The first application of quantum mechanics to molecules and chemistry was the study of the simplest covalent bond, namely, the hydrogen molecular cation (1) and molecule (2). For the neutral molecule, Heitler and London (2) started from the product of the ground states of the hydrogen atoms, putting the two electrons in each of the 1s orbitals. They obtained a rea-sonable binding energy and bond distance and initiated a way of thinking about the molecular wavefunction that was called the valence-bond (VB) theory, (for a recent review on VB the reader is referred to ref 3). In this model the bond electron pairs, in agreement with Lewis’s basic idea, were described by a bielec-tronic local wavefunction, essentially built from two atomic orbitals. These orbitals are either the spectroscopic valence orbitals of s, p, or d character or hybrid atomic orbitals directed along the bond axis. This model became less popular than the molecular orbital (MO) based pictures. The MO approach was more flexible and gave useful information concerning the elec-tron spectroscopy, the ionization potentials (through the Koop-mans’ theorem; ref 4), a compact zeroth-order description of the visible and ultraviolet spectra, of the spin-density distribution in free radicals, and so forth. The interpretation of the stereo-specificity of concerted reactions by Woodward and Hoffman (5) convinced chemists to assimilate and teach the MO method (for some recent examples of didactic considerations regarding the MO approach, see refs 6–9).
In the meantime quantum chemists developed intellectual and technological efforts to obtain sufficient accuracy and to give quantitative answers to the experimentalists. Most of their sophisticated codes are conceived along the MO line. The VB way of thinking, however, never died. Some efforts, such as the generalized VB method (GVB) (10, 11), introduced flexibility in the original model and improved its numerical efficiency. But the main benefit of the VB philosophy is conceptual, it makes clear the so-called non-dynamical correlation effects, it immediately goes beyond the independent particle or mean field approximation, it traces the physics of the bond (either simple or multiple) in local terms: the electrons of the bond jump from one atom to the neighboring atom, but their interatomic delocalization takes into account the energetic preferences of the atoms to remain in their lowest states. The fluctuation of the positions of the electrons is controlled in a pictorial way, which is much less transparent in the MO approach. Some ef-fects such as spin polarization in free radicals receive a clear and pedagogical interpretation. Moreover the physics of the bond
breaking and formation frequently proceeds through diradical transition states that are qualitatively correctly described by VB approaches. The qualitative differences between the neutral and ionic valence excited states is directly accessible from a study of the VB contents while it is obscure in MO descriptions. Molecu-lar and solid-state magnetism, which are receiving an increasing interest, pertain to the VB approach, the MO approach being there essentially irrelevant. We therefore believe that VB pictures must be retained in the theoretical chemistry courses and even-tually obtain a larger role.
The present article concerns a fundamental bifurcation in the VB method; should one introduce non-orthogonal orbitals and VB functions or orthogonal orbitals and VB configurations? Should one think with largely overlapping neutral and ionic components or with exclusive, orthogonal components? The choice is a matter of epistemological debate. But we shall show that the use of the orthogonal decomposition of the ground state wavefunction introduces configurations from which the valence excited states are immediately identified and characterized. The orthogonal VB (OVB) decomposition of the popular valence complete active space self-consistent field (CASSCF, see ref 12) wavefunction is straightforward and enables the physical con-tent of the correlated wavefunction to be understood in terms of the possible distributions of the valence electrons in a set of molecularly optimized atom-centered orthogonal orbitals. The coupling between these localized distributions is responsible for the electron delocalization but the energetic hierarchy between them reduces the charge fluctuation with respect to what it is in the MO picture. Hence the VB reading of the CASSCF functions shows the convergence of two historical models in a modern combination of numerical efficiency and intelligibility. All these aspects cannot be illustrated in this article, which serves as a simple introduction to the topics briefly evoked.
The Non-Orthogonal VB Approach
The VB method is usually presented starting from the consideration that for two non-interacting hydrogen atoms in which the Hamiltonian reduces to the sum of two terms with separate variables (each term is the Hamiltonian for one hydro-gen atom). In this case the ground state (GS) wavefunction of the H2 system is the product of the GS wavefunctions of the two separate atoms. If the antisymmetry principle is considered, two wavefunctions are possible,1
On the Relative Merits of Non-Orthogonal and Orthogonal Valence Bond Methods Illustrated on the Hydrogen MoleculeCelestino Angeli* and Renzo CimiragliaDipartimento di Chimica, Università di Ferrara, Via Borsari 46, I–44100 Ferrara, Italy; *[email protected]
Jean-Paul MalrieuIRSAMC, Laboratoire de Physique Quantique, Université Paul Sabatier, 118 route Narbonne, 31062 Toulouse cedex, France
Advanced Chemistry Classroom and Laboratory edited by
where 1sA and 1sB are the atomic orbitals centered on the two hydrogen atoms (hereafter indicated with A and B), S is their overlap, S = ⟨1sA|1sB ⟩, and 1 and 2 indicate electrons 1 and 2, and α and β are the spin-up and spin-down functions, respec-tively. In the following the explicit indication of the variables is avoided, with the convention that in a product of two one-electron wavefunctions, the first refers to electron 1 and the second to electron 2. The wavefunctions are indicated with 1Ψn
N and 3Ψn
N to stress that both have “neutral” (N) nature, in the sense that each H atom bears one electron, that both are built using non-orthogonal (n) atomic orbitals, and that the first is a “singlet” and the second is a “triplet”. The two wavefunctions have different spatial symmetry, 1Ψn
N being of Σ+g type and 3Ψn
N of Σ+
u type.These two wavefunctions are exact for two non-interacting
hydrogen atoms in their ground states. The basic hypothesis of the VB approach is to use them also for the molecular system where they are only approximate. For this reason the dissocia-tion energy (De) computed in this approach must be less than the exact value, owing to the variational principle that ensures that at the equilibrium geometry, where an approximate wave-function is used, the energy is greater than the exact one. In the remainder of this section and in the next one, we shall concen-
s sNn A B1 1 1 1 2 1 1 1 2
2 1
1 2 1 22
2
s sB A
S (1)
3
2
1 1 1 2 1 1 1 2
2 1
1 2
1 2 1
Nn A B B As s s s
S
22
1 2
(2)
trate on the singlet wavefunction of Σ+g symmetry, which is an
approximation for the GS. The triplet wavefunction is further considered later.
Following the formalism reported in the chapter devoted to the hydrogen molecule of Slater’s book (13), the energy for the VB structure, using atomic units, can be written as
where quantities K, J, J′, and K′ are defined in the online supple-ment and the analytical expression (13) of their dependence upon the internuclear distance R is also reported for the sake of completeness. The energy of the neutral VB singlet configura-tion (eq 3) is reported in Figure 1 (“neutral” curve). The qual-ity of the GS description associated with this wavefunction is described in most quantum chemistry books and is therefore only summarized here: the agreement with the exact GS curve is reasonable, with De = 3.156 eV and Re = 1.643 a0 (exact values are De = 4.748 eV and Re = 1.401 a0, see Table 1).
To improve the GS description a second wavefunction can be considered:
(4)
This wavefunction has an “ionic” nature (in each of the two spatial terms, one H atom bears 2 electrons and the other none). Its energy can be easily computed
(5)
while the coupling between the two VB configurations is (see the online supplement for the definition of abaa)
11 2 2
1
1 1
2
Nn
NnH
RJ J K
SK S
11 sIn A11 1 1
2 1 22
s s s
S
A B B
11 2 2
1 1
58
H
R
K S J K
In
Inˆ
11 2S
21
11
1 1
2
H
SK J S S
Rabaa
In
Nnˆ
Ene
rgy
/ Eh
Internuclear Distance / a0
ES
coupling
neutral
GS
ionic
0.5
1 2 3 4 5 6
0.6
0.7
0.8
0.9
1.0
1.1
Figure 1. Energy for the neutral and ionic VB structures and their electronic coupling in the non-orthogonal formulation using the minimal basis (1sA and 1sB). The two eigenvalues (ground state, GS, and excited state, ES) of the Hamiltonian operator projected onto this space are also reported.
(3)
(6)
Table 1. Equilibrium Geometries and Dissociation Energies of the H2 Molecule at Various Level of Calculation
Method Re/a0 E(Re)/Eh De/Eh
Minimal Basis Set VB: 1Ψn
N 1.643 ‒1.115969 0.115969 MO-SCF 1.603 ‒1.099080 0.411579 CIa 1.668 ‒1.118650 0.118650
One can note, in passing, that the energy difference between the two VB forms has a simple expression:
(7)
A better approximation to the GS wavefunction can be expressed as a linear combination of the two singlet wavefunc-tions
GS Nn
I In1 1 1c cN (8)
where the optimal coefficients cN and cI are obtained, using the variational principle, from the diagonalization of the Hamilto-nian matrix built on the {1Ψn
N , 1Ψn
I } basis. A second result of this approach is the description of the “ionic” excited state. The two potential energy curves (indicated with “GS” and “ES”), togeth-er with the neutral and ionic VB energies and their electronic coupling are shown in Figure 1. The computed spectroscopic constants for the GS obtained in this way (Table 1) show that the description is similar to the one obtained with the neutral VB configuration. One can also see in Figure 1 that the ionic VB configuration gives a poor description of the excited state.
At this point the didactic presentation of the VB approach normally stops. Our aim in this article is to analyze the nature of the VB wavefunctions more deeply and to present an alterna-tive approach. To do that, it is helpful to compute the overlap between the two VB wavefunctions:
(9)
This result shows that the neutral and ionic VB structures are not orthogonal if S is non-vanishing (that is when the two H atoms are at finite distance). The overlap between the two VB wavefunctions as a function of the internuclear distance is shown in Figure 2 and one notes that in the bond region (1 < R < 3 a0) the overlap has a sizeable value. For R approaching zero the over-lap becomes 1 (1sA and 1sB are the same function for R = 0).
The non-orthogonality of the VB wavefunctions has two consequences:
Nn
Nn
In
In1 1 1 1
5821
ˆ ˆH HJ
S
In
Nn1 1
22
1S
S
• Practical:the{1ΨnN ,
1ΨnI } basis is not orthonormal, so the
working equation for the diagonalization of the Hamilto-nian matrix is a “generalized eigenvalue equation”
where M is the metric matrix
• Philosophical: theoverlapbetween the two forms in-dicates that one cannot describe the system with either structure in an exclusive manner. In other words in this approach the neutral form has partially ionic nature and vice versa. Contrarily to the previous item, this is a relevant consequence.
The difficulty in ascribing a clear physical nature to the VB structures affects also the description of the origin of the chemi-cal bond in the H2 molecule. The standard didactic approach is based on the observation that the GS is fairly well-described by the neutral VB configuration, whose energy is analyzed in terms of the Coulomb and exchange integrals (see, for instance, ref 14, chapter 13 and ref 15, chapter 7). In this scheme, the stabiliza-tion of the molecule with respect to the free atoms is ascribed to the exchange integral. The simplest VB wavefunction can then be improved by using the ionic structure, which brings a mod-est improvement of the computed dissociation energy. Taking into consideration that the ionic structure does not change the qualitative picture given by the neutral configuration and is supposed to be marginal in the formation of the chemical bond, such an interpretative model becomes questionable if the nature of the VB components is not unequivocally defined. In the next section we present an alternative approach that allows this point to be discussed in more depth.
The Orthogonal VB Approach
In the orthogonal VB (OVB) approach, one first defines two orthogonal “atomic” orbitals (OAO) that are essentially centered on one atom, but with a tail on the other atom to be orthogonal. They can be obtained by a Löwdin orthogonaliza-tion of the two atomic (non-orthogonal) 1sA and 1sB orbitals. Among the infinite possibilities for the orthonormalization of two functions, the Löwdin procedure (16) emerges as the natural choice, given that it provides two functions as close as possible to the starting ones. Moreover the resulting OAOs are equivalent in the sense that they are transformed into each other by a reflection through the xy plane orthogonal to the internuclear axis. The use of other orthogonalization procedures does not ensure these properties: for instance, the OAOs ob-tained with the well-known Gram–Schmidt technique are not equivalent, the first being one of the two 1s AO and the other being the appropriate linear combination of both AO.
The low dimensionality of the problem in this case allows one to derive the full Löwdin procedure easily by hand. Given the didactic importance of such a procedure, we consider the detailed development worth reporting in the online supplement.
c E cH M (10)
M2
2
12
12
11
SS
SS
Neu
tral
–Ion
ic O
verla
p
Internuclear Distance / a0
1 2 3 4 5 6 7 8 9 100.0
0.2
0.4
0.6
0.8
1.0
Figure 2. Overlap of the neutral and ionic VB structures in the non-orthogonal formulation using the minimal basis (1sA and 1sB).
The OAOs are indicated in the following with a and b (A and B being the atom on which they have the largest coefficient) and their expressions are
One notes that with S << 1 one has a ≈ 1sA − 0.5S1sB and b ≈ 1sB − 0.5S1sA. As can be seen in Figure 3 the OAOs present a large oscillation at the interatomic distance of 1.67 a0. This is due to the large overlap of the 1s orbitals at this distance (S = 0.678). At a somewhat larger distance (2.1 a0) where the overlap has a smaller value, more typical of the overlap between the valence orbitals entering a normal covalent bond, the OAOs are more clearly concentrated on a given atom. With this choice, the neutral and ionic VB structures are (o indicating that they are built using OAOs)
(15)
and one can promptly verify that in this case the two wavefunc-tions are orthogonal.
By replacing in eqs 14 and 15 the expressions for the a and b OAOs (eqs 12 and 13) one gets a relation between the non-
s
s
A
B
aS S
S S
12
11
11
1
12
11
11
1
s
s
A
B
bS S
S S
12
11
11
1
12
11
11
1
N1 oo a b ba
2 2(14)
aa bbIo
2 21
orthogonal and the orthogonal wavefunctions
11
12
21 1S
SSN
oIn
Nn
(16)
oI
1 111
2
21 1S
SSI
nNn
An important comment is in order here: the use of OAOs is intrinsically ambiguous in ascribing an electron in a given OAO to a given atom and therefore in defining the nature of the VB configurations in a clear-cut way. The same ambiguity is however present also in the non-orthogonal VB approach, since the non-vanishing overlap between the two 1s atomic orbitals implies an “invasion” of the orbital of one atom in the space of the other. In the next section the problem of the definition of the nature of the VB wavefunctions is analyzed in detail by comparing them with the Σ+
u VB structures and with some excited states of H2. A clear advantage of the orthogonal approach is however evident: the orthogonality of the VB structures reported in eqs 14 and 15 means that they do not have components on each other; that is, they are not “mixed” as happens in the non-orthogonal VB approach.
The energies of the two orthogonal VB wavefunctions can be computed to be
As in the non-orthogonal approach, the difference of the energy of the two configurations is quite simple and similar to the one obtained in the non-orthogonal approach (eq 7):
In this case too, the diagonalization of Ĥ in the basis of the two VB wavefunctions gives the energy of the GS and of the excited state: these energies coincide with those obtained in the previous section (the space spanned by the two VB wavefunctions is the same for both approaches) and is reported, together with the
abaa
No
No1 1
2
2 58
2
2
11
21
1 4
1
HR
J KSS
K S S J S
S22
(18)
1 12
2 2 58
2 2
11
21
1 4
1
Io
Io
abaa
HR
J KSS
K S S S J
S
(19)
aba
Io
No1 1
2 2
2
2
1 25
16
21
2
HS
SK
J
K J SS
aa 1
1
2
2 2
S
S
(20)
JNo
No
Io
Io
11 1 1 1
582H H
Sˆ ˆ (21)
(z)
z / a0
AO
GBV AO
OAO0.6
0.4
0.2
0.0
0.23 2 1 0 1 2 3
Figure 3. Amplitude of the atomic orbitals for the localized 1sA AO, for the orthogonal “atomic” orbital a (OAO, see eq 12) and for the partially delocalized “atomic” orbital a′ (GVB AO, see the online supplement for its definition, b′ can be obtained from a′ by a reflection through the xy plane) computed on the internuclear axis z. The two hydrogen atoms are at the minimal basis CI equilibrium geometry (Re = 1.67 a0, see Table 1) and the zero of the z axis is the nuclear center.
energy of the OVB configurations and their electronic coupling, in Figure 4.
In Figures 5, 6, and 7 the comparison of the two VB ap-proaches is reported for the neutral VB form, for the ionic form, and for their coupling, respectively. As far as the GS is concerned, the most striking difference between the VB and OVB energy curves is that in the OVB case the neutral curve is dissociative and therefore the formation of the bond is due to the mixing with the bound ionic state, while the VB neutral curve is bound and close to the GS one. To proceed in the comparison of the non-orthogonal and orthogonal VB approaches, we need to discuss the Σ+
u states.
The 𝚺+u States
In the case of the Σ+u states too, one can build a neutral and
an ionic configuration. In the non-orthogonal VB approach the neutral VB configuration has been reported in eq 2, while the ionic one is
where we have added the letter u to stress that this wavefunc-tion is ungerade for the inversion symmetry operation (the same
11
2
1 1 1 1
2 1 2I, u
A A B Bs s s sn
S (22)
Ene
rgy
/ Eh
Internuclear Distance / a0
ES
ioniccoupling
neutral
GS
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1 2 3 4 5 6
Figure 4. Energy for the neutral and ionic VB structures and their electronic coupling in the orthogonal formulation using the minimal basis (1sA and 1sB). The two eigenvalues (ground state, GS, and excited state, ES) of the Hamiltonian operator projected onto this space are also reported.
Ene
rgy
/ Eh
Internuclear Distance / a0
orthogonal
non-orthogonal
GS
0.95
1.00
1.05
1.10
1.151 2 3 4 5 6
Figure 5. Energy for the neutral VB structure in the non-orthogonal and orthogonal formulations using the minimal basis (1sA and 1sB). The lowest eigenvalues (GS) of the Hamiltonian operator projected onto the VB basis is also reported (the same result is obtained for the two formulations).
Ene
rgy
/ Eh
ES
non-orthogonal
orthogonal
0.5
0.6
0.7
0.8
0.9
1.0
0.4
1 2 3 4 5 6
Internuclear Distance / a0
Figure 6. Energy for the ionic VB structure in the non-orthogonal and orthogonal formulations using the minimal basis (1sA and 1sB). The highest eigenvalues (ES) of the Hamiltonian operator projected onto the VB basis is also reported (the same result is obtained for the two formulations).
Ene
rgy
/ Eh
Internuclear Distance / a0
non-orthogonal
orthogonal
0.0
0.2
0.4
0.6
0.8
1.0
1.21 2 3 4 5 6 7 8 9
Figure 7. Comparison of the neutral–ionic electronic coupling in the non-orthogonal and orthogonal formulations using the minimal basis (1sA and 1sB).
must be done for the wavefunction in eq 2). Firstly, one notes that in this case the spatial parts of the two wavefunctions have a different symmetry for the interchange of the two electrons, being symmetric in the ionic form, and antisymmetric in the neutral one. Therefore 1Ψn
I,u is a singlet and 3ΨnN,u a triplet. The
two wavefunctions are trivially orthogonal and non-interacting (they have different spin multiplicity).
Following the formalism introduced in the previous sec-tions, in the OVB approach one has
3NN, uo ab ba
2 2(23)
1
2I, uo a a bb
22 (24)
For these states the two VB approaches are strictly equivalent, as can be easily verified. This means that in both approaches the two wavefunctions have an unequivocally defined nature. For these states we therefore drop the superscripts o and n. The energies associated with the two VB configurations are
N, u N, u
3 3
11
H
R2 2
1 2KS J J K
S
(25)
11 2 2
1
1 1
58
H
R
KS J KI, u I, u
ˆ
SS 2
(26)
Let us now compute the energy difference between the Σ+g and
the Σ+g OVB configurations of the same nature. Using eqs 18, 19,
25, and 26 these differences are seen to be coinciding:
The same results can be obtained using Slater’s rules for the energy of a determinant and for the interaction of two deter-minants
where Kab = ⟨ab|1∙r12|ba⟩ is the exchange integral. It can be computed replacing the a and b OAOs in this equation (see the online supplement for its expression). This interatomic exchange integral remains small whatever the interatomic distance, as will be shown later on.
Equation 27 also indicates that the two Σ+u VB configura-
tions have the same energy difference as the two Σ+g (orthogonal)
configurations:
Comparison of the Non-Orthogonal and Orthogonal VB Approaches
The potential energy profile for the GS (17) and for the 3Σ+
u, 1Σ+u, 1Σ+
g excited states (18–20) obtained by Kolos and Wol-nievicz from accurate ab-initio calculations (they can be consid-ered “exact” curves) are shown in Figure 8. It is well known that the two lowest states (the GS 1Σ+
g and the 3Σ+u) both have valence
character; that is, they are well-described by the minimal valence basis set (1sA and 1sB): at long distances they are fully neutral, while at short distances one of them (1Σ+
g) contains an increasing ionic component while the triplet remains neutral. On the con-trary the two highest states (1Σ+
u and 1Σ+g) have valence character
at short internuclear distances (having essentially ionic nature), while they dissociate to a limit where one H atom is in its ground state and the other is in the 2s excited state.
E E
SS S S J
2 22 21
1
58
4 2abaa
g u N g u I
KK(27)
E E
abr
ba Kab
g u N g u I
21
212
(28)
3NN, u N, u I, u I,u
ˆ ˆH HJ
S3 1 1
5821
(29)
1g
1u
3u
1g
Ene
rgy
/ Eh
Internuclear Distance / a0
1.2
0.6
0.7
0.8
0.9
1.0
1.1
1 2 3 4 5 6
Figure 8. Energy of the 1Σ+g ground state (19) and of the 1Σ+
u and 1Σ+
g excited states (20 and 22, respectively) obtained with accurate calculations.
Ene
rgy
/ Eh
Internuclear Distance / a0
OVB
0.5
0.6
0.7
0.8
0.9
1.01 2 3 4 5 6
OVB 1u
1u
OVB 1g
3u
OVB 3u
1g
Figure 9. Energy of the 3Σ+u (neutral) and 1Σ+
u (ionic) excited states (19 and 20, respectively) obtained with accurate calculations and with the orthogonal VB approach using the minimal basis (1sA and 1sB). The energy of the first two 1Σ+
g VB structures (neutral and ionic) computed in the orthogonal VB approach with the same basis are also reported for comparison.
In the previous section it has been shown that the two 1Σ+u
VB configurations have a well-defined nature (neutral and ionic) and it is therefore worth comparing the energy curves of these VB configurations with those reported in Figure 8. This com-parison is presented for the OVB approach in Figure 9. For the neutral OVB curves (1Σ+
g and 3Σ+u) the agreement with the exact
3Σ+u curve is excellent. The singlet and triplet neutral OVB curves
are very close and only slightly higher in energy than the exact triplet curve. Remembering that they are split by the quantity 2Kab, the small magnitude of this interatomic exchange integral is noted. As far as the non-orthogonal VB is concerned, recall that for the triplet state the orthogonal and non-orthogonal VB configurations coincide, while the neutral non-orthogonal VB singlet curve (see Figure 1) shows a shape that is even qualita-tively different from the exact 3Σ+
u curve.In the case of the ionic OVB curves (1Σ+
g and 1Σ+u) the
comparison with the 1Σ+u exact curve is meaningful only for
short internuclear distances (where it is ionic in nature). The agreement in this case is satisfactory apart from an energy shift. This behavior is due to the inadequacy of the 1s hydrogen atomic orbitals to describe the H‒ anion. Moreover the presence of the two electrons on the same atom involves a relevant dynamical correlation energy, which cannot be described with a single OVB configuration. Both effects make the OVB energies of the ionic wavefunctions too high. Again with the non-orthogonal VB approach the ionic 1Σ+
g state is poorly described (see Figure 1). Remember that for the 1Σ+
u state orthogonal and non-orthogonal VB give the same result.
The considerations reported here show the validity of the orthogonal approach and its superiority (with respect to the non-orthogonal one) in the physical interpretation of the nature of the VB forms since the neutral singlet and triplet con-figurations on one hand, the ionic gerade and ungerade singlet configurations on the other hand, have similar behaviors in the orthogonal approach, as one may desire.
The Optimized Two-Electrons-in-Two-Orbitals Descrip-tion, the VB Reading of a CASSCF Wavefunction
The results reported in the previous sections regard the VB approach and are obtained starting from the minimal basis set, spanned by the two 1s H atomic orbitals. In this section we discuss the relation between the VB and the MO approaches and their extension for general basis sets (for a more exhaustive discussion, see ref 21). This relation allows one to define a simple way to obtain optimized OVB configurations and energies from standard MO packages.
The previously considered atomic orbitals 1sA and 1sB (or a and b, see eqs 12 and 13) define a gerade σg and an ungerade σu MO:
These MOs enable one to define two Σ+g determinants, namely,
gA Bs s
Sa1 1
2 1
b2 (30)
Sa b1 1
2 1 2s s
uA B
(31)
2g g g g g (32)
which represents the usual single-determinantal description of the ground state and a double-excited determinant of much higher energy:
Here a line over an orbital indicates that the orbital is multiplied by the β spin function. Φg and Φg* are in-phase and out-of-phase combinations of the previously introduced neutral and ionic VB configurations:
g
Io
No
In
2
12
11
1 1
21 1S
S Nn
(34)
g
Io
No
In
Nn
*1 1
21 1
2
12
11
SS
(35)
The GS wavefunction is a linear combination of Φg and Φg* ,
where λ, μ > 0. Therefore this treatment is totally equivalent to the use of the neutral and ionic VB or OVB configurations given that in all cases the vector space used to expand the wavefunc-tion is the same. The analytical expression for the energy of the ||σg σg
−|| and ||σu σu−|| determinants and for their electronic cou-
pling as a function of the internuclear distance R are reported in the online supplement. These relations establish the connections between the MO-CI and the VB approaches.
The Σ+u excited states also receive an MO expression
g (ionic) OVB configurations built up using OAO orbitals a and b defined by eqs 43 and 44, where σg and σu are the active MOs of a 2e‒/2MO CASSCF calculation of the GS using the cc-pVQZ basis (24) (see text). The “exact” 1Σ+
g GS energy is also reported. For the sake of comparison the 3Σ+
u (neutral) and 1Σ+u (ionic) excited states
(19 and 20, respectively) obtained with accurate calculations are also shown.
The crucial benefit of the VB way of thinking is illustrated on this elementary problem. The MO approach associates the triplet and singlet Σ+
u excited state as having the same space con-figuration and differing only by the spin part of their wavefunc-tion. The VB transcription of these two functions shows their intrinsically different physical nature. In the triplet state each atom keeps one electron without any fluctuation of the atomic charge, while in the singlet state the two electrons jump simulta-neously from one atom to the other, that is, with the maximum charge fluctuation. The difference in the physical content of the excited states is an important feature, of major consequences, and understanding this difference requires proceeding through a VB-type analysis.
One may now release the minimal basis set constraint and consider larger basis sets of AO, {χr}, centered on atoms A and B (or elsewhere). One may define the σg and σu MOs on this basis
g cr rr
(41)
u cr rr
* (42)
and minimize the energy of a two-electrons in two-orbitals wavefunction (keeping the form of eq 36) with respect to both the λ∙μ ratio and the MO coefficients cr and cr*. This is precisely the so-called (2e‒/2MO) complete active space self-consistent field (CASSCF) description of H2, which defines the best mini-mal valence description of this simple bond (within a given basis set), and the best definition of the symmetry-adapted valence MOs σg and σu, from which one immediately obtains the best valence orthogonal localized orbitals a and b
(43)
(44)
ga u
2
g u
2b
and an OVB transcription of the CASSCF wavefunction through the definition of the neutral and ionic orthogonal configuration and the application of eqs 34 and 35:
This approach is relevant in the context of the VB descrip-tion because it allows for the definition of optimized OVB configurations through a VB reading of an optimized MO wave-function, which can be easily obtained with standard quantum chemistry packages. Moreover, the optimized σg and σu MOs enable one to define the 3Σ+
u and 1Σ+u spin-adapted excited con-
figurations through eqs 39 and 40 or, equivalently, using eqs 23 and 24 with the a and b atom-centered orbitals defined in eqs 43 and 44. Figure 10 shows the potential energy curves of • theCASSCFGS • theneutralandionicOVB1Σ+
g configurations • the3Σ+
u and 1Σ+u OVB configurations
The OVB configurations are built up using the OAO orbitals a and b defined by eqs 43 and 44, where σg and σu are the active MOs of a 2e−∙2MO CASSCF calculation of the GS using quite a large basis set (termed “cc-pVQZ”; ref 22). Also in this case the ionic OVB curves dissociate at too high an energy. This is due to the fact that the σg and σu MOs (and consequently the a and b OAOs) are optimized to describe the GS, which at long internuclear distances has pure neutral nature. The a and b OAO are therefore not well-suited for a H− anion. In general the description of the VB structures obtained with this approach closely resembles those obtained in the minimal basis set, but the increased flexibility of the basis set leads to a closer agreement of the GS wavefunction with the exact results, as indicated by the values reported in Table 1.
Conclusions
This article has discussed the relative advantages of using either non-orthogonal pure atomic orbitals and the traditional non-orthogonal valence bond configurations or orthogonal atom-centered orbitals and an orthogonal VB formalism. The first advantage of the former is that it is closer to intuition since one tackles the molecular problem starting from the orbitals of the atoms in their ground states. The second advantage is that the neutral VB configuration brings a major contribution to the binding energy, the ionic configuration contribution being non-negligible but smaller. The first advantage of the orthogonal formulation is conceptual. The neutral singlet configuration is really neutral and close to the intrinsically pure neutral triplet state, while the ionic configuration of gerade symmetry is close to the intrinsically pure ionic singlet state of ungerade symmetry. The bond building appears to be due to the mixing between the ionic and neutral configurations, the ionic configuration allowing the electrons to jump from one atom to the other and the atomic charge to fluctuate. Of course it also presents the advantage of thinking in terms of non-overlapping, exclusive situations, which does not hold in the non-orthogonal formalism. Another advantage of the orthogonal formulation is its generality and
flexibility. The modern CASSCF codes variationally define the best valence molecular orbitals, either bonding or antibonding. These orbitals are the best valence MOs, from an energetic point of view. From them a localizing unitary transformation defines orthogonal atom-centered orbitals, which are in a one-to-one correspondence with the valence atomic orbitals of the free atoms but which have been optimized in the molecular field. These procedures are general and valid for extended basis sets, but the valence space is as simple as the usual minimal basis set valence space and may perfectly receive a valence bond interpretation in terms of neutral, singly, or multiply ionic configurations. In this basis the interpretation of the non-dynamical (i.e., internal to the valence space) electronic correlation becomes straightforward, as being a compromise between the reluctance of the atoms to leave their low energy states and the interatomic delocalization of the electrons, which allows them to move in broader domains of the space. The correlation in the valence space partly restores the preferences of the atoms that had been denied by the inde-pendent particle or mean-field single determinantal description. As shown elsewhere (21) even the dynamical correlation is more easily understood when starting from the OVB decomposition of the CASSCF wavefunction. This represents a point of conver-gence of the two main streams of quantum chemistry, namely, the dominant MO formulation and the still living VB picture.
The results presented in this article are partly used in the physical chemistry course for the graduate students of the first year of the master in physics at the University of Ferrara. The authors believe that it could be an extension of the standard presentation of the VB and MO methods in the quantum chem-istry courses or a part of more advanced courses in quantum chemistrycurriculaorPhDintheoreticalchemistry.
Acknowledgments
The authors wish to thank the reviewers for careful reading the original manuscript and for their comprehensive suggestions. This work has been financed by the University of Ferrara (nano & nanoproject)andbytheItalianMIURthrough itsPRINfunds.The “LaboratoiredePhysiqueQuantique” is “UnitéMixte de Recherche” (UMR5626) of the CNRS.
Note
1. The form in the text is a shorthand notation for the full expressions of the three spin components of the triplet function, which for eq 2 are
1
1 1 1 2 1
3
s s
Nn
A B
Sz
ss sB A1 1 2
2 11 2
2S
s s
Nn A B0
1 1 1 23 Sz1 1 1 2
2 1
1 2 1 22
2
s sB A
S
3Nn Sz 1
1 1 1 2 1 1 1 2
2 11 2
2
s s s sA B B A
S
The shorthand notation is also used in eqs 23 and 39.
Literature Cited
1. Burrau, O. Kgl. Danske Vid. Selskal. Math-fys. 1927, 7, 14. 2. Heitler W.; London, F. Zeits. f. Physik. 1927, 44, 455. 3. Shaik,S.;Hiberty,P.C.InReviews in Computational Chemistry, Vol. 20;
Lipkowitz, K. B., Larter, R., Cundary, T. R., Eds.; Wiley-VCH: New York, 2004; p 1.
4. Angeli, C. J. Chem. Educ. 1998, 75, 1494–1497. 5. Woodward, R. B.; Hoffmann, R. J. Am. Chem. Soc. 1965, 87, 395–
397. 6. Magnasco, V. J. Chem. Educ. 2004, 81, 427–435. 7. Magnasco, V. J. Chem. Educ. 2005, 82, 1311. 8. Harrison,J.F.;Lawson,D.B.J. Chem. Educ. 2005, 82, 1205–1209. 9. Matito,E.;Duran,M.;Solà,M.J. Chem. Educ. 2006, 83, 1243–
Supplement Analytical expressions of the relevant integrals Löwdin orthogonalization of the 1sA and 1sb orbitals The intermediate results in the derviation of the energy expressions The relation between the non-orthogonal and the orthogonal VB
wavefunctions and energies The energy expression in the MO approach The generalized valence bond method
