Visualizations of transition dipoles, charge transfer, and electron-holecoherence on electronic state transitions between excited statesfor two-photon absorption
Mengtao Sun,1,a� Jianing Chen,1,2 and Hongxing Xu1,3
1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,Chinese Academy of Sciences, P.O. Box 603-146, Beijing, 100080, People’s Republic of China2School of Physics and Optoelectronic Technology, Dalian University of Technology,Dalian 116024, People’s Republic of China3Division of Solid State Physics, Lund University, Lund 22100, Sweden
�Received 24 October 2007; accepted 4 December 2007; published online 12 February 2008�
Molecular two-photon absorption1 �TPA� has gained in-terest over recent years owing to its applications in variousfields, including spectroscopy, optical data storage,2 opticalpower limitation,3 microfabrication,4 and three-dimensionalimaging.5 Various design strategies have been employed tosynthesize organic molecules with large two-photon absorp-tion cross sections, such as donor-bridge-acceptor �D-�-A�and donor-bridge-donor �D-�-D� derivatives, donor-acceptor-donor �D-A-D�, acceptor-donor-acceptor �A-D-A�and D-�-A-�-D, and star structure.6–16
TPA is a nonlinear absorption process wherein two pho-tons are absorbed simultaneously. Characteristic features areadherence to even-parity selection rules and quadratic inten-sity dependence, while one-photon absorption processestypically conform to odd-parity selection rules and linear in-tensity dependence. The two-photon absorption coefficient�2 is proportional to the imaginary part of the third-ordersusceptibility tensor.17 At the molecular level, the macro-scopic �3 can be replaced by the third-order molecular non-linearity. Thus, the molecular two-photon absorption crosssection can be characterized by the imaginary part of themolecular third-order nonlinear polarizability, defined at anabsorption frequency of � as �2���� Im ��−��.18,19 It canalso be determined by computing the TPA transition matrixelements between the initial and finial states based on the
sum-over-state formalisms.20–24 With the recent advances inthe theory and application of DFT to time dependent proper-ties and quadratic response functions, such applications havebecome a practical proposition in two-photonabsorption.25–29
Visualization tools and techniques have become veryuseful tools to understand the excited state properties.30 Wehave developed visual methods �two dimensional �2D� siteand three dimensional �3D� cube representations�31–33 forone-photon absorption �OPA�. The 2D site representation re-veals visually the electron-hole coherence during electronictransitions from the ground state. The 3D cube representa-tions show visually the orientation of transition dipole mo-ment by transition density, and the charge redistribution onthe excited states by charge difference density.
The purpose of this study is to further develop the 2Dsite and 3D cube representations for TPA on electronic tran-sitions between excited states. With which the orientation oftransition dipole moment, the charge redistribution, and theelectron-hole coherence for TPA of molecules �with nonlin-ear properties� on electronic transitions between excitedstates were visualized. It should be noted that our 2D and 3Drepresentations are specifically designed to be used withsum-over-state formalisms where all states are constructed.In this paper, the D-�-A-�-D-type 2,1,3-benzothiadiazoles�BTD� �see Fig. 1� was investigated to examine our newdeveloped methods, as an example, since D-�-A-�-D-typeBTD have shown the excellent nonlinear optical excited stateproperties.14
a�Author to whom correspondence should be addressed. Electronic mail:[email protected].
THE JOURNAL OF CHEMICAL PHYSICS 128, 064106 �2008�
The OPA properties of charge transfer and electron-holecoherence on the electronic transition from the ground statewere detailed described in Ref. 31–33. In this paper, we fo-cus on the TPA properties on the electronic transitions amongexcited states with the visualized 2D site and 3D cube rep-resentations.
The TPA cross section, directly comparable with experi-mental measurement, is defined as20–22
�tp =4�2a0
5�
15c0
�2g���� f
tp, �1�
where �0 is the Bohr radius, c0 is the speed of light, � is thefine structure constant, � is the photon energy of the incidentlight, and g��� denotes the spectral line profile, here it isassumed to be a function. � f is lifetime broadening of thefinal state.9 The TPA probability �tp in Eq. �1�� of moleculesexcited by a linearly polarized monochromatic beam can becalculated by
tp = 6�Sxx + Syy + Szz�2
+ 8�Sxy2 + Sxz
2 + Syz2 − SxxSyy − SxxSzz − SyySzz� , �2�
where S� a is two-photon matrix element for the two-photonresonant absorption of identical energy. S� can be calculatedwith sum-over-state formulas,
S� = �j� �f ����j�j���g
� j − � f/2 − i� f+
�f ���j�j����g� j − � f/2 − i� f
, �3�
where �g and �f denote the ground state and final states,respectively, �j means all the states, � j is the excited energyof excited states, and � is electronic dipole moment. If wetake into account only the products of two-photon transitionmatrix elements with the same intermediates state �j= j�� andneglect the products having different intermediate states �j� j��, tp is expected as22,24
tp = 8 �j�g
j�f
��f ���j�2��j���g�2
�� j − � f/2�2 + � f2 �1 + 2 cos2 � j�
+ 8� � fg�2��f ���g�2
�� f/2�2 + � f2 �1 + 2 cos2 �� , �4�
where � j is the angle between the vector �f ���j and �j���g, � fg= �f ���f− �g���g is the difference between the perma-nent dipole moments of the excited and ground states, andthe � is the angle between the two vectors � fg and �f ���g.In the derivation of Eq. �4�, the two-photon resonance con-dition,� j =� f /2, is used. The first and the second terms arecalled the three state term and the dipole �two states� terms,respectively. For the central symmetrical molecules, the di-pole terms is vanishingly small, because of � fg= �f ���f��g���g�0.23 � can be obtained by a finite field methodon the excitation energy. The transition energy dependenceon the static electric field F can be expressed as34
Eexc�F� = Eexc�0� − �F − 12 �F2, �5�
where Eexc�0� is the excitation energy at zero field, and � isthe change in polarizability.
B. 3D cube representation for TPA
The singlet Bu �OPA allowed� and Ag �TPA allowed�excited states �see Fig. 2� can be written as, respectively,
�� j�Bu� = ���unocc
o�occ
Cj�o�au ← bg�a�+�au�ao�bg���g�Ag�
�6�
and
��k�Ag�� = �m�unocc
n�occ
�Ckmn�bg ← bg�am+ �bg�an�bg�
� Ckmn�au ← au�am+ �au�an�au����g�Ag� . �7�
In Eq. �7�, ��k�Ag�− and ��k�Ag�+ stand for the second andthe higher Ag �two-photon allowed� excited states,respectively.23 So, the electronic state dipolar transitionsfrom �� j�Bu� to ��k�Ag� can be written as
FIG. 1. �Color online� �a� The chemical structure of D-�-A-�-D-type 2,1,3-benzothiadiazoles. �b� It is classified for A–J units for Figs. 6 and 9.
FIG. 2. �Color online� The schematic representation of the electronic con-figurations from Ref. 23.
064106-2 Sun, Chen, and Xu J. Chem. Phys. 128, 064106 �2008�
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���k�Ag����� j�Bu� = �m,��unocc
Ckmn�bg ← bg�
�Cj�o�au ← bg��m←u�bg ← au�
� �n,o�occ
Ckmn�au ← au�
�Cj�o�au ← bg��n←o�au ← bg� .
�8�
The relation between transition dipole moment and transitiondensity is35
�k,j = e� r�k,j�r�d3r , �9�
and the transition density contains information about the spa-tial location of the excitation. Thus, the transition density onelectronic transition between excited states for TPA can bewritten as
�� = �m,��unocc
Ckmn�bg ← bg�Cj�o�au ← bg��m�u
� �n,o�occ
Ckmn�au ← au�Cj�o�au ← bg��n�o, �10�
where the �m��u� and �n��o� stand for the unoccupied andoccupied molecular orbitals �cubes�. The charge differencedensity on the electronic transition from the j excited state tothe k excited states for TPA can be written as
��,k,j = ��,m,u � ��,n,o, �11�
where ��,m,u and ��,n,o are
��,m,u = �m,m�,��unocc
Cj�o�au ← bg�Ckmn�bg ← bg�
�Ckm�n�bg ← bg��m�m�
− �m,�,u�,�unocc
Cj�o�au ← bg�
�Cj��o�au ← bg�Ckmn�bg ← bg��u�u� �12�
and
�n,o = �n,n�o�occ
Ckmn�au ← au�Ckmn��au ← au�
�Cj�o�au ← bg��n�n�
− �n,o,o��occ
Ckmn�au ← au�Cj�o�au ← bg�
�Cj�o��au ← bg��o�o�. �13�
C. 2D site representation
The electron-hole coherence on the electronic transitionfrom the j excited state to the k excited states for TPA can beanalyzed by the transition density matrix in a site represen-tation. We characterize the nature of the various excitedstates by calculating the electron-hole two-particle wavefunctions at the INDO/SCI level. This leads to a two-dimensional grid running over all the carbon sites along each
axis. Each data point �x ,y� gives the probability ���x ,y��2 offinding one charged particle on site x and the second one onsite y. The probability amplitudes �� j�Bu ,q ,r� �OPA al-lowed� and ��k�Ag ,q ,r�� ���k�Ag ,q ,r�− and ��k�Ag ,q ,r�+
stand for the second and the higher Ag two-photon allowedexcited states, respectively� have one charged particle inatomic orbital q �which can be a 2s, 2px, 2py, or 2pz orbital�on site x and the second in atomic orbital r on site y given by
�� j�Bu,q,r� = �l
ClCI�au ← bg�Cq
LCAO�au,h+�CrLCAO�bg,e−�
�14�
and
��k�Ag,q,r��
= �l
ClCI�bg ← bg�Cq
LCAO�bg,h+�CrLCAO�bg,e−�
� �l
ClCI�au ← au�Cq
LCAO�au,h+�CrLCAO�au,e−� , �15�
where CqLCAO�h+� and Cr
LCAO�e−� are the corresponding linearcombination of atomic orbitals coefficients in the occupiedand unoccupied molecular orbitals involved in the lth con-figuration. The Cl
CI are the associated configuration-interaction �CI� expansion coefficient. The orthonormaliza-tion of the basis set imposed by the ZDO approximationensures that
Pk←j�Ag ← Bu� = ����k�Ag,x,y��� j�Bu,x,y��2
= �q�x
�r�y
����k�Ag,q,r��� j�Bu,q,r��2.
�16�
D. Quantum chemical calculations
All the quantum chemical calculations were performedwith GAUSSIAN 03 suite.36 The ground state geometry wasoptimized with PM3 method.37 The excited state propertieswere done with ZINDO method.38 The ZINDO calculationconsists of a Hartree-Fock calculation on a parameterizedSTO basis and a subsequent diagonalization of the single-configuration-interaction �single-CI� Hamiltonian. All inte-grals are parameterized except for the exchange integrals in
TABLE I. Transition energies �eV, nm� and their oscillator strengths �f� andCI coefficients for the orbital transitions for OPA.
064106-3 Visualizations of transition dipoles J. Chem. Phys. 128, 064106 �2008�
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the two-electron interaction among orbitals belonging to dif-ferent centers. This parameterization implicitly takes into ac-count a part of the electron correction, which is not includedin ab initio single-CI calculation.
The parameters for simulating the OPA and TPA spectraare the transition energies, the transition moments betweenthe ground state and excited states, and the transition mo-ments between the different excited states �with keywordZINDO=All TransitionDensities�, which were listed inTables I–III. For simulating the OPA spectrum, the lowest 20excited states were taken into account. This choice is enoughsince the energy of 20th excited state is up to 250 nm. Forsimulating the TPA spectrum, the lowest seven excited stateswere taken accounted, since the transition dipole moments�excitation from the ground state� for higher excited statesare small.
III. RESULTS AND DISCUSSION
A. One-photon absorption
Before discussing the TPA properties, we firstly discussthe OPA properties obtained from the calculation. From thesimulated OPA spectrum �the damping constant is set to3000 cm−1�, there are three strong electronic states below4.133 eV �300 nm� in �see Fig. 3�a��. The orbital transitionsfor the first ten excited states were listed in Table I, whichwere also used in the calculations for TPA. The observedabsorption peak is 426 nm, while the calculated S1 is at456 nm. The shifts of 30 nm between experimental and the-oretical result from the ignorance of solvent effect. The cal-culated transition dipole moments were listed in Table II.The orientations of three transition dipole moments wereshown in Fig. 4, which is consistent with the calculated re-sults in Table II. Furthermore, the transition dipole momentfor the neutral species �see Fig. 4� is constitute with a seriesof small transition dipoles–one permonomeric unit.32,35 Thecharge difference density allows us to follow the change of
the static charge distribution upon excitation. From Fig. 5, allof these three excited states with strong absorption are thecharge transfer excited states, and all the electrons transferfrom the donor or bridge to the acceptor, but with differentcharge transfer capabilities. For example, for the first excitedstates, the electrons on the BTD �the acceptor� transferredonly from the nearest bridges, while the electrons on theBTD transferred from all the four bridge units �except thatfrom the donor units� for S5, which means that the electronoscillation of S5 is stronger than that of S1. From Fig. 5, S7 isof the strongest charge transfer capability �or the largest sizeof electron oscillation� among these three excited states, andthe donor units also provide electrons. The charge transferproperties of these three excited states were also studied with2D site representation �see Fig. 6�, with which the electron-hole coherence on the excitation can also be clearly ob-served. The exciton sizes �largest off-diagonal extent of thenonzero matrix area�39 for these excited states are also con-sistent with the results of charge difference densities.
B. Two-photon absorption
The simulated TPA spectrum is shown in Fig. 3�b�,where the damping constant is set to 0.1 eV, which is thecommonly used value,9 and corresponds to lifetimes of a fewfemtoseconds. In the simulation, the transition dipole mo-ments between excited states were listed in Table III. From
TABLE II. The transition dipole moments �a.u.� for three strong absorptionpeaks for OPA, and the permanent static dipole moment on the ground state.
TABLE III. The transition dipole moments �a.u.� between excited states forthree strong absorption peaks of TPA.
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the TPA spectrum, the absorption strengths of lowest twosinglet excited states are very weak. For the first singlet ex-cited state of TPA, the only contribution is from the dipolarterm in Eq. �4�. The �10 is very small, since the electronicstatic permanent dipole moments at the ground and the ex-cited states are very small, respectively, becauseD-�-A-�-D-type BTD is the central symmetricalmolecule.23 So, the absorption of the first excited state isvery weak. The second singlet excited state of TPA is mainly
contributed from the three state term in Eq. �4�. The transi-tion dipole moment �1���0 is large �see data in Table II�,while the transition dipole moment �2���1 is vanishinglysmall �see data in Table III�. So, tp�S2�= ���2���1�2��1���0�2 / ��1−�2 /2�2+�2��1+2 cos2 �1� isvery small. The reason why �2���1 is small can be inter-preted with Eq. �10�. In Eq. �10�, the transition dipole mo-ment for S2 is �−�S2�, and the first term is almost equal to thesecond term in �−�S2�, according to the alternancy symmetry�hole-particle symmetry�.40–44 In case of neutral alternant�-conjugated hydrocarbon systems, this symmetry strictlyholds in the framework of �-electron theory such as simpleHückel or Pariser-Parr-Pople-CIS methods, which gives riseto the simple rules, �−�S2�=0, due to a selection rule calledpseudoparity selection rule.42 For the ZINDO calculation, thecalculated orientation of transition density for S2←1 is from Fto E unit of BTD �see Fig. 7�, which is consistent with theresult in Table III. Since the orientation of transition densityis perpendicular to the molecular axis from F to E unit ofBTD, the electrons should transfer from F to E unit, whichcan be seen from the charge difference density in Fig. 8. Thecontour plot of transition density matrix of S2 �see Fig. 9�supports the above conclusion, since the electron-hole co-heres between E and F units of BTD. For S2←1←0 in TPA,the first step �S1←0� is that some electrons transfer from E toF unit, while for the second step �S2←1�, the transferred elec-trons on F unit move back to E unit. So, the second stepcontraflows the orientation of electron transfer in the firststep for TPA.
From Fig. 3�b�, the third excited state is the strong ab-sorption in TPA, and there are two channels for this excitedstate, tp�S3←S2←S0� and tp�S3←S1←S0�. �2←0 is small,though �3←2 is large �see data in the Table III�, so tp�S3
←S2←S0� should be small. Since �1←0 and �3←1 are large�see data in Tables II and III�, the strong absorption of S3 forTPA is mostly contributed from tp�S3←S1←S0�. For OPA,the transition density, charge difference density, and electron-hole coherence for the electronic state transition from theground state to the first excited state have been visualized in
FIG. 3. �Color online� The �a� one-photon absorption and �b� two-photonabsorption spectra.
FIG. 4. �Color online� The transition densities for OPA on the electronicstate transition from the ground state. The green and red colors stand forhole and electron, respectively. The isovalue is 4�10−4 in a.u.
FIG. 5. �Color online� The charge difference densities for OPA on the elec-tronic state transition from the ground state. The green and red colors standfor hole and electron, respectively. The isovalue is 4�10−4 in a.u.
064106-5 Visualizations of transition dipoles J. Chem. Phys. 128, 064106 �2008�
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Figs. 4–6. For TPA, it is of great significant to visualizethem, since they can promote deeper understanding to theoptical and electronic properties for TPA. The transition den-sity of �3←1 can be seen from Fig. 7. By comparing the �3←1
and �1←0, one can see that they are of the opposite orienta-tion, and �3←1 should be larger than �1←0, since the elec-trons and holes for �3←1 are localized at both ends of themolecule, while the electrons and holes for �1←0 are local-ized at center of the molecule. �3←1��1←0 can be sup-
ported by the calculated results in Tables II and III. Theorientation and strength of �3←1 derived from �3←1 are con-sistent with the calculated results in Table III. Because�3←1��1←0, the charge transfer capability for S3←1 is largerthan that of S1←0, which can be supported by the chargedifference densities listed in Fig. 8. For S1←0, the electronstransferred from the nearest bridges to BTD, while for S3←1
the electrons transferred from the outer bridges to the centralbridge and BTD. From Fig. 9, the exciton size of S3←1 islarger than that of S1←0. Furthermore, from Fig. 8, the elec-trons on BTD for S3←1 are all transferred from the outerbridge, since there are no electron-hole pairs on the BTD,while for S1←0 there is much localized excitation on BTD,since there are much electron-hole pairs on the BTD.
FIG. 6. �Color online� Contour plots of transition density matrix for OPA.The color bar is shown at the right of last figure, and the absolute values ofmatrix elements are scaled to a maximum value of 1.0.
FIG. 7. �Color online� The transition densities for TPA on the electronicstate transition between excited states. The green and red colors stand forhole and electron, respectively. The isovalue is 4�10−4 in a.u.
FIG. 8. �Color online� The charge difference densities for TPA on the elec-tronic state transition between excited states. The green and red colors standfor hole and electron, respectively. The isovalue is 4�10−4 in a.u.
064106-6 Sun, Chen, and Xu J. Chem. Phys. 128, 064106 �2008�
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IV. CONCLUSION
Based on our previous development on the visualizationmethods of 2D site and 3D cube representations for OPA onthe electronic transition from the ground state, the visualiza-tion methods of 2D site and 3D cube representations for TPAon the electronic transitions between excited states have beensuccessfully further developed, which promote deeper under-standing to the optical and electronic properties for TPA.
V. PROSPECTIVE
The 2D and 3D representations in this paper are specifi-cally designed to be used with sum-over-state formalismswhere all states are constructed. Equation �3� is often calledtransition polarizability45–47 or �first order� transitionhyperpolarizability,48,49 which is directly connected with theTPA probability. The visualizations of the transition polariz-ability or �first order� transition hyperpolarizability with 2Dand 3D representations are on the way.
The advantage of response theory25–29 is that intermedi-ate states are never explicitly constructed, so one does nothave to worry about convergence issues with respect to num-ber of intermediate states. We are also trying to implementour 2D and 3D representations in conjunction with quadraticresponse function techniques in the DALTON program.50
ACKNOWLEDGMENTS
M. T. Sun thanks Dr. Koji Ohta �Photonics ResearchInstitute, National Institute of Advanced Industrial Scienceand Technology, Japan� for the calculations about the transi-tion dipole moments between excited states in TPA. Thiswork was supported by the National Natural Science Foun-dation of China �Grant Nos. 10625418 and 20703064�, theSino-Swedish Collaborations about nanophotonics and nano-electronics �No. 2006DFB02020�, the National Basic Re-search Project of China �Grant Nos. 2007CB936801 and2007CB936804�, and the “Bairen” projects of CAS.
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FIG. 9. �Color online� Contour plots of transition density matrix for TPA,which are S1,0 and S3,1. The color bar is shown at the right of last figure, andabsolute values of matrix elements are scaled to a maximum value of 1.0.
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