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A Hamiltonian matrix correction based density functional valence bond method Chen Zhou, Yang Zhang, Xiping Gong, Fuming Ying, Peifeng Su, and Wei Wu J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b01144 • Publication Date (Web): 19 Dec 2016 Downloaded from http://pubs.acs.org on December 21, 2016

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Journal of Chemical Theory and Computation

A Hamiltonian matrix correction based density functional valence bond method Chen Zhou,† Yang Zhang,† Xiping Gong,† Fuming Ying,† Peifeng Su,* ‡ Wei Wu‡ † Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China; ‡ The State Key Laboratory of Physical Chemistry of Solid Surfaces, iChEM, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China. E-mail: [email protected] Abstract: In this work, a valence bond type multi-reference density functional theory (MRDFT) method, called Hamiltonian matrix correction based density functional valence bond method (hc-DFVB), is presented. In hc-DFVB, the static electronic correlation is considered by the valence bond self-consistent field (VBSCF) strategy, while the dynamic correlation energy is taken into account by Kohn-Sham density functional theory (KS-DFT). Different from our previous version of DFVB (J. Chem. Theory Comput. 2012, 8, 1608), hc-DFVB corrects the dynamic correlation energy with a Hamiltonian correction matrix, improving the functional adaptability and computational accuracy. The method was tested for various physical and chemical properties, including spectroscopic constants, bond dissociation energies, reaction barriers, and the singlet-triplet gaps. The accuracy of hc-DFVB matches KS-DFT and high level molecular orbital (MO) methods quite well. Furthermore, hc-DFVB keeps the advantages of VB methods, which are able to provide clear interpretations and chemical insights with compact wave functions.

*

To whom correspondence should be addressed. 1

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1. Introduction Kohn-Sham density functional theory (KS-DFT) has become the most widely used method for electronic structure calculations.1-3 In general, KS-DFT mainly focuses on dynamic correlation while fails in the description of static correlation, which can be well considered by the multi-reference wave function methods. A possible way to consider dynamic and static correlations simultaneously is to develop multi-reference density functional theory (MRDFT), in which the dynamic correlation is considered by KS-DFT, while the static correlation is treated by a multi-configuration wave function method. Various MRDFT strategies have been proposed by incorporating multi-reference schemes into KS-DFT approaches.4-33 Compared to the accurate but expensive high level multi-reference wave function methods, such as multi-reference configuration interaction (MRCI),34 multi-reference perturbation theory (CASPT2,35 MRMP236 etc.), and their valence bond (VB) analogues, valence bond configuration interaction (VBCI)37,38 and valence bond perturbation theory (VBPT2),39,40 MRDFT methods provide satisfactory accuracy with relatively low computational cost. One of the critical issues of MRDFT is the double counting error. It is still challenging due to the fact that dynamic and static correlations cannot be separated exactly. Recently, a VB-based MRDFT scheme, called density functional valence bond (DFVB) method, was presented.41 This method combines the valence bond self-consistent field (VBSCF)42,43 wave function with pure correlation functionals, and provides an economic route for adding the dynamic correlation effects into the framework of ab initio VB theory. On one hand, DFVB improves the accuracy over VBSCF by taking dynamic correlation into account with EC correlation functional. On the other hand, it overcomes some difficulties of KS-DFT due to the use of a single Kohn-Sham determinant. The computational cost of DFVB is approximately the same as that of VBSCF, and much cheaper than the other high-level VB methods, such as VBCI and VBPT2. Despite the partial success, there is room for improving the DFVB method. Firstly, the method is still limited to pure correlation functionals, such as LYP44 and PW91c.45,46 Given the fact that the aim of DFT functional parameterization is to obtain accurate estimation of the sum of exchange and correlation 2


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energies EXC, taking EC only to compute the correlation correction would introduce error to some extent. Secondly, the DFVB method takes dynamic correlation into account simply by adding the correlation correction to the total energy of system. Consequently, the dynamic correlation could not be explicitly described for individual VB structures. The motivation of this paper is to address the two issues of DFVB mentioned above and to further improve the computational accuracy. A new version of DFVB, Hamiltonian matrix correction based density functional valence bond method (hc-DFVB), is presented.

It extends DFVB to

exchange-correlation and hybrid DFT functionals. To this end, the dynamic correlation energy is taken into account by correcting Hamiltonian matrix elements of VBSCF method, through KS-DFT functional. It must be stressed that the scheme that modifies Hamiltonian matrix elements including information from KS-DFT has also been employed by Grimme5, Gao25 and Wu47,48 etc. To distinguish from the hc-DFVB method, the previous DFVB method is named as dynamic correlation corrected DFVB (dc-DFVB) in this paper.

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2. Methodology In valence bond theory, the wave function of a many-electron system can be expressed as a linear combination of Heitler−London−Slater−Pauling (HLSP) functions,4,49-51

Ψ = ∑ CK ΦK ,

(1)

K

where ΦK and CK is a HLSP function corresponding to a specific VB structure and its coefficient respectively. A HLSP function is expressed as a linear combination of VB determinants that are built upon atomic orbitals:

Φ K = ∑ dκK Dκ ,

(2)

κ

where coefficient dκ K is 1, -1, or 0. The coefficients, CK, in eq. 1 can conveniently be determined by solving the secular equation:

HC = EMC,

(3)

where H, M and C are the Hamiltonian, overlap, and coefficient matrices respectively. Hamiltonian and overlap matrix elements are defined as: H KL = Φ K H ΦL = ∑ dκK dλL Dκ H Dλ ,

(4)

κ ,λ

and

M KL = ΦK ΦL = ∑ dκK dλL Dκ Dλ κ ,λ

.

(5)

VB structural weights are usually used to compare the relative importance of individual VB structures. They are helpful in the understanding of the relationship between molecular structure and reactivity. One of the definitions is provided by the Coulson-Chirgwin formula:52

WK = ∑ CK CL M KL .

(6)

L

In dc-DFVB, the electronic energy of a system is given by adding DFT correlation energy to the VBSCF energy, E dc-DFVB[ ρ VB ] = E VBSCF + EC [ρ VB ] 4


,

(7)

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where EC[ρVB] is obtained from a pure correlation functional with the density of the VBSCF wave function. As a MRDFT approach, the dc-DFVB method suffers from the size-inconsistency problem. To remedy this problem, the strategy based on natural orbitals and their occupation numbers proposed by Jordá and coworkers 8,9 can be applied in the dc-DFVB method. In hc-DFVB, the dynamic correlation energy is taken into account by correcting the VBSCF Hamiltonian matrix elements, hc - DFVB VBSCF corr , H KL = H KL + H KL

(8)

corr where the correlation correction matrix elements, H KL , are obtained from DFT calculation. corr can be expressed as In a similar fashion to VBSCF, H KL corr corr , H KL = ∑ dκK d λL H κλ

(9)

κ ,λ

corr where H κλ are the correlation correction matrix elements between determinants Dκ and Dλ . Eq. 9

indicates that the dynamic correlation is corrected through the corrected Hamiltonian matrix, which is related to the density of each VB determinant, instead of the total density in dc-DFVB. corr In hc-DFVB, H κλ is defined as bellow:

For the diagonal elements, κ = λ , corr Hλλ = Ecorr[ρλ ] = EC [ρλ ] + (1- a)(EX [ρλ ] - Kλ ) ,

(10)

where ρλ is the electronic density of determinant Dλ , EC [ρλ ] and EX [ρλ ] are the correlation and exchange energies computed from DFT functional, respectively. K λ is the exact exchange energy of the determinant Dλ . The parameter a is the portion of K λ , ranging from 0 to 1.0. For example, a = 0.2 for B3LYP, and a = 1.0 if only EC correlation functional is employed. Eq. 10 shows the correlation correction of a single determinant. corr For the off-diagonal elements, κ ≠ λ , H κλ , correlation correction of the coupling between two

determinants, cannot be directly calculated from DFT functionals. There is no exact expression for this 5



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term. According to the works of Grimme5 and Gao25 etc., it can be treated approximately. One of the possible way is to express it as a function related to correlation correction energies of two determinants. corr Here we present two definitions for H κλ . The first one is:

  H VBSCF corr Hκλ ≈  VBSCFκλ VBSCF  E corr [ρκ ] + E corr [ρλ ] ,  Hκκ + H λλ 

(

)

(11)

It should be pointed out that eq. 11 retreats to eq. 10 if κ = λ . The second definition is expressed as: H κλcorr ≈

1 Sκλ ( E corr [ ρκ ] + E corr [ ρ λ ]) 2

(12)

Where Sκλ denotes the overlap matrix of VB determinants Dλ and Dκ. In this paper, these two definitions are used and compared. Table S1 in supporting information shows that the hc-DFVB energies with the two definitions are almost the same. Test calculations show that the two definitions provide almost the same results. Both of them can be used in the current version of hc-DFVB. For brief, the hc-DFVB results with the first definition are shown in the main text, while those with the second definition are shown in supporting information. corr Given the correlation correction matrix elements, H κλ , the hc-DFVB energy is obtained by solving

the secular equation, eq. 3, as in VBSCF, with the hc-DFVB Hamiltonian matrix by eq. 9. And thus, the energy is given as: hc - DFVB . E hc -DFVB = ∑ ∑ C K C L H KL K

(13)

L

The procedure of hc-DFVB can be summarized as follows, (1) VBSCF wave function is obtained by a normal VBSCF calculation. (2) The density of each VB determinant is computed based on VBSCF wave function. (3) According to eqs. 10 and 11 (or 12), the correlation correction matrix elements of determinants are constructed by using the density of VB determinant. (4) The correlation corrections for VB structures are computed by using eqs. 8 and 9. After diagonalization, the hc-DFVB energy is achieved. 6


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3. Computational Details The hc-DFVB method has been implemented in the XMVB module53-55 for the GAMESS package.56 All the VB calculations are performed by XMVB while all calculations with the molecular orbital (MO) method and KS-DFT are carried out by GAMESS. As one of the most accurate post-VBSCF methods, VBCI is used for comparison. This method employs a configuration interaction technique to consider the dynamic correlation energy. VBCI has various truncation levels of CI excitations. In this work, VBCISD, which involves single and double excitations, is applied. For dc-DFVB, pure correlation functional LYP44 is employed. For hc-DFVB, pure correlation functional LYP, hybrid GGA functional B3LYP,57,58 hybrid meta-GGA functional M06,59 and meta-GGA functional revTPSS60 are used. The functionals used in DFVB calculations are notated in parentheses. For example, dc-DFVB(LYP) stands for the dc-DFVB calculation with LYP functional. Test calculations are summarized as follows: Firstly, the potential energy curves of H2 molecule are investigated. Secondly, several physical and chemical properties, including the spectroscopic constants of a series of diatomic molecules, the reaction barriers of two Diels-Alder (D-A) reactions, the rotation barrier of Fe(CO)4 (CH2=CH2) and the singlet-triplet gaps of two diradicals, trimethylenemethane and carbine, are calculated. Finally, the issue of size-consistency is discussed. The Stuttgart RSC 1997 ECP basis set is used for bond dissociation energies of Cu2, Ag2 and Au2,61 while the cc-pVTZ (CCT) basis set is employed for the rest diatomic molecules and diradicals. The 6-31G* basis set is used for the D-A reactions. For the rotation barrier of Fe(CO)4 (CH2=CH2), the basis set Lanl2dz is used for Fe atom while the basis set 6-31G* for the other atoms. The geometries of reactants and transition states for Fe(CO)4 (CH2=CH2) are taken from dc-DFVB paper.41 The geometries for the diradicals, the reactants and transition states in the D-A reactions are optimized by M06.

8


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4. Results and Discussion 4.1. Potential energy curves of H2 Figure 1 plots the potential energy curves (PECs) of H2 with various methods, while Table 1 shows the full CI (FCI) energies and the relative errors of various VB methods to FCI at different bond distances, together with the non-parallelity errors (NPEs). In all the VB computations, three VB structures (one covalent and two ionic) with hybrid atomic orbitals (HAOs) are used. As can be seen from Figure 1, the uppermost PEC comes from VBSCF, due to the lack of dynamic correlation. The VBCISD curve is lower than the VBSCF one but is higher than the FCI curve, as expected. The dc-DFVB curve is the lowest one because of the severest double counting error. The hc-DFVB PEC locates between those of FCI and dc-DFVB, showing that the double counting error in hc-DFVB is reduced to some extent. All PECs share correct dissociation behavior, indicating that both of dc-DFVB and hc-DFVB overcome the KS-DFT problem for bond dissociation. However, the gradient of the dc-DFVB(LYP) curve from 2.0 Å to 5.0 Å is different from the others. This is because the size-consistent correction in dc-DFVB tends to enlarge the double counting error. Table 1 provides the more detailed information on the error analysis. It can be seen that the error of VBSCF is the largest. VBCISD improves the VBSCF energy with the relatively small errors. The deviations of dc-DFVB and hc-DFVB are negative, due to the existence of the double counting error. The NPE values of hc-DFVB (LYP) and hc-DFVB (B3LYP), 5.5 and 5.2 kcal/mol respectively, are smaller than VBSCF and dc-DFVB. It shows that the accuracy of hc-DFVB is further improved compared to dc-DFVB. The energy difference between VBSCF and post-VBSCF, such as VBCISD and hc-DFVB, can be regarded as the correlation correction energy. The correlation correction energies of the individual structures and the total wave function for H2 are shown in Table 2. It can be found that for dc-DFVB, all the three values (covalent, ionic, and the total) keep almost the same, showing that the correlation correction in dc-DFVB is inert to the variation of VB wave function. The hc-DFVB values are in the same trend as the VBCISD ones, showing that the ionic structures tend to gain more correlation energy compared to the covalent structure and total wave function. As such, with the dynamic correlation 9



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incorporated, the ionic structures in hc-DFVB become more important, which can be further confirmed by the VB weights shown in Table 3. As can be seen, the weights of dc-DFVB are almost the same as the VBSCF ones, while the hc-DFVB ones are close to VBCISD values, showing that the correlation correction of the ionic structure is larger than that of the covalent structure.

4.2 Spectroscopic constants The spectroscopic constants for a series of diatomic molecules, including equilibrium bond distances (Re, in Å), vibrational frequencies (ωe, in cm-1), and bond dissociation energies (BDE, in kcal/mol), calculated by various methods, are presented in Tables 4-6 respectively. Covalent structures with overlap enhanced orbitals (OEOs) are used in all the VBSCF and DFVB calculations. As can be seen from Tables 4-6, generally speaking, the accuracy of VBSCF is the worst with the big deviations compared to the experimental data, while the dc-DFVB results are improved. The MAE values of hc-DFVB are smallest, showing that the accuracy of hc-DFVB is better than VBSCF and dc-DFVB. It is found that the accuracy of hc-DFVB(LYP), which is worst among all the hc-DFVB calculations, is close to dc-DFVB(LYP), showing that the pure correlation correction only is still not satisfactory for hc-DFVB. The hc-DFVB(B3LYP) results are better than hc-DFVB(LYP). Among these hc-DFVB results, those with M06 and revTPSS are the best. For example, the MAE value of hc-DFVB (M06) for BDE is 5.9 kcal/mol, very close to the value of M06, 4.8 kcal/mol. In general, the accuracy of hc-DFVB is able to match DFT with meta-GGA and hybrid meta-GGA functionals. A further discussion for the BDE of N2 is necessary. From Table 6, it seems that dc-DFVB is able to provide the satisfactory binding energy compared to hc-DFVB and DFT. The bond dissociation energy (relative energy) curves of N2 by various VB methods are shown in Figure 2. Around the equilibrium bond distance, the VBSCF curve is the uppermost. The hc-DFVB(LYP) and MRMP2 curves almost coincide with each other. While the VBCISD and hc-DFVB(B3LYP) curves, which are very close to each other, are lower than that of MRMP2. It makes sense because the performances of high level multi-reference methods are highly basis set dependent. The hc-DFVB results with B3LYP, M06 and 10


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revTPSS are better than the MRCI/CCT result and close to the MRCI/complete basis set (CBS) estimation.62 Finally, among all the curves, the dc-DFVB one is the nearest to the experimental data. However, the dc-DFVB curve around the equilibrium bond length are lower than the line of the experimental data. Similar to Figure 1, the dc-DFVB curve shows the large deviation to the other curves from 1.1 Å to 3.5 Å. It shows that the double counting error of dc-DFVB(LYP) leads to the good BDE result. However, the error is not always helpful for BDE estimation. As for C2, this double counting error results in the great positive error, 28.7 kcal/mol, compared to the experimental value.

4.3 Reaction barriers and singlet-triplet gaps The reaction barriers for the two Diels-Alder reactions (Scheme 1) are shown in Table 7. Five covalent VB structures (shown in Figure 1 of supporting information) with OEOs are used in the VB calculations. Six valence electrons are considered as the active electrons. For comparison, CASSCF and MRMP2 are employed. It can be seen that for reaction (a), VBSCF and CASSCF provide the highest barriers due to the lack of dynamic correlation. The MRMP2 result is close to the experimental data (23.3 ± 2.0 kcal/mol).63 On the other hand, most of DFT functionals underestimate the barrier, as expected. For example, revTPSS, which predicts a value of 13.4 kcal/mol, obviously underestimates the reaction barrier. Although dc-DFVB shows some improvement compared to VBSCF, it overestimates the barrier. Except hc-DFVB(LYP), all the other hc-DFVB results are able to describe the reaction barrier quite well. For example, the value of hc-DFVB(M06) is 23.4 kcal/mol, which matches the MRMP2 and experimental data quite well. For reaction (b), it is generally accepted that electron-withdrawing groups of the dienophile tends to reduce the barrier. However, it can be seen that VBSCF, CASSCF, and most of DFT functionals are unable to describe the effect of electron-withdrawing groups, while MRMP2 acknowledges this effect. By hc-DFVB, the substituent effect has been described qualitatively right. For example, the hc-DFVB (M06) barrier of reaction (b) is lower than the value of reaction (a) by 3.3 kcal/mol. The weights of the five covalent VB structures for the transition states of the two reactions are 11



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presented in Table 1 of supporting information. It is shown that the hc-DFVB weights are somewhat different from the VBSCF ones, showing the effect of dynamic correlation on the individual VB structures. In the recent VB review,49 the rotation of the complex Fe(CO)4(CH2=CH2), which is shown in scheme 2, is regarded as an example of the current computational capabilities of VB method for organometallic systems. For hc-DFVB and dc-DFVB calculations, four valence electrons are considered as the active electrons, two covalent VB structures (shown in Figure 2 of supporting information) with OEOs are used. Its rotation barriers by various methods are shown in Table 8. It is found that the dc-DFVB(LYP) and hc-DFVB(LYP) barriers are similar to that of CASSCF, while those of hc-DFVB(B3LYP) and hc-DFVB(revTPSS) are close to the CASPT2 result. The singlet-triplet gaps of the diradicals, trimethylenemethane (TMM), and carbine (CH2), are shown in Table 9. Due to the non-dynamic characteristic, DFT functionals fail to give satisfactory values for the singlet-triplet gaps. The VBSCF method is able to handle the cases of the multi-reference systems. The VBSCF values of 18.6 and 39.0 kcal/mol for TMM and CH2, are close to the corresponding experimental values, 18.1 and 32.9 kcal/mol respectively. dc-DFVB underestimates the ∆EST values. All the hc-DFVB results are close to the experimental data quite well. With M06 and revTPSS, the hc-DFVB results are even more accurate than the VBSCF ones.

4.4 Resonance energy In VB theory, the resonance energy, which is defined as the energy difference between the wave function with the full structure set and the lowest energy structure(s), is a useful concept for understanding the nature of chemical bond, especially for charge-shift bond. The value of the resonance energy, B, depends on the level of ab initio VB methods. It is known that VBCISD provides the most accurate description for the B value. The resonance energies of H2, F2, Li2, and HF by various VB methods are shown in Table 10. It is found that the DFVB computed resonance energies located between the VBSCF and VBCISD values. While dc-DFVB and hc-DFVB provide the similar resonance energy. 12


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For example, for HF molecule, the DFVB computed resonance energies, varying from -49.4 ~ -50.1 kcal/mol, larger than the VBSCF result but slightly smaller than the VBCISD value. The VB resonance energy and the VB weights show that hc-DFVB is able to provide clear interpretations and chemical insights with compact wave functions.

4.5 Size-consistency of hc-DFVB The size-consistency error is defined as the energy difference between the computed energy of the combined system in infinite separation and the sum of the calculated energies of two isolated open-shell subsystems. The size-consistency errors of hc-DFVB for H2, Li2, F2, O2, C2 and N2 are shown in Table 11. For single bond breaking, off-diagonal Hamiltonian elements at the dissociation limit are zero. According to eq. 11, the corresponding correction matrix elements are also zero. Therefore, hc-DFVB is size consistent. For multiple bonding breaking, some off-diagonal Hamiltonian elements at the dissociation limit are not zero. Then hc-DFVB is not size-consistent because the current definition of correlation correction is determined by the density of a single VB determinant, which does not contain the information of spin configuration. From Table 11, it can be seen that hc-DFVB is size-consistent for the H2, Li2, and F2 molecules while is not for the remaining molecules, which belong to multiple bonding breaking. It is found that the hc-DFVB errors using M06 are smaller than those of LYP and B3LYP, showing that the size-consistency error of hc-DFVB can be small.

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Conclusion In this work, a Hamiltonian matrix correction based density functional valence bond method, called hc-DFVB, is presented. In hc-DFVB, the dynamic correlation energy is taken into account by correcting the VBSCF Hamiltonian matrix elements. The hc-DFVB method shares the following advantages over the previous DFVB method, dc-DFVB: Firstly, the hc-DFVB method breaks through the limit of DFT functionals in the dc-DFVB method, in which only pure correlation functionals can be employed. Either exchange-correlation or hybrid DFT functionals can be used in hc-DFVB calculations. Secondly, with the correction for Hamiltonian matrix, individual VB structures share their own dynamic correlation corrections, which are more physical. VB structural weights of hc-DFVB are in good agreement with the other high level VB methods such as VBCISD. Test calculations show that the hc-DFVB method improves the accuracy compared to dc-DFVB. Moreover, hc-DFVB results are in good agreement with high level VB and MO approaches. In the comparison with DFT calculations, on one hand, for the molecules that share one-determinant character, the hc-DFVB results match their DFT analogs quite well. On the other hand, for the molecules that are multi-reference characteristic, hc-DFVB overcomes the problems that arise from the use of a single determinant in DFT. hc-DFVB still suffers from the double counting error. It is size-consistent for single bond breaking, but not for multiple bond breaking.

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Acknowledgements: This project is supported by the Natural Science Foundation of China (Nos. 21373165, 21120102035, 21273176, 21573176, 21503172) and the Fundamental Research Funds for the Central Universities, China (Nos. 20720150037, 20720150146).

Supporting Information Supporting information include: the hc-DFVB energies of diatomic molecules with two definitions (eqs 11 and 12); the computational results with the definition of eq 12; the VB informations and the Cartesian coordinates of the two chemical reactions. This information is available free of charge via the Internet at http://pubs.acs.org/.

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Figure 1. The potential energy curves of H2 molecule.

-1

-1.05 E (a.u.)

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-1.1

VBSCF VBCISD dc-DFVB(LYP) hc-DFVB(LYP) hc-DFVB(B3LYP) full CI

-1.15

1

2

。3 R (A)

4

5

16


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Figure 2. The relative energy curves for N2 molecule. The dot line in the small graph is the location of the experimental data.

0 -0.05 Relative Energy (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-0.1 -0.15 -0.2 VBSCF MRMP2 VBCISD dc-DFVB(LYP) hc-DFVB(LYP) hc-DFVB(B3LYP)

-0.25

-0.25 -0.3

-0.3 -0.35

-0.35 1

1 1.1 1.2

1.5

2

。

2.5

3

3.5

R (A)

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Scheme 1. The two Diels-Alder reactions. (a)

(b)

Scheme 2. The rotation of Fe(CO)4 (CH2=CH2)

18


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Page 19 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 1. The total energies and nonparallelity errors (NPE) of the ground state of H2 calculated by various VB methods.* VBCIS FCI VBSCF dc-DFVB(LYP) hc-DFVB (LYP) R(Å) D -1.10108711 15.6 8.3 -9.2 -5.5 0.50 -1.15366750 14.6 7.9 -9.7 -5.2 0.60 -1.17113799 13.6 7.5 -10.2 -4.9 0.70 -1.17242965 13.1 7.3 -10.5 -4.7 0.74 -1.17052566 12.6 7.0 -10.7 -4.5 0.80 -1.16053514 11.6 6.5 -11.3 -4.2 0.90 -1.14587760 10.5 5.9 -11.9 -3.8 1.00 -1.11215769 8.5 4.7 -13.0 -3.3 1.20 -1.08018862 6.4 3.5 -14.2 -2.9 1.40 -1.05383588 4.5 2.4 -15.3 -2.5 1.60 1.5 -15.1 -2.1 -1.03409643 3.0 1.80 -1.02047579 1.8 0.9 -14.7 -1.7 2.00 -1.01172902 1.1 0.5 -13.6 -1.3 2.20 -1.00643631 0.6 0.3 -12.0 -1.0 2.40 -1.00469120 0.4 0.2 -11.1 -0.8 2.50 -1.00337767 0.3 0.1 -10.1 -0.7 2.60 -1.00239597 0.2 0.1 -9.2 -0.6 2.70 -1.00072740 0.1 0.0 -6.6 -0.3 3.00 -0.99967531 0.0 0.0 -1.6 0.0 4.00 -0.99962558 0.0 0.0 0.0 0.0 5.00 15.6 8.3 15.3 5.5 NPE * FCI values are given in total energies (in a.u.), all others are relative values (in kcal/mol) with respect to FCI ones.

19


hc-DFVB (B3LYP) -1.7 -1.5 -1.2 -1.0 -0.8 -0.5 -0.2 0.2 0.6 0.9 1.3 1.7 2.1 2.4 2.6 2.7 2.8 3.1 3.5 3.5 5.2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Table 2. The correlation energy of H2 at the equilibrium bond distance (kcal/mol). VB wave function VBCISD dc-DFVB(LYP) covalent -6.3 -24.0 ionic -11.9 -23.8 Total -6.2 -24.0

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hc-DFVB(LYP) -18.1 -23.8 -20.3

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 3. The weights of the covalent and ionic structures for H2. Structure covalent ionic

VBSCF 0.821 0.089

VBCISD 0.808 0.096

dc-DFVB(LYP) 0.821 0.090

21


hc-DFVB(LYP) 0.771 0.114


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 22 of 32

Table 4. The equilibrium bond distances (Re) of diatomic molecules by the various VB methods (Å).a Re

Expt.b

VBSCF

B3LYP

M06

revTPSS

dc-DFVB LYP

LYP

Li2

2.673

0.260

0.028

0.035

0.063

-0.064

0.078

0.058

0.075

0.026

F2

1.412

0.057

-0.016

-0.037

0.001

0.014

0.001

-0.009

-0.013

-0.020

C2

1.243

0.005

0.004

0.005

0.007

-0.010

-0.022

-0.014

-0.012

-0.012

N2

1.098

0.002

-0.007

-0.009

-0.001

-0.009

-0.015

-0.008

-0.010

-0.005

O2

1.208

-0.004

-0.004

-0.014

0.010

-0.019

-0.021

-0.007

-0.013

-0.004

Cu2

2.203

0.224

0.044

0.019

-0.007

0.129

0.121

0.057

0.029

0.014

Ag2

2.530

0.237

0.068

0.060

0.011

0.132

0.118

0.070

0.053

0.020

Au2

2.472

0.204

0.108

0.121

0.065

0.130

0.120

0.097

0.100

0.049

0.124

0.035

0.038

0.028

0.063

0.062

0.040

0.038

0.031

MAE a b

hc-DFVB B3LYP M06

revTPSS

Deviations from experimental values are listed. The experimental values of Li2, F2, N2 and O2 are taken from ref 65, that of C2 is from ref 64, while those for Cu2, Ag2 and Au2 are from ref 65.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 5. The vibrational frequency (ωe) of diatomic molecules by various VB methods (cm−1).a ωe

Expt.b

VBSCF

B3LYP

M06

revTPSS

Li2

351

-78

-9

6

F2

917

-191

138

C2

1855

26

N2

2359

O2

b

hc-DFVB

LYP

LYP

B3LYP

M06

revTPSS

-16

27

-7

11

68

13

159

97

-85

-55

87

2

117

24

33

20

135

-150

210

122

64

173

95

115

51

-32

81

91

95

91

1580

-1

56

129

-36

116

112

11

15

23

Cu2

266

-95

-5

11

24

-65

-34

-13

65

-21

Ag2

192

-64

-14

-2

5

-43

-33

-8

0

12

Au2

191

-62

-28

-27

-15

-40

-43

-23

-18

-8

86

46

60

33

76

64

57

48

44

MAE a

dc-DFVB


23



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Page 24 of 32

Table 6. The bond dissociation energies of diatomic molecules by various VB methods (kcal/mol).a De

Expt.b

VBSCF

B3LYP

M06

revTPSS

Li2

24.4

-14.2

-3.7

0.2

F2

38.2

-21.4

2.3

C2

148.0

-35.9

N2

228.5

O2 Cu2

b

hc-DFVB

LYP

LYP

B3LYP

M06

revTPSS

11.4

-9.2

-5.4

-4.9

-8.4

0.9

-3.2

11.4

-9.9

-7.2

-3.0

-2.4

4.8

-17.9

-20.9

-17.3

28.7

-12.6

-8.8

-9.2

-5.4

-58.4

3.1

-6.3

3.3

3.3

-18.1

-8.2

-8.5

-6.3

120.3

-53.0

4.3

-0.2

8.8

-17.9

-29.5

-14.3

-5.8

-7.4

-28.6 -25.4

-3.2 -6.1

1.4 0.1

6.6 1.6

-19.7 -17.1

-13.9

-6.0

-0.7

0.3

Ag2

46.4 41.0

-12.6

-7.3

-1.1

-1.0

Au2

53.0

-30.2

-11.0

-5.8

-0.8

-21.2

-17.8

-13.0

-8.6

-5.1

33.4

6.5

4.8

7.7

15.9

14.6

8.2

5.6

3.9

MAE a

dc-DFVB


24


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 7. The barriers of the two Diels-Alder reactions (kcal/mol).

∆E≠ *

Method Reaction(a)

CASSCF[6,6] MRMP2[6,6] VBSCF DFT

dc-DFVB hc-DFVB

Expt.

Reaction(b)

41.3 23.5 44.0 B3LYP 21.3 M06 19.7 revTPSS 13.0 LYP 34.5 LYP 34.2 B3LYP 29.3 M06 23.3 revTPSS 21.5 23.3±2 63

* For all the VB results, the zero-point energy correction is calculated at the M06/6-31G* level.

25


40.3 18.4 43.5 21.3 17.0 13.1 33.4 32.5 28.3 19.9 19.8


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Table 8. The rotation barrier of Fe(CO)4 (CH2=CH2) (kcal/mol).* Method CASSCF CASPT2 VBSCF B3LYP M06 DFT revTPSS LYP dc-DFVB LYP B3LYP hc-DFVB M06 revTPSS *The computed barriers of CASSCF, CASPT2 and dc-DFVB are taken from the ref 41.

∆E≠ 13.9 15.5 10.6 10.0 9.4 10.2 12.5 12.4 14.1 19.1 14.1

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Table 9. The singlet−triplet energy gaps of the diradicals TMM and CH2 (kcal/mol).

Exp. TMM 18.167,68 CH2 32.969 a a 3 gap of B1 → 1B1

VBSCF

B3LYP

M06

revTPS S

18.6 39.0

42.0 29.8

41.4 27.0

38.7 35.6

hc-DFVB

dc-DFVB LYP

LYP

B3LYP

M06

12.0 25.9

20.0 32.2

18.3 28.3

18.3 33.4

27


revTPS S 17.9 32.3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Table 10. Resonance energies (B, in kcal/mol) of H2, Li2, F2 and HF with the cc-pVTZ basis set. H2 Li2 VBSCF 1.6 0.0 VBCISD 2.8 1.3 dc-DFVB(LYP) 1.6 0.0 hc-DFVB(LYP) 2.5 0.5 hc-DFVB(B3LYP) 2.3 0.5 hc-DFVB(M06) 3.1 2.5 hc-DFVB(revTPSS) 2.1 0.7

F2 39.9 57.7 51.1 50.3 49.4 50.8 49.8

Table 11. The size-consistency error of hc-DFVB (kcal/mol).

H2 Li2 F2 O2 C2 N2

LYP 0.0 0.0 0.0 -5.4 -13.7 -11.2

hc-DFVB B3LYP M06 0.0 0.0 0.0 0.0 0.0 0.0 -8.6 -1.3 -5.5 -5.8 -17.1 -4.8

28


revTPSS 0.0 0.0 0.0 -6.7 -5.2 -9.0

Page 28 of 32

HF 36.0 32.3 35.7 36.9 39.5 38.4 35.6

Page 29 of 32

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TOC Graphic

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