has not been studied so far ourprediction must be put to a future experimental test.

In contrast to the absence of experimental informa-tion on IR data of transient tetrapyrrole compoundsthe time-resolved resonance Raman (TR3) spectra havebeen studied for many triplet-excited porphyrins[50,86,51,52]. An available experimental TR3 studyclose to our interest is connected with free-base meso-tetraphenylporphin (H2TPP) [51]. Since phenyl-substitu-tion does not change many vibrations of the tetra-pyrrolering we can make a comparison of H2TPP and H2P res-onance Raman TR3 spectra. For example, the intenseRaman band m91 = 1604 cm�1 (ag) in the ground stateof H2P (Table 6) is shifted in our calculations to 1590cm�1 in the triplet state; its intensity increases from292 to 841 A4/amu without significant change in polar-ization. This band is known as m2 using the notations of[43] and it corresponds mostly to the Cb–Cb stretch ofdeprotonated pyrrole rings in the ground state (proton-ated rings are less involved in contrast to discussion of[52] for the H2TPP molecule). In the 3B2u state this modeis located entirely on the Cb–Cb vibrations of the depro-tonated rings (C17–C20, Fig. 1). In the resonance Raman(TR3) spectrum of the triplet-excited H2TPP molecule[51] this band is shifted to lower frequency by 16cm�1, which can be compared with our prediction forthe corresponding H2P shift (14 cm�1).

A large increase of the Raman scattering intensity(two-three orders of magnitude) is predicted for manyb1g vibrational modes of the H2P molecule (startingfrom m42 vibration) upon the S–T excitation. This isprobably connected with a small 3B3u–

3B2u energy gap(Table 2). The outstanding mode in this respect is m92in Table 6. Its intensity rises from 0.26 A4/amu to1070 A4/amu. Its frequency decreases by 90 cm�1. Thisvibration includes Ca–N–Ca asymmetric stretches inprotonated pyrrole rings with notable N–Hdeformations.

The m42 (b1g) vibration is absolutely inactive in theground state Raman scattering (Table 6). In the excited3B2u state its activity rises by 105 times without any fre-quency shift. The near-lying m44 (b1g) frequency (Table 6)with low Raman activity in the ground state is shifteddown by 166 cm�1 upon the S–T excitation and its scat-tering activity inceases thousand times to 1122 A4/amu.Such great perturbations of the Raman-active b1g modesin H2P molecule upon the S–T excitation have neverbeen addressed before since experimental and thereticalTR3 studies are usually restricted to ag modes in therange 960–1600 cm�1 [51,52]. Comparison of only agvibrational modes in the two states can obviously mis-lead in interpretation of the Raman spectra. For exam-ple, interpretation of the H2TPP Raman scattering [52]for the m4 (ag) mode (in notation of [43]) considers its

strong increase upon the S–T excitation; on the otherhand our calculation predicts only 1.7 times increaseof its activity. In our study of the H2P molecule thisvibration corresponds to the m80 (ag) mode (Table 6).At the same time the close-lying b1g mode m79 increasesits Raman activity 50 times to 1452 A4/amu and its fre-quency is still in the same region. Thus, we can reinter-pret the resonance Raman spectra of the H2TPPmolecule in the range 1369 cm�1 in terms of the ag–b1gchange. An experimental resonance Raman (TR3) studyof the H2P molecule could be very desirable to check ourprediction.

4.2. Vibronic structure of H2P phosphorescence

Neither phosphorescence microwave double reso-nance experiments, nor highly resolved vibronic struc-ture in the absence of the EHA effect have beenpublished so far for the free-base porphin T1 ! S0 spec-trum. Thus, our results for the H2P vibronic activity forselected spin sublevels can be considered as a predictionfor a future experimental test. At the moment we cancompare results of our calculations with the low-resolu-tion phosphorescence spectrum of H2P in an octane–benzene mixture at 77 K presented in [12]. A highresolution vibronic spectrum of H2P phosphorescencehas been obtained by Radziszewski et al. [24] in a Xematrix. They detected single-site selected emission byreplacing the inner hydrogen atoms with deuterium;the fast photoinduced double-proton-transfer reactionleads to depopulation of the selectively pumped site inthe matrix and the deuterium substitution can slow downthis photochemical transformation of the triplet excitedsites. The 0–0 band of the H2P phosphorescence occursat 12,680 and 12,578 cm�1 for two sites in the Xe matrix.Comparison of this spectrum with our results on vib-ronic intensity in the T1 ! S0 band system requires someprecautions because of the strong SOC perturbation byXe atoms. We used the first site emission presented inFig. 3 of [24] for comparison with our DFT QR results.

The symmetry of the triplet spin in the D2h grouptransforms as B1g(T

z), B2g(Ty), B3g(T

x) irreducible repre-sentations; the corresponding spin sublevels of the 3B2u

state have the total symmetry B3u, Au, B1u, respectively.Thus, the 0–0 band of the 3B2u–

1Ag phosphorescencefrom the Tz spin sublevel is x-polarized; it is found tobe extremely weak in our QR calculation in all basis sets(Table 3). The phosphorescence transition from the Ty

spin sublevel is forbidden even under the SOC accountat the D2h geometry, and the transition from the Tx spinsublevel is z-polarized. This is the most intense phospho-rescence transition in H2P as follows from our QR cal-culation (Table 3). Account of the ag vibrationsprovides additional T1–S0 allowed transition moments(Table 10).

Table 10QR DFT/3-21G results for the T1–S0 transition calculated along the ag (in-plane) vibrational displacements in the H2P molecule

mi (cm�1) mj (cm

�1)a Transition moments (10�5 ea0) Rate constants (10�5 s�1) sp (s)

Mz(Tx) Mz(T

y) My(Tz) Mx(T

z) kz ky kx

m13 = 310.4 m8 = 309 2.4 0.1 0.0 0.0 182.0 0.00 0.06 1649.5m32 = 734.0 m16 = 723 2.4 0.5 0.0 0.0 186.6 0.00 0.05 1607.2m54 = 976.8 m15 = 952 2.4 0.6 0.0 0.0 196.7 0.00 0.05 1525.0m94 = 1654.8 m10 = 1614 2.6 1.9 0.1 0.0 325.3 0.21 0.08 921.3S0(0–0) 0.0 2.4 0.0 0.0 0.0 183.0 0.00 0.06 1618.7

MzðTxÞ ¼ h3Wx1jzj1W0i; kx rate constant for emission polarized along x-axis; sp – the phosphorescent radiative lifetime at high-temperature limit (s).

a Experimental frequency with numeration of the in-plane vibrational modes accepted in [43].


4.2.1. Vibronic activity of the ag modes in H2P

phosphorescence

We have chosen to study a number of ag vibrations(m13 = 310 cm�1, m33 = 738.5 cm�1, m54 = 977 cm�1,m94 = 1654 cm�1; Table 6) in the range of intense vib-ronic activity in the H2P phosphorescence [12,24]. Inthe highly resolved phosphorescence spectrum of H2Pin a Xe matrix [24] the corresponding lines 309, 723,952 and 1614 cm�1 are well recognized. We have foundnegligible changes in the emission rate along the dis-placements of the m13 mode (Table 10). This vibrationis very active in the Raman spectrum (Table 5) and wellseen in fluorescence of H2P [24,45]. This mode can be de-scribed as a uniform breathing of the whole tetrapyrrolering. The N–H bonds come very close at the inner site ofthe amplitude and this provides a strong change in thedipole moment of the S–S transition. This fundamentalline is strongly polarized in the normal Raman spectrumas the mode remains totally symmetric under D4h pseu-do-symmetry [45]. The intensity of the Raman lines canbe induced by the Herzberg–Teller (HT) and by theFranck–Condon (FC) mechanisms [39,45,84]. The HTmechanism is determined by vibrationally induced cou-pling between excited singlet states. In nonresonanceRaman spectra the largest contribution comes fromthose S states to which electronic transitions are stronglyallowed and which are close in their excitation energy tothe energy of the scattering light. In a normal Ramanspectrum of H2P, where the incident photon energy isabout 1 eV below the onset of the Q band, all four Q

and B states (1,21B2u,3u) will contribute comparably. Ifwe limit our consideration to these excited singlet states,then only ag and b1g vibrations should be active in theRaman spectrum of H2P. The totally symmetric modesgain Raman intensity through the HT and FC mecha-nisms, while the b1g modes (and also the b2g and b3gout-of-plane modes) gain their activity entirely fromthe HT mechanism [45]. As follows from our calcula-tions for the m13(ag) mode (Table 6) the gradient of theexcited states 3B2u,

1B3u potential along this mode is verylarge. Thus, both FC and HT mechanisms are expectedto contribute to the Raman and fluorescence intensity ofthe m13 = 310 cm�1 mode. The line 309 cm�1 is observedin H2P phosphorescence in Xe matrix with very high rel-

ative intensity (the second intense line after the 0–0 tran-sition) [24]. As follows from Table 10 this line should benegligible. One has to note that the HT mechanism inphosphorescence is very much different from HT vib-ronic theory of fluorescence and Raman scattering, sinceadditional account of SOC and spin-sublevel symmetryleads to a new selection rule [58]. We come to the con-clusion that the line 309 cm�1 observed in H2P phospho-rescence in Xe matrix is induced by the EHA effect.Since the calculated FC factor is very high for overtonesof this vibration, h0|1i = 0.25, the corresponding line618 cm�1 is well seen in the T1 ! S0 spectrum inducedby xenon, though the authors of [24] have not men-tioned such an assignment. The other two vibrationsin Table 10 are predicted to be moderately active inphosphorescence, though they are relatively intense inthe Xe matrix measurements.

The only large exception is provided by vibration m94,for which we get quite large nonzero derivative for theMz(T

y) transition moment (Table 10). This vibration(1614 cm�1 in the experiment) behaves very differentlyin fluorescence and phosphorescence of H2P in Xe ma-trix [24]. While fluorescence is typical for porphyrins inthat no strong progressions are observed, phosphores-cence reveals a nice progression based on a 1614 cm�1

mode. Even the second overtone of this vibration is de-tected in deuterated free-base porphin [24]. We can re-late this to strong vibronic activity of the m94 mode inthe entire phosphorescence spectrum of free-base por-phin itself. We get also the largest gradient of the 3B2u

potential along the displacement of this mode, whichprovides appreciable FC factors for overtones:h0|0i = 0.88, h0|1i = 0.12 and h0|2i = 0.01.

It was stressed that only totally symmetric vibrationsare active in the phosphorescence spectrum detected inXe matrix [24]. This gives us additional confidence thatthe phosphorescence spectrum of H2P in Xe matrix [24]is dominated by the EHA effect. A strong enhancementof the 0–0 band has been explained recently [36] by cal-culations of the vertical T1 ! S0 transition probabilityin the complex of H2P with Xe. Spin–orbit coupling inthe complex is dominated by one-center contributionsfrom the xenon atom [36], the 0–0 band is much strongerthan all vibronic bands [24] and this spectrum cannot be

Table 11QR DFT/3-21G results for the T1–S0 transition calculated along the b1g (in-plane) vibrational displacements in the H2P (D2h) molecule

mi (cm�1) mj (cm

�1)a Transition moments (10�5 ea0) Rate constants (10�5 s�1) sp (s)

Mz(Tx) Mx(T

z) kz kx

m4 = 98.9 m35 = 109 2.6 0.0 228.2 0.06 1314.4m18 = 395.9 m25 = 389 2.1 0.0 150.0 0.04 2003.8m19 = 420.1 m33 = 418 2.4 0.1 183.0 0.08 1639.7m42 = 797.1 m32 = 786 2.2 0.0 155.0 0.05 1932.5m44 = 817.4 m24 = 805 3.0 0.1 288.7 0.09 1038.8m56 = 999.0 m22 = 976 2.6 0.0 228.2 0.06 1314.4m65 = 1169.6 m21 = 1138 2.4 0.0 172.4 0.01 1740.1m69 = 1222.8 m34 = 1182 3.2 0.0 353.5 0.08 848.6m71 = 1261.9 dNH = 1226 2.2 0.0 155.9 0.04 1923.6m75 = 1357.3 m26 = 1316 2.1 0.0 185.3 0.06 1618.7m79 = 1422.4 m20 = 1388 1.8 0.1 90.9 0.09 3297.9m86 = 1538.6 m28 = 1497 2.1 0.1 141.4 0.18 2118.7m92 = 1640.7 m19 = 1600 1.6 0.0 56.9 0.02 5267.5m96 = 3196.7 m27 = 3042 2.4 0.0 189.1 0.07 1586.0m99 = 3230.4 m31 2.4 0.0 193.8 0.06 1547.7m102 = 3249.1 m23 2.3 0.0 173.6 0.07 1727.4S0(0–0) 0.0 2.4 0.0 183.0 0.06 1618.7

MzðT xÞ ¼ h3Wx1jzj1W0i; kx rate constant for emission polarized along x-axis; sp – the phosphorescent radiative lifetime at high-temperature limit (s).

a Experimental frequency with numeration of the in-plane vibrational modes accepted in [43].


compared with the entire H2P phosphorescence. Wehave restricted our further search of the ag vibronicactivity as the b1g modes are more important and there-fore studied all of them.

4.2.2. Vibronic activity of the b1g modes in H2P

phosphorescence

For the other in-plane vibration, b1g, which is also ac-tive in inducing vibronic phosphorescence bands the dis-placed geometry still belongs to the D2h symmetry pointgroup. The results of phosphorescence calculationfor displacements along the b1g modes are presented inTable 11. Vibronic activity of emission from the T z spinsublevel is very low for these in-plane vibrations. Tran-sitions from the T x spin sublevel with the out-of-plane(z) polarization are more active. Thus, all b1g vibrations

Table 12QR DFT/3-21G results for the T1–S0 transition calculated along the b2g (ou

mi (cm�1) Transition moments (10�5 ea0)

My(Ty) Mx(T

x) Mz(Tx) Mx(T

z)

m5 = 125 0.1 0.3 2.4 0.0m8 = 188 0.1 0.2 2.4 0.0m20 = 430 0.4 0.5 2.6 0.1m23 = 630.5 0.2 0.1 2.4 0.0m26 = 680 0.0 0.2 2.4 0.0m30 = 713 0.2 0.5 2.4 0.0m39 = 793 0.1 0.6 2.4 0.0m47 = 862 0.1 0.6 2.4 0.0m50 = 912.9 0.2 0.1 2.4 0.0S0(0–0) 0.0 0.0 2.4 0.0

MzðT xÞ ¼ h3Wx1jzj1W0i; kx rate constant for emission polarized along x-axis; s

Mx(Tz) remains negligible as for the 0–0 transition; Mz(T

x) remains constan

in the H2P phosphorescence vibronic spectrum shouldhave the same polarization as the 0–0 band.

The most intense line is m69 = 1223 cm�1 (Tables 6and 11) which should correspond to 1182 cm�1 in theexperimental spectrum. There is a wide vibronic featurein this region in the H2P phosphorescence spectrum inhydrocarbon solvents at 77 K [12]. The next intense 0–1 line in the H2P phosphorescence spectrum is predictedfor the m44 vibration, Table 11; it corresponds to 805cm�1 frequency and such a line is really observed in aShpolskii matrix [12]. A little bit less active is the m56vibration (976 cm�1 in experiment, Table 11), which liesin the region of a broad emission band in hydrocarbonsolvents [12]. The C–H stretching vibrations in thehigh-frequency region (about 3200 cm�1, Table 11) arenon-active in agreement with observations [12].

t-of-plane) vibrational displacements in the H2P (D2h) molecule

Rate constants (10�5 s�1) sp (s)

Mz(Tz) kx kz ky

0.1 3.82 183.2 0.33 1602.90.0 1.49 183.0 0.26 1624.30.1 6.88 222.6 4.58 1281.70.0 0.29 183.5 0.48 1626.30.0 0.93 183.1 0.06 1632.00.2 9.01 186.7 1.70 1520.10.0 13.7 181.6 0.63 1531.00.1 12.2 183.0 0.46 1532.80.0 0.27 182.1 1.47 1632.20.0 0.00 183.5 0.06 1618.7

p – the phosphorescent radiative lifetime at high-temperature limit (s).t as for the 0–0 transition for all displacements.

Table 13The QR DFT/3-21G results for the T1–S0 transition calculated along the b3g (out-of-plane) vibrational displacements in the H2P molecule

mi (cm�1) Transition moments (10�5 ea0) Rate constants (10�5 s�1) sp (s)

Mz(Tx) My(T

x) Mx(Ty) Mx(T

z) kz ky kx

m6 = 133.2 2.4 0.0 0.2 0.0 183.0 0.00 1.10 1629.7m9 = 207.8 2.4 0.1 0.0 0.0 184.5 0.30 0.09 1622.6m21 = 443.2 2.4 0.2 0.2 0.0 181.0 0.82 1.72 1635.3m25 = 679.2 2.6 0.4 0.0 0.1 217.5 5.77 0.19 1342.2m31 = 714.6 2.4 0.6 0.2 0.0 183.3 10.3 1.92 1533.9m37 = 789.2 2.4 0.2 0.1 0.0 180.9 1.05 0.50 1644.5m46 = 860.6 2.4 0.1 0.2 0.0 181.6 0.11 0.81 1643.9m51 = 917.9 2.4 0.2 0.2 0.0 181.5 1.44 1.27 1628.3S0(0–0) 2.4 0.0 0.0 0.0 183.0 0.0 0.06 1618.7

MzðT xÞ ¼ h3Wx1jzj1W0i; kx rate constant for emission polarized along x-axis; sp – the phosphorescent radiative lifetime at high-temperature limit (s).


4.2.3. Vibronic activity of the out-of-plane vibrations

(b3g and b2g modes) in H2P phosphorescence

Out-of-plane vibrations were not considered by Liand Zgierski [43], thus we present only our throughnumeration from Table 6. All eight vibrational modesof the b3g symmetry are studied and presented in Table13. One can anticipate that these out-of-plane vibrationscan be active in the H2P phosphorescence vibronic spec-tra with such a weak 0–0 band, since the out-of-planedisplacements induce one-center SOC integrals betweensinglet and triplet pp* states. In contrast to this expecta-tion there is no large vibronic activity of the b3g modesin the free-base porphin phosphorescence as one can seefrom Table 13.

The modes m25 = 679 cm�1 and m31 = 715 cm�1 canbe noticed as more intense bands. They correspondto the frequencies 660 and 700 cm�1 calculated withproper scaling factors [45]. Such lines have not beenfound neither in Raman spectra, because of their neg-ligible intensity (Table 6), nor in highly resolved lumi-nescence spectra of H2P in noble gas matrices [24].Both lines reside in the region of a wide band 600–800 cm�1 seen in the low-resolution phosphorescencespectrum of H2P in hydrocarbon solvent at 77 K[12]. Since the EHA-induced phosphorescence in Xematrix [24] is probably much more intense than thelimit of the predicted intensity of these two lines, ourpredictions do not contradict available experimentalinformation. More sensitive PMDR spectra in low tem-perature neutral matrices are necessary to check ourpredictions.

In Table 12, we present our QR DFT calculations fordisplacements along the b2g modes. Accounting for thetotal symmetry of spin sublevels presented above onecan see that the b2g vibration induces x-polarized emis-sion from the Tx spin sublevel, y-polarized emissionfrom the Ty sublevel, and z-polarized emission fromthe Tz sublevel in addition to those transitions whichhave been obtained for the ‘‘vertical’’ 0–0 band: Mz(T

x)and Mx(T

z).

Only one vibration induces nonzero derivatives for allsymmetry-allowed T1 ! S0 transitions. This is them20 = 430 cm�1 vibrational mode (Table 12). At thesame time only the m20 mode provides a considerablechange in the Mz(T

x) transition dipole moment, whichis the main source of the phosphorescence intensity. This0–1 line is clearly seen as the most intense one in theexperimental spectrum [12] at the frequency about 440cm�1. It is more intense than the 0–0 line [12] in goodagreement with our QR DFT calculation. It should bestrongly depolarized as also follows from the QR DFTcalculation.

The transition moments in Table 12 (except Mz(Tx))

are equal to the derivatives since the displacement isnot scaled. Among other b2g modes the 0–1 line is themost intense one for the m30 = 713 cm�1 vibration; it isemitted from all three spin sublevels with the strongestemission from Tx. The line m30 has a mixed polarizationin which the NH� � �HN direction (x-axis) strongly pre-vails. The next relatively intense 0–1 line correspondsto the m39 = 793 cm�1 vibration; it has strong x-polariza-tion (along the NH� � �HN axis) and corresponds to emis-sion from the Tx spin sublevel. Both 0–1 lines, m30 andm39, are in the range of a wide band 600–800 cm�1 de-tected in the low-resolution phosphorescence spectrumof H2P in Spolskii matrix (hydrocarbon solvent at 77K) [12]. They are close to the maximum of this bandat about 770 cm�1. A similar result is obtained for them47 = 862 cm�1 mode (Table 12), which is close to theneighbor band (860 cm�1) in the low-resolution phos-phorescence spectrum [12]. Thus, the calculated activityof the b2g modes agrees quite well with the experimentalH2P phosphorescence spectrum [12].

Concluding the above discussion of H2P spectra wehave to mention a preliminary comparison of vibronicpatterns in phosphorescence and fluorescence, which re-veals a big difference between these two types of emis-sion spectra in agreement with experimental data[12,10,24]. The phosphorescence spectrum of H2P differsfrom the fluorescence spectrum in the increased intensity


of vibronic bands with frequencies in the 160–700 cm�1

range and also in the low intensity of the vibronic bandin the 1500–1600 cm�1 range which are very intense influorescence. The most active modes in the fluorescencevibronic spectra are two vibrations involving themethine bridge at about 1600 cm�1 of ag and b1g symme-try [38]. The ag band observed in the nonresonance Ra-man spectrum of H2P at 1609 cm�1 has been identifiedwith the 1614 cm�1 band in fluorescence [24,45]; in our6-31G** calculation without any scaling it is obtainedat 1655 cm�1 (m94 in Table 10). As follows from the pres-ent and previous calculations [39,43,45] this mode in-cludes mostly the CaCm stretching of the methinebridges (m10 in notations of Li and Zgierski [43]).Though almost all ag vibrations of the H2P moleculehave low depolarization ratio (Table 6), the band m94in Table 6 has a large calculated depolarization ratiofor plane polarized light: Dep(P) = 0.684. This is closeto 0.75, which is typical for b1g vibrations; they haveopposite signs for the in-plane components of the polar-izability derivatives. According to vibronic theory ofRaman spectra [39] vibrational modes forming vibronicbands are active in resonance Raman spectra. This is nottrue for phosphorescence, the intensity of which is deter-mined by SOC and its specific selection rules for vib-ronic bands. Phosphorescence emission of H2P issimilar to fluorescence only in Xe matrix, where the for-mer is dominated by totally symmetric ag modes, whichare efficient mediators for the EHA effect.

Table 14Electronic g-tensors of triplet porphyrins (Dgab = gab � 2.0023 in ppt)

Molecule Dgab LDA B3LYP BP Exp.a,b

ZnP Dgxx 101 �90 88 �300a

Dgyy 1654 1100 977 �300Dgzz �7244 �3400 �5231 �13300

MgP Dgxx 550 210 700a

Dgyy �450 �122 �300Dgzz �1320 �760 –

H2P Dgxx 1481 800b

Dgyy 716 100Dgzz �1192 �1700

a Ref. [80].b Ref. [79].

4.3. Phosphorescence and EPR spectra of the

metalloporphyrins

In a system with D4h symmetry vibrations of the b1gand b2g types can be Jahn–Teller active [77]. We havecalculated the phosphorescence emission from the low-est component of the degenerate 3Eu state which corre-sponds to the b1g distortion. We thus analyze thevibrational modes calculated for the optimized fourfoldsymmetry of the ground state: the a1g,b1g and b2g in-plane vibrations and eg out-of-plane vibrational modes.All these vibronically active modes in the range 400–1600 cm�1 were accounted for in our vibronic structurecalculations of phosphorescence of MgP and ZnP. Fulldetails will be discussed elsewhere [75]; here, we presentonly short qualitative features.

Phosphorescence of Mg-porphyrin in EPA at 77 Kdiffers from that of free-base porphin by the fact thatthe 0–0 line is the strongest one in the MgP spectrum[12]. This finding is supported by our calculations. Wehave also found the eg out-of-plane vibrational modem31,32 = 722 cm�1 to be the most active one among 0–1bands. The b2g vibration m77 = 1390 cm�1 providesweaker intensity. The low-resolution phosphorescencespectrum [12] does not contradict these predictions.

The phosphorescence spectrum of ZnP in n-octanesingle crystal at 1.2 K [77] indicates the strong 0–0 lineto be at 14,666 cm�1. Two other lines of appreciableintensity are shifted 1316 and 1575 cm�1 to the red.We have interpreted these lines as induced by 0–1 tran-sitions of m77 and m85 vibrational modes (Table 8). Allother vibrations are quite inactive in ZnPphosphorescence.

Since the parameters of zero-field splitting for the T1

state of free-base porphin have been calculated and dis-cussed before [37], we focus here on our results from cal-culations of spin-Hamiltonian parameters in connectionwith the EPR spectra of MgP and ZnP in the tripletstate [80]. The observed hyperfine structure (HFS) inZnP closely resembles that in MgP and originates fromfour equivalent methine C–H fragments and twoinequivalent pairs of nitrogen atoms [80]. Our resultsare in a good agreement with these EPR data. At theoptimized geometry of the lowest triplet state of ZnP,3B1u (b1g distortion from the D4h structure), we have ob-tained almost equal isotropic HFS constants for themethine carbons 13C: a9 = 28.84 MHz. For the fourmethine hydrogens isotropic HFS constants are identi-cal: a25 = �13.6 MHz. For the elongated nitrogens(yN = 2.085 A) the isotropic HFS constant is 8.14MHz, but for those nitrogen atoms which are closer tothe central ion (xN = 2.077 A) the isotropic HFS con-stant is rather small (1.24 MHz). No isotropic HFS onthe Zn-ion is found; the anisotropic HFS tensor at the67Zn nucleus has small nonzero components: Bxx =0.56 MHz, Byy = �0.13 MHz, Bzz = �0.43 MHz. (y isan axis of elongation of the rectangle, z-axis is perpen-dicular to the plane). The largest anisotropy of HFS isfound for the methine carbons 13C: Bzz = 55.69 MHz,Byy = �27.31 MHz, Bxx = �28.38 MHz.

We have calculated anisotropy of g-factor in the low-est triplet state of ZnP and MgP (Table 14) by the re-stricted DFT linear response method of Rinkeviciuset al. [62]; the best results have been obtained with theLDA functional and the scaled atomic nuclear chargeapproximation for the SOC integrals. For ZnP we have:


gzz = 1.995 (1.989), gyy = 2.0033 (2.002), gxx = 2.0039(2.002) (Experimental data [80] in parentheses.) ForMgP an almost isotropic g-factor has been obtained(Table 14) in agreement with experiment [80]. We alsotried to estimate the g-tensor deformation along somevibrational distortions. For those modes which are ac-tive in phosphorescence no appreciable derivatives ofthe g-tensor components have been found indicatingthat vibronic corrections to the g-factor could be negli-gible for ZnP and MgP triplet states.

4.4. Comparison of vibronic patterns in the intersystem

crossing

The non-radiative decay of spin sublevel Ta can beanalyzed by two terms [13]: one depends on the FC fac-tor between the singlet and triplet vibronic levels con-nected by the ISC process (it is the same for all threespin sublevels), the other term is a vibronic SOC matrixelement which is highly spin-selective. The rate of non-radiative decay of the spin sublevel Ta to the groundstate S0 can be estimated by [13,87]

ka ¼ 2p3

� �Xi

ohS0jHSOjT aioQi

��2

ðFCÞi. ð15Þ

The Franck–Condon factors for vibrational mode Qi de-pend on the energy gap between the states. In practice,only few ‘‘promoting modes’’ of the molecule prove tobe dominant in the non-radiative process [13]. We havecalculated a number of derivatives of SOC integrals inthis equation for the H2P and ZnP molecules. As alreadymentioned above we have paid attention to the m107vibrational mode of b3u symmetry (Table 6), which cor-responds to a high-frequency N–H out-of-phase vibra-tion of free-base porphin. Since this mode is absent inZnP it may account for the faster relaxation of the T1

state in H2P than in metalloporphyrins. In contrast toour expectation we get very small spin-vibronic activityof this mode. At the maximum of displacement alongthis normal mode we have found

h1AgjHzSOj3B

z2ui ¼ 7.5� 10�5 cm�1; ð16Þ

thus a negligible derivative has been obtained(4.7 · 10�4 cm�1/A). By comparison with this and someother b3u modes we have found that the au vibrations aremore active in promoting the ISC process in the H2Pmolecule. For example, for the m52 mode we get

h1AgjHySOj3B

y2ui ¼ 3.89� 10�2 cm�1. ð17Þ

This vibration is silent in the IR spectrum (Table 6), butits overtones should contribute to the non-radiative de-cay of the triplet state. The most strong spin-vibronicinteraction is found for the b1u vibrations. For the m43mode which is very active in the IR spectrum (Table6), we get

h1AgjHxSOj3Bx

2ui ¼ 7.12� 10�2 cm�1. ð18Þ

We get a similar large SOC integral for the m48 mode(5.62 · 10�2 cm�1). Though we cannot estimate the totalnon-radiative rate constant of the T1–S0 relaxation wecan predict a large contribution to the ISC process ofthese two vibrational modes. Such analysis is in agree-ment with experimental data from optical detection ofmagnetic resonance (ODMR) [13]: the non-radiative de-cay of the Tx spin sublevel of H2P is the most effectiveone. It is 38 times faster than that of the Tz subleveland 3 times faster than that of the Ty sublevel [13].

The a2u vibrations of ZnP can promote the T1(3Eu)

mixing with the ground state by x,y components of theSOC operator. We have checked the vibronic activityof the m6 mode in inducing the T1 ! S0 transition. Asmentioned above, this mode is very specific for ZnPsince it includes the out-of-plane motion of the Zn-ion.No appreciable activity of the m6 mode has been foundin contrast to expectation. At the same time other vibra-tions of a2u type in ZnP, which are similar to the b1umodes of H2P (m38 and m49, Table 5), are found activewith the same order of magnitude. Thus, the Zn ion doesnot enhance this promoting modes in comparison withfree-base porphin and the Tx,Ty sublevels are almostequally active in Zn-porphyrin [13].

The in-plane eu vibrations in ZnP can induce nonra-diative decay of the Tz sublevel of the phosphorescenttriplet 3Eu state. In contrast to the very weak spin-vibronic activity of the b3u modes of H2P we have foundthat some of the eu modes (m87, m71 and m67 in Table 5)provide much bigger SOC integrals than that of Eq.(16). These Cb–H vibrations can even induce SOC mix-ing of the Tz

1 and S0 states in ZnP. Thus, the Tz spin sub-level in ZnP is much more involved in the non-radiativerelaxation of the triplet state than in H2P in agreementwith ODMR results [13].

The high-frequency stretching C–H vibrations (m95–m105) are not active in triplet decay of either ZnP orH2P molecules. They do not induce large SOC inte-grals. Because of that only low-frequency C–H modesare involved for which the FC factor strongly dependson the energy gap between the T1 and S0 states. Thus,the energy gap law for radiationless transitions [13,88]is the dominant factor for porphyrins. According tothis law the radiationless decay rate T1 ! S0 dependson the energy gap DE = E(T1) � E(S0) via a factorDE�1/2exp(�cDE/hmi), where c is an empirical constantand mi is a frequency of the promoting mode. In accordwith our calculation and experimental data (Table 2)the gap DE in free-base porphin is about 2200 cm�1

smaller than the corresponding gap in zinc porphyrin.This is comparable with the frequencies of the promot-ing modes and provides the faster T1 ! S0 radiation-less decay in the H2P molecule in comparison withZn-porphyrin.


4.4.1. S1 ! T1 radiationless transition

The Minsk group [12,25] has shown that the main de-cay route of the first excited singlet state S1 of free-baseporphin is the intersystem crossing to the lowest tripletstate T1 with a very high quantum yield (/ISC = 0.9).This is in general agreement with many observationsin spectroscopy of porphyrin free bases such as the trip-let–triplet absorption (TTA) measurements [10] and theoptical detection of magnetic resonance [13].

There are four triplet states which are lower in energythan the S1(

1B3u) state (only two of them are shown inTable 2). Direct SOC integrals between these S and Tstates of pp* type are negligible for the z-componentof the SOC operator and it should therefore be spin-vibronic interaction which promotes the most efficientmixing between S1(

1B3u) and lower-lying triplet statesof 3B2u and 3B3u symmetry. From vibronic and SOCselection rules one can see that the b2g modes can be ac-tive in the S1–Tn mixing. Spin–orbit coupling integralshTn|HSO|S1i are calculated for n = 1–4 in the H2P mole-cule under vibronic distortions of all eight b2g modes gi-ven in Table 12.

Some of the low-frequency vibrations of this typeindicate appreciable radiative activity in phosphores-cence (Table 12). These are m5 = 125 cm�1 (out-of-planeand out-of-phase bending protonated rings) andm20 = 430 cm�1 (out-of-plane and out-of-phase twist ofnon-protonated rings). All b2g modes are not active inSOC-induced mixing between the S1 and T2–T4 states.The strongest interaction is induced for the direct S1–T1 spin-vibronic mixing: it means promoting spin–orbitinteraction between the 11B3u and 13B2u states under b2gdistortion. Vibration m20 provides the following SOCintegrals

h11B3ujHxSOj13B

x2ui ¼ 2.13� 10�2 cm�1; ð19Þ

h11B3ujHySOj13B

y2ui ¼ 2.41� 10�2 cm�1. ð20Þ

The next active mode is m47 = 862 cm�1 (out-of-planeand out-of-phase methine bridge Cm–H vibration)which induces the corresponding SOC integrals8.34 · 10�2 and 1.53 · 10�2 cm�1. Other b2g distortionsare not so active.

Thus, only the hS1jHxSOjT x

1i integral increases substan-tially along these vibrations. This is in a good agreementwith ODMR studies [13] where the Tx sublevel has beenfound to indicate the largest probability to be pumpedduring the intersystem crossing. The x-axis is an in-plane axis perpendicular to the direction of the nitrogenlone pairs (y-axis). The rotation around the x-axis trans-forms the lone pair orbitals to p-orbitals and such arotation creates a torque to induce a spin flip duringthe non-radiative S1 ! T1 transition. The lower ISCprobability has been found for the Ty spin sublevel[13]. The populating rate of the Tz spin sublevel has beendetected as completely negligible [13]. All these findings

are in a good agreement with our calculation of the b2gvibrational activity. Thus, very high spin-sublevel selec-tivity of the singlet–triplet ISC transitions is predicted.The free-base porphin molecule is a highly anisotropictop with a large momentum of inertia. Rotational diffu-sion in a viscos solvent cannot average to zero theanisotropy of the ISC S1 ! Ta

1 process [90]. In this casean external magnetic field will mix spin sublevels andinfluence the total rate of the S1 ! T1 process. Thus,we come to the conclusion that the magnetic field caninfluence the T–T absorption (at least at low tempera-tures) which is important for optical limiting devicesbased on porphyrins in viscos solvents, in crystals orin thin films. One should thus be able to manipulatethe T–T absorption by an external magnetic field andthis can be important in non-linear optics and opticalstorage applications.

5. Conclusions

Metalloporphyrins with a closed-shell ground stateexhibit an internal heavy atom effect which enhancesan extremely weak phosphorescence of the free-baseporphin. An aim with the present study was to presenta theoretical explanation of this phenomenon includingthe vibronic structure of the T1 ! S0 emission. The qua-dratic response time-dependent DFT method was usedfor calculations of the phosphorescence spectra of free-base porphin, magnesium- and zinc-porphyrins. VerticalT–S transitions were used as a first estimation of the 0–0band intensity and radiative lifetimes in the high- andlow-temperature limits. Vibronic activity of all phospho-rescence active out-of-plane vibrations (b2g and b3gmodes in H2P and eg modes in ZnP) was studied by di-rect calculations of the T–S transition dipole momentsalong the nuclear displacements. A strong activity ofthe m20 = 430 cm�1 (b2g) mode is obtained which agreeswith measurements of the first intense vibronic band ofthe H2P phosphorescence spectrum in solid hydrocar-bon solvent without heavy atoms [12].

A number of active b3g modes are calculated in therange 680–920 cm�1 which coincide with the observedwide bands in the low-resolution phosphorescence spec-trum of H2P in hydrocarbon glass at 77 K. Highly re-solved phosphorescence spectra of free-base porphinwhich have been detected at 4–10 K in noble gas matri-ces [24] cannot be compared with our results, while a xe-non matrix gives strong H2P phosphorescence asanalyzed by Radziszewski et al. [24]. A number of in-plane ag vibrations lack activity in the calculated T–Stransition moment. For example, the line m13 = 310cm�1 is predicted to be very weak; at the same time itis the most intense line (besides the 0–0 one) in the ob-served spectrum in a xenon matrix. Mostly, totally sym-metric vibrations are active in such a spectrum [24]


which means that they are induced by an external heavyatom (EHA) effect.

No polarization measurements have been reported sofar for the phosphorescence of H2P since the emission isvery weak in the absence of EHA. Our calculations pre-dict that the emitted light of the 0–0 band should becompletely polarized in the direction perpendicular tothe molecular plane. The vibronic lines should havemixed or in-plane polarization. A similar polarizationpattern is predicted for MgP phosphorescence. In con-trast, the ZnP phosphorescence has the strongest 0–0line with large in-plane polarization determined by emis-sion from the Tz spin sublevel. These results are in per-fect agreement with ODMR data [13] and polarizationmeasurements [82].

Tsvirko et al. [12] used a highly sensitive spectromet-ric apparatus to detect phosphorescence of porphins inthe absence of the EHA effect. They found the radiativelifetime of phosphorescence equal to 128 s for the H2Pmolecule (in octane–benzene) and about 48 s for Mg-porphyrin (in petroleum ether–ethanol) [12]. Muchshorter radiative lifetime has been estimated for Zn-por-phyrin [80]. Though our calculations predict larger spvalues they reproduce this trend.

The electronic g-tensor in triplet excited porphyrinswith and without transition metals is well reproducedby spin restricted DFT linear response. The spin-sub-level selectivity of the singlet–triplet transitions allowsto predict a magnetic field effect in non-linear opticsand optical storage applications.

We find that the DFT method provides explanationsof a large number of triplet state properties of the vari-ous porphyrins. An increase of the metal atomic numberleads to a clear enhancement of the phosphorescencerate constant and anisotropy of the g-tensor; these inter-nal heavy atom effects observed earlier in EPR andODMR spectra of porphyrins have found a comprehen-sive explanation by means of the DFT methodology.

Acknowledgements

The authors acknowledge a grant from the photonicsproject run jointly by the Swedish Materiel Administra-tion (FMV) and the Swedish Defense Research Agency(FOI). This work has also been supported by the Swed-ish Reseach Council. Collaboration with Yanhua Wangand Yi Luo who provided the Franck–Condon factors isgreatly acknowledged.

References

[1] J.A.A. Krasnovsky, Photosynth. Res. 76 (2003) 389.[2] G. Gurinovich, A. Sevchenko, K. Solovjev, in: Spectroscopy of

Chlorophyll and Related Compounds, Science and Technology,Minsk, 1968.

[3] P. Siegbahn, M. Blomberg, Chem. Rev. 100 (2000) 421.[4] J. Rogers, K. Nguyen, D. Hufnagle, D. McLean, W. Su, K.

Gossett, A. Burke, S. Vinogradov, R. Pachter, P. Fleitz, J. Phys.Chem. A 107 (2003) 11331.

[5] J.A.A. Krasnovsky, M. Bashtanov, N. Drozdova, O. Yuzhakova,M. Lukyanets, Quantum Electron. 32 (2002) 83.

[6] C. Schweitzer, R. Schmidt, Chem. Rev. 103 (2003) 1685.[7] M. Gouterman, J. Mol. Spectrosc. 6 (1961) 138.[8] K. Nguyen, P. Day, R. Pachter, J. Phys. Chem. A 104 (2000)

4748.[9] K. Nguyen, R. Pachter, J. Chem. Phys. 118 (2003) 5802.

[10] M. Gouterman, G. Khalil, J. Mol. Spectrosc. 53 (1974) 88.[11] A. Gradyushko, V. Mashenkov, A. Sevchenko, K. Solovjev, M.

Tsvirko, Sov. Phys. Dokl. 13 (1969) 869.[12] M. Tsvirko, K. Solovjev, A. Gradyushko, S. Dvornikov, Opt.

Spectrosc. 38 (1975) 400.[13] W. van Dorp, W. Schoemaker, M. Soma, J. van der Waals, Mol.

Phys. 30 (1975) 1701.[14] R. Connors, W. Leenstra, in: Triplet State ODMR Spectroscopy,

Wiley, New York, 1982, p. 258.[15] R. Bauernschmitt, R. Ahlrichs, Chem. Phys. Lett. 256 (1996) 454.[16] H. Nakatsuji, J. Hasegawa, M. Hada, J. Chem. Phys. 104 (1996)

2321.[17] S. van Gisbergen, A. Rosa, G. Ricciardi, A. Baerends, J. Chem.

Phys. 111 (1999) 2499.[18] K. Nguyen, P. Day, R. Pachter, J. Chem. Phys. 110 (1999) 9135.[19] D. Sundholm, Phys. Chem. Chem. Phys. 2 (2000) 2275.[20] A. Jarzecki, P. Kozlowski, P. Pulay, B. Ye, X. Li, Spectrochim.

Acta A 53 (1997) 1195.[21] B. Minaev, J. Appl. Spectrosc. (USSR) 42 (1985) 518.[22] L. Andersen, P. Ogilby, Photochem. Photobiol. 73 (2001) 489.[23] S.P. McGlynn, T. Azumi, M. Kinoshita, Molecular Spectroscopy

of the Triplet State, Prentice-Hall, Engelwood Cliffs, NJ, 1969.[24] J.G. Radziszewski, J. Waluk, M. Nepras, J. Michl, J. Phys.

Chem. 95 (1991) 1963.[25] M. Tsvirko, K. Solovjev, A. Gradyushko, S. Dvornikov, Zh.

Prikl. Spektrosk. 20 (1974) 528.[26] R. Clarke, in: Triplet State ODMR Spectroscopy, Wiley, New

York, 1982.[27] A. Parusel, S. Grimme, J. Porphyrins, Phtalocyanines 5 (2001)

225.[28] R. Becker, M. Kasha, J. Am. Chem. Soc. 77 (1955) 3669.[29] J. Allison, R. Becker, J. Chem. Phys. 32 (1960) 1410.[30] R. Becker, J. Allison, J. Phys. Chem. 67 (1963) 2269.[31] R. Becker, J. Allison, J. Phys. Chem. 67 (1963) 2262.[32] W. van der Poel, D. Singel, J. Schmidt, J. van der Waals, Mol.

Phys. 49 (1983) 1017.[33] H. Agren, O. Vahtras, B. Minaev, Adv. Quantum Chem. 27

(1996) 71.[34] I. Tunnell, Z. Rinkevicius, P. Salek, O. Vahtras, B.T. Helgaker,

H. Agren, J. Chem. Phys. 119 (2003) 11024.[35] B. Minaev, I. Tunnell, P. Salek, O. Loboda, O. Vahtras, H.

Agren, Mol. Phys. 102 (2004) 1391.[36] B. Minaev, Spectrochim. Acta A 60 (2004) 3213.[37] O. Loboda, I. Tunnell, B. Minaev, H. Agren, Chem. Phys. 312

(2005) 299.[38] L. Gladkov, A. Gradyushko, A. Shulga, K. Solovyov, A.

Starukhin, J. Mol. Struct. (TEOCHEM) 45 (1978) 463.[39] K. Solovyov, L. Gladkov, A. Gradyushko, A. Shulga, N.

Ksenofontova, A. Starukhin, J. Mol. Struct. (TEOCHEM) 45(1978) 267.

[40] L. Gladkov, K. Solovyov, Spectrochim. Acta A 41 (1985) 1437.[41] J.G. Radziszewski, M. Nepras, V. Balaji, J. Waluk, E. Vogel, J.

Michl, J. Phys. Chem. 99 (1995) 14254.[42] J.G. Radziszewski, J. Waluk, J. Michl, J. Mol. Spectrosc. 140

(1990) 373.[43] X.-Y. Li, M. Zgierski, J. Phys. Chem. 95 (1991) 4268.


[44] P. Kozlowski, A. Jarzecki, P. Pulay, J. Phys. Chem. 100 (1996)7007.

[45] P. Kozlowski, A. Jarzecki, P. Pulay, X. Li, M. Zgierski, J. Phys.Chem. 100 (1996) 13985.

[46] A. Becke, J. Chem. Phys. 98 (1993) 5648.[47] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A.

Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery Jr.,R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D.Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V.Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C.Adamo, S. Clifford, J. Ochterski, G.A. Petersson, P.Y. Ayala, Q.Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghava-chari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, B.B. Stefanov, G.Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L.Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A.Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B.Johnson, W. Chen, M.W. Wong, J.L. Andres, C. Gonzalez, M.Head-Gordon, E.S. Replogle, J.A. Pople, Gaussian-98, Gaussian,Inc., Pittsburgh, PA, 1998.

[48] W. Hehre, R. Ditchfield, J. Pople, J. Chem. Phys. 56 (1972) 2257.[49] A. Schafer, H. Horn, R. Ahlrichs, J. Chem. Phys. 97 (1992) 2571.[50] S. Sato, T. Kitagawa, Appl. Phys. B 59 (1994) 415.[51] S. Bell, A. Al-Olaidi, M. Hegarty, J. McGarvey, R. Hester, J.

Phys. Chem. 99 (1995) 3595.[52] R. Oakes, S. Bell, J. Phys. Chem. A 107 (2003) 10953.[53] I. Tunnell, Z. Rinkevicius, P. Salek, T. Helgaker, H. Agren, J.

Chem. Phys. 119 (2003) 11024.[54] T. Helgaker, H.J.A. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H.

Agren, T. Andersen, K.L. Bak, V. Bakken, O. Christiansen et al.,DALTON Release 2.0 (2005).

[55] W. Kohn, L.J. Sham, Phys. Rev. Lett. 52 (1984) 997.[56] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133.[57] J. Olsen, P. Jørgensen, J. Chem. Phys. 82 (1985) 3235.[58] A. Albrecht, J. Chem. Phys. 38 (1963) 354.[59] B. Minaev, S. Knuts, H. Agren, O. Vahtras, Chem. Phys. 175

(1993) 245.[60] R. Krishnan, J. Binkley, R. Seeger, J. Pople, J. Chem. Phys. 72

(1980) 650.[61] P. Macak, Y. Luo, H. Agren, Chem. Phys. Lett. 330 (2000) 447.[62] Z. Rinkevicius, L. Telyatnyk, P. Salek, O. Vahtras, H. Agren, J.

Chem. Phys. 119 (2003) 10489.[63] S. Koseki, M. Schmidt, M. Gordon, J. Phys. Chem. A 102 (1998)

10430.[64] C. Marian, U. Wahlgren, Chem. Phys. Lett. 251 (1996) 357.

[65] J. Hoard, in: K.M. Smith (Ed.), Porphyrins and Metaloporphy-rins, Elsevier, New York, 1975, p. 317.

[66] C. Weiss, H. Kobayashi, M. Gouterman, J. Mol. Spectrosc. 16(1965) 415.

[67] S. Langhoff, E. Davidson, C. Kern, J. Chem. Phys. 63 (1975) 4800.[68] J. Hasegawa, Y. Ozeki, M. Hada, H. Nakatsuji, J. Phys. Chem. B

102 (1998) 1320.[69] L. Serrano-Andres, M. Merchan, M. Rubio, B. Roos, Chem.

Phys. Lett. 295 (1998) 195.[70] D. Sundholm, Chem. Phys. Lett. 302 (1999) 480.[71] D. Sundholm, Chem. Phys. Lett. 317 (2000) 392.[72] M. Perrin, M. Gouterman, C. Perrin, J. Chem. Phys. 50 (1969)

4137.[73] B. Kim, J. Bohandy, J. Mol. Spectrosc. 73 (1978) 332.[74] A. Gradyushko, K. Solovyov, A. Shulga, A. Starukhin, Opt.

Spectrosc. 43 (1977) 37.[75] B. Minaev, Y. Wang, Y. Luo, M. Lindgren, H. Agren, Spectro-

chim. Acta A 000 (2005) 000.[76] M. El-Sayed, W. Mesman, J. Chodak, J. Chem. Phys. 57 (1972)

4061.[77] I. Chan, W. van Dorp, T. Schaafsma, J. van der Waals, Mol.

Phys. 22 (1971) 741.[78] W. van Dorp, T. Schaafsma, M. Soma, J. van der Waals, Chem.

Phys. Lett. 21 (1973) 221.[79] W. van Dorp, M. Soma, J. Kooter, J. van der Waals, Mol. Phys.

28 (1974) 1551.[80] J. Kooter, J. van der Waals, Mol. Phys. 37 (1979) 997.[81] N.V. Dijk, J. van der Waals, Mol. Phys. 38 (1979) 1211.[82] G. Zagusta, V. Kotlo, K. Solovjev, S. Shkirman, Zh. Prikl.

Spektrosk. 24 (1976) 352.[83] B.F. Minaev, A.B. Minaev, Opt. Spectrosc. 98 (2005) 243.[84] L. Gladkov, K. Solovyov, Spectrochim. Acta A 42 (1986) 1.[85] M. Tazi, P. Lagant, G. Vergoten, J. Phys. Chem. A 104 (2000) 618.[86] H. Petek, A. Bell, R. Christensen, K. Yoshihara, J. Chem. Phys.

96 (1992) 2412.[87] B. Minaev, Fizika Molecul, Naukova Dumka, Kiev 7 (1979) 34.[88] R. Englman, J. Jortner, Mol. Phys. 18 (1970) 145.[89] U. Even, J. Jortner, J. Chem. Phys. 77 (1982) 4391.[90] Y. Serebrennikov, B. Minaev, Chem. Phys. 114 (1987) 359.[91] B. Chen, A. Tulinsky, J. Am. Chem. Soc. 94 (1972) 4144.[92] R.A Schadee, Thesis, University of Leiden, The Netherlands,

1978.[93] L. Edwards, D. Dolphin, M. Gouterman, A. Adler, J. Mol.

Spectrosc. 38 (1971) 16.

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Theoretical DFT study of phosphorescence from porphyrins

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