Valence Bond Based Energy Decomposition Analysis Scheme and Its

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A Valence Bond Based Energy Decomposition Analysis Scheme and Its Application to Cation-# Interactions Yang Zhang, Sifeng Chen, Fuming Ying, Peifeng Su, and Wei Wu J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b04201 • Publication Date (Web): 14 Jun 2018 Downloaded from http://pubs.acs.org on June 18, 2018

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The Journal of Physical Chemistry

A Valence Bond Based Energy Decomposition Analysis Scheme and its Application to Cation-π π Interactions Yang Zhang, Sifeng Chen, Fuming Ying, Peifeng Su,∗ Wei Wu The State Key Laboratory of Physical Chemistry of Solid Surfaces, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, iChEM, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China Abstract: A new energy decomposition analysis (EDA) scheme based on valence bond (VB) wave function, called VB-EDA, is presented. In VB-EDA, the total interaction energy is decomposed into frozen, charge transfer, polarization and dynamic correlation terms based on valence bond calculations. The frozen term is the energy variation of the unrelaxed VB wave function according to the change of an interaction distance. The charge transfer term is the contribution of the additional VB structures while the polarization term is due to the relaxation of VB orbitals. Dynamic correlation term is computed by post-VBSCF methods. Different from other existing VB based EDA schemes, which were used to analyze non-covalent interactions for some specific complexes, the newly developed VB-EDA is designed for the general use. Using VB-EDA, the bonding nature of cation-π interactions in a series of cation-π complexes (cations: Li+, Na+, K+, Mg2+ and Ca2+; π systems: ethylene and benzene) is explored. Furthermore, a new covalency index, which demonstrates the covalency of cation-π interactions, is presented based on the VB-EDA results. The VB-EDA analysis reveals that the cation-π interactions in the Li+, Na+ and K+ complexes belong to the typical ionic bonds while the Mg2+ and Ca2+ complexes have the relatively large covalent characteristics. However, only the C2H4-Mg2+ complex can be regarded as a covalent bonding complex while the other complexes belong to the typical ionic complexes. Thereupon, it must be careful in the cognition for the covalency of intermolecular interaction. Large non-electrostatic interaction component does not always correspond to a covalent bond.

∗

Author to whom correspondence should be addressed. Email: [email protected]. 1

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1. Introduction Energy decomposition analysis (EDA) method is a powerful approach for understanding intermolecular interactions based on quantum mechanical calculations. It divides the total interaction energy into several interaction terms with physical meaning. Since 1970s, various EDA methods have been proposed,1-16 which are widely used for analyzing various intermolecular interactions. Considering the efficiency and computational cost, most of EDA schemes are designed for the single determinant based molecular orbital (MO) or density functional theory (DFT) method. These MO-based or DFT-based EDA approaches, which provide quantitative analysis with delocalized orbitals, tends to require large basis sets for the computational accuracy. Compared to MO and DFT methods, one of the advantages of classical valence bond (VB) theory is its intuitive wave function, which is expressed as a linear combination of chemically meaningful VB structures with localized VB orbitals.17-19 Though MO and DFT methods currently dominate the electronic structure calculations, ab initio VB methods are becoming invaluable tools for providing insights into various chemical problems quantitatively, thanks to advances in methodologies and algorithms that have been achieved since 1980.20-31 Different from the traditional MO-based or DFT-based EDA approaches, the development of VB-based EDA, combining VB theory with EDA approach, is still a challenge. This is due to the fact that VB theory is a multi-reference approach, and VB orbitals are not orthogonal, which result in the complexity of defining the interaction components in EDA. Recently, several VB theory based analysis schemes are proposed for CH…HC interactions, halogen bonds, blue-shift hydrogen bonds and so on.32-34 In these studies, the origin of non-covalent intermolecular interactions can be revealed with the compact VB wave function. For example, in the study of hydrogen bonds in F3C−H…Y complexes, with several VB structures, the relationship between the origin of the F3C−H bond and the hydrogen bonding interaction was discussed. It is concluded that the covalent state of the F3C−H bond tends to blue-shift while the ionic state responds to the red-shifting.34 Despite of some successful applications, the current VB based analysis schemes are not for the general 2


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use because their interaction terms are defined on a case by case basis. For example, in the analysis scheme for halogen bond,33 the polarization term arises from the contribution of some ionic VB structures, while in the analysis scheme for the CH···HC Interactions,32 the polarization term is absorbed into a frozen energy term, computed by the VB wave function correspond to the perfectly paired wave function of the two CH moieties. The analysis results depend on the definition of intermedia-state wave function. It is difficult to apply the existing VB-based schemes to other interaction systems. Developing a general VB based EDA scheme is the first purpose of this paper. Cation-π interaction is a strong attractive interaction between the cation and π moiety, playing a key role in molecular recognition, self-assembly, material design, protein–ligand interaction and so on.35-42 The strength of strong cation-π interaction is greater than the classical single σ bond while the weak one is similar to a typical vdW interaction. The cation-π interaction in alkali/alkaline earth metal cation complexes is well documented and explored extensively.43-56 The importance of induction was highlighted by Luque et al51 and Jiang et al.52,53 Jiang and coworkers even concluded that the alkaline-earth-metal cation/benzene complex belong to a chemical bond, named as the cation-π bond because of the large induction energy. On the contrary, some other investigators claimed that electrostatic is the most important. Tsuzuki reported that the electrostatic interaction and induction are mainly responsible for the alkali-metal cation/benzene interactions, and this kind of interaction is not a chemical bond.55 Considering the variation of electron densities upon complexation, Mohajeri et al also found that the cation–benzene interaction has less covalent character (donor–acceptor interaction) and is mostly electrostatic in nature.54 Clearly, it is still worthwhile to explore the nature of cation-π interaction with an unambiguous answer, which is another purpose of this paper. In this paper, a new valence bond based energy decomposition analysis scheme, named VB-EDA, is presented and employed for the nature of the intermolecular cation-π interactions between the π molecules (benzene and ethylene) and the alkali (Li+, Na+ and K+) /alkaline-earth (Mg2+ and Ca2+) metal cations. To our best knowledge, it is the first VB study for cation-π interactions. The bonding nature in these complexes will be discussed in details. 3



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This paper is organized as follows: First, the methodology is introduced and the performances of the VB-EDA scheme for various weak and strong molecular interactions are investigated, including covalent bond, hydrogen bond and ligand bond. Second, the VB-EDA results of a series of cation-π interactions are shown and discussed. Finally, the paper ends with the conclusion and further perspective.

2. Methodology and Computational Details 2.1 VBSCF and hc-DFVB methods In VB theory, a many-electron wave function is expressed by the linear combination of VB structures,17-19

Ψ = ∑ CK Φ K ,

(1)

K

where CK is the coefficients of the VB structure ΦK. The total energy and the structural coefficients can be obtained by solving the secular equation below: HC = EMC,

(2)

where H, M and C are the Hamiltonian, overlap and coefficient matrices respectively. VBSCF is the basic method in ab initio classical VB theory. In the VBSCF method, the VB structure coefficients CK and VB orbitals {φi} are optimized simultaneously. Analogous to the MO-based CASSCF method, VBSCF mainly considers the static correlation with lack of dynamic correlation. To cover dynamic correlation, various high level VB methods have been proposed.22-30 Among them, hc-DFVB is a recently developed VB method that incorporates DFT into VB,28 where VBSCF wave function is used as the electronic wave function. Thus hc-DFVB is regarded as a multi-reference DFT method. In hc-DFVB, DFT correlation energy is added to the Hamiltonian matrix elements, i.e., hc- DFVB VBSCF corr H KL = H KL + H KL ,

(3)

corr where the correlation correction matrix elements, H KL , are obtained from DFT calculation. The

hc-DFVB energy is obtained by solving the secular equation, eq. 2, with the hc-DFVB Hamiltonian 4


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matrix, eq. 3. Thus, the hc-DFVB energy is given as: hc-DFVB E hc-DFVB = ∑∑ CK CL H KL . K

(4)

L

Various DFT functionals can be used in the hc-DFVB method, including GGA, meta-GGA and hybrid functionals. The accuracy of hc-DFVB is satisfactory, and computational cost is lower, compared to high level molecular orbital (MO) methods. Similar to other modern VB methods, hc-DFVB has capability of presenting clear interpretations and chemical insights with compact wave functions. The weights of VB structures are usually used to show the relative importance of individual VB structures. By the Coulson-Chirgwin definition,57 weights of VB structures are expressed as:

WK = ∑ CK CL M KL = CK2 + ∑ CK CL M KL ,

(5)

L≠ K

L

where MKL is the overlap between structures Kth and Lth. The Coulson-Chirgwin weights can be regarded as an equivalence of the Mulliken population analysis in MO theory.

2.2 VB-based energy decomposition analysis Considering a complex formed by monomers A and B, the total interaction energy is divided into the following terms by VB-EDA:

∆E int = ∆E fro + ∆E CT + ∆E pol + ∆E corr .

(6)

To compute these interacting terms, analogous to the other multi-reference based EDA schemes,32,58 all VB structures involved in the total wave function, eq. 1, are grouped into two sets, the sets SM and ST. The set SM is composed of VB structures in which there is no electron transfer between monomers A and B. The wave function of SM is:

ΨM =

∑C

K

ΦM K .

(7)

K ∈S M

The set exists in all interacting distances. For example, as shown in Table 1, for a single bonding X-X molecule (X is an atom or a fragment with an unpaired valence electron), the set SM is only one covalent structure X-X. For the hydrogen bond X-H…Y (X is usually an electronegative atom or group and Y is an 5



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atom or group with excessive electrons), the set SM is a linear combination of the three VB structures that describe the covalent and ionic forms of X-H along with the hydrogen atom acceptor Y. For the BH3-NH3 molecule with the ligand bond, set SM is only one structure describing the neutral form of B-N bond. Set ST includes all the structures Φ that describe electron transfer from one monomer to the other, and thus the wave function of ST is expressed as:

ΨT =

∑C

K

K∈S

ΦTK .

(8)

T

Clearly, set ST contributes to total wave function only at short distances. The interaction terms in eq. 6 can be obtained sequentially by four intermediate states. In the first state, the complex is totally dissociated (R = ∞), and there is no interaction between monomers A and B. In this state, the total wave function is exactly identical to ΨM and thus the total energy is expressed as:

E∞ (S M ,ϕ ) = Ψ M H ∞ Ψ M ,

(9)

where H ∞ is the Hamiltonian at the dissociation limit (R = ∞), and ϕ indicates that the wave function is built with the monomer optimized orbitals . In the second state, monomers A and B are placed in a given distance R, and the form of wave function and orbitals are kept unchanged from state one. Thus, the total energy is now written as: E R (S M , ϕ ) = Ψ M H R Ψ M .

(10)

The frozen term is defined as the energy difference between the two states:

∆E fro = E∞ (S M ,ϕ ) − E R (S M ,ϕ ) .

(11)

The frozen term gives the energy variation of the unrelaxed wave function, resulting from the variation of the distance R, which can be regarded as the sum of electrostatic and exchange repulsion energies in cation-π interactions. This term also involves some covalent interactions if the two monomers are free radicals. The negative value of frozen term, showing the attractive interaction, indicates that electrostatic interaction is larger than exchange repulsion interaction and vice versa. In the third state, both SM and ST sets are involved in the wave function, while the orbitals are kept 6


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unchanged. Thus, the energy is now expressed as:

E R (S M + S T ,ϕ ) = Ψ (ϕ ) H R Ψ (ϕ ) ,

(12)

where Ψ(ϕ) denotes the total VB wave function with the monomers’ optimized orbitals. The charge transfer (CT) term is defined as:

∆E CT = ER (S M ,ϕ ) − E R (S M + S T ,ϕ ) .

(13)

This CT term is the contribution of the additional VB structures describing electron transfer between monomers A and B without the orbital optimization. In the final state, the VB wave function and orbitals are optimized simultaneously with the current geometry. The energy with the optimized orbitals for the complex, is expressed as:

(

)

E R S M + S T ,φ = Ψ(φ ) H R Ψ(φ ) ,

(14)

where φ indicates that the wave function is built with the complex optimized orbitals. The polarization term is defined as:

∆E pol = E R (S M + S T ,ϕ ) − E R (S M + S T ,φ ) .

(15)

This term denotes the contribution for the relaxation of all the orbitals from {ϕi} to {φi}. Similarly, the contribution of the relaxation from assigned orbitals, ∆Epol(ζ), can be computed as:

∆E pol (ζ ) = E R (S M + S T ,ϕ ) − E R (S M + S T , ζ ) ,

(16)

where ζ can be intermediate orbitals in which the assigned orbitals are optimized while the other orbitals are kept unchanged. The ∆Epol(ζ) value can be used to estimate the energy contribution from one or several assigned orbitals. VB-EDA can be performed at various VB methods. If VBSCF is employed, only ∆Efro, ∆Epol and ∆ECT are provided. If a high-level VB method that covers dynamic correlation is used, an additional correlation term ∆E corr is included, defined as

(

)

∆E corr = EScorr - EAcorr + EBcorr , where Ecorr is defined as the energy difference between VBSCF and the post-VBSCF method. In this 7


(17)


paper, the correlation term is obtained by hc-DFVB, while the other three terms of hc-DFVB are the same as the corresponding VBSCF terms, as hc-DFVB employs the VBSCF orbitals without any further optimization. In VB-EDA, geometry, structures and orbitals are changed sequentially to compute frozen, CT and polarization respectively. All the terms are computed with the VB orbitals strictly localized on each monomer. It helps VB-EDA to avoid the basis set superposition error (BSSE), which should be considered by the EDA schemes with delocalized orbitals. In VB-EDA, CT and polarization are well separated because they have different origins. CT and polarization are contributed from the additional VB structures and from the VB orbitals relaxation respectively. Polarization in VB-EDA is a purely induction. For the MO or DFT based EDA schemes, it is not easy to separate polarization and CT terms exactly, due to the fact that they are both related to the variation of delocalized orbitals. A general definition of CT is the energy contribution of the electrons transfer from an occupied MO on a monomer to a virtual orbital associated with another monomer, or vice versa. However, the magnitudes of the two terms are heavily basis set dependent. In a calculation using a

large basis set, the distinction between polarization (or induction) and CT is ambiguous, as discussed by Herbert.59 Several schemes have been proposed to separate these two terms. For example, Head-Gordon and coworkers developed a fragment electrical response function model to try to divide polarization and CT.60 To show the performance, a series of intermolecular interactions, including water dimer, CH3-CH3, BH3-NH3, and ethylene-Mg2+, are tested by VB-EDA with various basis sets, including 6-31G*, 6-311G*, 6-311G**, cc-pVDZ and cc-pVTZ. In VB-EDA calculations, two DFT functionals, B3LYP and M06, are used in hc-DFVB. The results are listed in Table 2. The VB sets for the first three molecules are shown in Table 1 while those for ethylene-Mg2+ will be discussed below. In general, except the hc-DFVB(M06) results for water-dimer, the total interaction energies are in good agreement with the experimental data or the computational results by others. For example, for CH3-CH3, the binding energy in the single C-C bond,

8


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-106.21 kcal/mol by hc-DFVB(M06)/cc-pVTZ, is similar to the computed result of RO-CCSD/aug-cc-pVQZ, -111.69 kcal/mol.4 Furthermore, the individual interaction terms are in accord with the results of the other EDA schemes. For example, the CT value in water dimer ranges from -0.71 to -1.12 kcal/mol, similar to the CT term in constrained density functional theory (CDFT), -0.79 kcal/mol, by de la Lande et al.61 The CT term for BH3-NH3, from -15.0 to -17.52 kcal/mol, is quite close to the CDFT value of about -14.6 kcal/mol with the aug-cc-pVXZ (X=D, T, Q and 5) basis sets.59 Finally, the results show that VB-EDA does not suffer from basis set dependent. The analysis results with different basis sets are in good agreement. For example, for ethylene-Mg2+, the VB-EDA results always show the order of polarization energy > CT > electrostatic interaction with all the basis sets. As such, different from MO based calculations, a relatively small basis set is adequate for VB-EDA calculations. In general, VB-EDA is able to perform analysis not only for weak intermolecular interactions but also for strong ligand bonds and covalent bonds. The test calculations show that VB-EDA shares satisfactory accuracy with insensitivity to basis sets. This scheme is capable of analyzing the nature of cation-π interactions. Meanwhile, the performance of hc-DFVB depends on the DFT functional that the method uses. hc-DFVB(B3LYP) is better for water-dimer while hc-DFVB(M06) is better for the others. For the cation-π interactions (vide infra), hc-DFVB is performed with the M06 functional.

3. VB-EDA application to the cation-π interactions 3.1 Computational details The geometrical optimizations were performed at the M06/6-311G** level by GAMESS.62 The zero-point energy (ZPE) correction is not considered. The LMO-EDA and SAPT calculations were carried out by GAMESS62 and Psi4 program63 respectively. All VB calculations were performed by XMVB program,64-66 which is an ab initio VB program. The current VB-EDA calculations for the cation-π complexes were done by the two levels, VBSCF and hc-DFVB, with the 6-311G** basis set. 9



All valence electrons are involved in the VB calculations. The VB orbitals are strictly localized on the metal cations or π donors. As shown in Figure 1, the π donors (the left moiety), ethylene and benzene, are located on the x-y plane. The central point of the π monomer lies on the z-axis. All the active electrons are provided by π donors, two active electrons for ethylene and six active electrons for benzene respectively. For benzene, the three valence orbitals are denoted as p1x, p1y, p1z orbitals respectively according to the symmetry. For ethylene, the valence orbital is set as the p1z orbital. By the way, orbitals of ethylene and benzene are nothing else than the bonding π MOs. The three valence orbitals in the metal cation (the right moiety) are denoted as p2x, p2y, and p2z. The pnx and pnz orbitals of both M+/2+ and π donors are in the plane of the paper, whereas the pny orbitals are drawn as circles with one lobe pointing at the observer (n = 1 or 2). For cation-π complexes, the set SM is the original ionic structure, which is denoted as Φion in Table 3. For the ethylene complexes, there are totally 10 VB structures for two electrons at four VB orbitals, while for benzene complexes, there are 175 VB structures for six electrons at six VB orbitals. Thus, the numbers of VB structures in the set ST are 9 and 174 for the ethylene complexes and benzene complexes respectively. Test calculations show that for the ethylene complexes, only one structure in the set ST is important, which is denoted as Φ cov in Table 3. For the benzene complexes, the three covalent VB structures (Φ cov1, Φcov2 and Φcov3 in Table 3) describing one electron transfers from benzene to the metal cation are relatively significant. For conciseness, only these important VB structures are shown and discussed below.

3.2 Results and discussions 1. The VB-EDA results The optimized geometries of the cation-π complexes in Figure 2 show good agreement with the references’ results. For example, the distance of benzene-Na+ is 2.486 Å, similar to the 2.390 Å value at the MP2/aug-cc-pVQZ level.45 In general, considering the same metal cation, the benzene complexes always have shorter binding distances compared to the ethylene ones. 10


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Table 4 shows the VB-EDA results of cation-π interactions by hc-DFVB (M06). In general, the total interaction energies are close to the experimental data and other computational results. For comparison, the LMO-EDA and SAPT results are shown in Tables S1 ~ S2 in supporting information (SI). The total interaction energies by experimental approaches and the other theoretical studies are listed in the final column of Table 4. In general, the computed binding energies accord with the results of high level methods quite well. For example, the binding energy of C2H4-Mg2 by hc-DFVB is -71.02 kcal/mol, very close to the result by DFT-SAPT at PBE0/aug-cc-pVDZ level, -71.64 kcal/mol.46 It is not surprising that the total binding energies of the M2+ complexes are larger than the M+ complexes. The total interaction energies follow the order: Mg2+ > Ca2+ > Li+ > Na+ > K+, showing in agreement with the results by references.43,44 As for the individual interacting terms, it is found that all the frozen terms are negative, showing that the electrostatic interactions are important for the cation-π interactions. These frozen terms in VB-EDA approximate to the sum of ∆Eele and ∆Eexrep values in LMO-EDA (Table S1) or the sum of ∆Epol(1) + ∆Eex in SAPT (Table S2). In general, the benzene complexes have larger ∆Epol values than the

ethylene complexes. The CT terms are sensitive to the cationic charges. Furthermore, it is found that in some complexes, the correlation (∆Ecorr) energies are positive, which can be verified in the LMO-EDA and SAPT results collected in SI. These positive contributions are also noticed in the previous studies,4,67 showing that the dynamic correlation does not always tend to enhance the total binding interactions. Among all the ethylene complexes, C2H4-Mg2+ has the largest values for VB-EDA terms. Among these terms, the largest one comes from the polarization interaction. For C2H4-Ca2+ and C2H4-Li+, the polarization interaction is largest. For C2H4-Na+ and C2H4-K+, both frozen and polarization interactions are important, but CT is quite small. For the benzene complexes, because of the larger contribution from the polarization term, each complex has larger binding energy compared to the corresponding ethylene complex. It is interesting that the ∆ECT value of benzene-Mg2+ is smaller than that of C2H4-Mg2+. For benzene-M+ (M = Li, Na and K), both the ∆Efro and ∆Epol terms are important while ∆ECT is small. It is shown that the benzene-M+ interactions are controlled by electrostatic and polarization. 11



Polarization arising from the relaxation of given orbitals provides useful insights into the covalency of cation-π interaction mentioned in the introduction. The active valence orbitals, pnx, pny and pnz (n = 1 or 2) shown in Figure 1, are occupied by the active π electrons, while the remaining orbitals, denoted as σ-core orbitals here, belong to the orbitals for σ back-bone and inner core of atoms. Polarization terms from the σ-core and the valence orbitals, denoted as ∆Epol(σ-core) and ∆Epol(val) respectively, are computed using

eq. 16 and collected in Table 5. It is surprising that in the cation-π complexes except C2H4-Mg2+, the polarization interactions from the valence orbitals are smaller than those from the σ-core orbitals. The two polarizations have different origins. ∆Epol(val) arises from the interplay of valence orbitals while ∆Epol(σ-core) is the relaxation of σ-core orbitals under the external electric field from the metal cation.

Generally speaking, ∆Epol(σ-core) is quite large, covering 30% ~ 50% of the total polarization interactions. As addressed by Grabowski et al,68 the covalency of intermolecular interaction can be related to the delocalization interaction energy (the sum of CT and polarization); while Jiang et al53 pointed out that the covalency is attributed by the non-electrostatic interactions. According to their criteria, with the VB-EDA results, it seems that all the cation-π interactions in this paper, which have large portions of CT and polarization, belong to covalent bonds. However, this judgment ignores the difference of origin in the three interaction terms. The frozen term, which is computed by the ionic structures shown in Table 3, is not directly related to the covalent bond. The CT term, coming from the contribution of covalent structures, does contribute to the covalent bonding. However, only the polarization of the valence orbitals, but not the total polarization, contribute to the covalent bond. Therefore, based on VB-EDA calculations, the covalency index, the portion of interaction terms for the covalent bond in total interaction energy, is written as:

I cov =

∆E pol (val) + ∆E CT ∆E int

The covalency indices of the cation-π complexes by VBSCF are listed in the last column of Table 5. It is found that only the C2H4-Mg2+ complex has a large value of Icov, showing that C2H4-Mg2+ can be 12


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considered as a covalent bond while the other complexes, whose Icov values are smaller than 0.5, cannot be regarded as covalent bonds. Furthermore, except benzene-Mg2+, the Icov values of the M2+ complexes are larger than the corresponding data of the M+ complexes, indicating that the benzene complexes have relatively large portion of covalent characteristic. The correctness of the covalent index can be validated by the VB wave function analysis in the next.

2. The VB wave function The VB weights of the cation-π complexes in their equilibrium distances are shown in Table 6. For the ethylene complexes, the weights of the two VB structures in Table 3 are shown. For all the ethylene complexes except C2H4-Mg2+, the ionic structure, which describes the ionic bonding interaction, is the most important. The weight of the covalent structure Φcov, which shows one electron transfers from ethylene to the metal cation, is small. It accords with the VB-EDA results, showing that the CT terms in C2H4-M (M = Li+, Na+, K+ and Ca2+) are small. For C2H4-Mg2+, the covalent structure is the most important while the ionic structure is the secondary, which highlighting the significant role of CT. Therefore, same to the covalency indices mentioned above, the C2H4-Mg2+ interaction can be regarded as a chemical covalent bond. For the benzene-M+/2+ complexes, the weights of the four important VB structures (Φion, Φcov1, Φcov2, and Φcov3) are shown. For all the benzene complexes except benzene-Mg2+, the total wave function is dominated by the ionic structure. The sum of weights for the three covalent structures cover only 9% ~ 42% of the total VB wave function. The weights of the covalent structures Φcov2 and Φcov3 are smaller than that of Φcov1, showing that the electron transfer via px and py orbitals is weaker than that by pz orbitals. The weights indicate that these benzene-M+/2+ interaction belongs to the ionic interaction. For benzene-Mg2+, the sum of the weights for the three covalent structures are larger than that of the ionic structure, showing that the CT interaction in benzene-Mg2+ is more important. However, it can be found that the most important structure in benzene-Mg2+ is still the ionic structure, covering 43% of the total VB wave function. As such, the CT in benzene-Mg2+ is smaller than that of ethylene-Mg2+, which agrees with the 13



VB-EDA result, shown in Table 4. As such, both the VB weights and the covalent index reveal that only the C2H4-Mg2+ interaction is a covalent bond while the others belong to ionic bonding complexes, governed by electrostatic interaction.

4. Conclusion To provide a widely used valence bond theory based analyses for various intermolecular interactions, ranging from weak non-covalent interactions to strong covalent bonds, a new energy decomposition analysis scheme, called VB-EDA, is presented. The total interaction energy is decomposed into frozen, charge transfer, polarization and correlation terms based on valence bond calculations with localized orbitals. Different from the currently existing MO- or DFT-based EDA schemes, the advantages of VB-EDA are as following: First, polarization and charge transfer in VB-EDA are well separated; second, the analysis results are not sensitive to basis sets, which means that a relatively small basis set is adequate for VB-EDA. Compared to the other VB based EDA schemes, the capability and robustness of VB-EDA is improved, which is validated by the test calculations of the hydrogen bond, ligand bond and covalent bond shown in Table 2. The currently developed VB-EDA scheme is applied to explore the origin of cation-π interaction in a series of cation-π complexes (cation: Li+, Na+, K+, Mg2+ and Ca2+; π = ethylene and benzene). Computational VB-EDA results are in very good agreement with those of LMO-EDA and SAPT, showing the satisfactory accuracy. As for the nature of the cation-π interactions, in the Mg2+ and Ca2+ complexes, polarization interaction is the most important while in the Li+, Na+ and K+ complexes, both frozen and polarization interactions are important. To study the covalent characteristic in cation-π interaction, according to the analysis for the individual VB-EDA terms, a new covalency index is proposed. Based on the covalency index and the VB wave function analysis, it is revealed that the cation-π interactions in the Li+, Na+ and K+ complexes belong to the typical ionic bond while the Mg2+ and Ca2+ complexes have the relatively large portions of the covalent characteristics. However, strictly speaking, only the C2H4-Mg2+ interaction can be regarded as a covalent bond while the other cation-π complexes are typical ionic 14


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complexes. It must be stressed that the covalency of intermolecular interaction cannot be simply determined by non-electrostatic interactions. A large portion of non-electrostatic interactions in total binding energy does not always indicate a covalent bond.

Supporting Information The LMO-EDA and SAPT results of the cation-π interactions.

Acknowledgement This project is supported by the National Natural Science Foundation of China (Nos 21573176 and 21733008).

15



Table 1. The SM and ST sets for single bonding molecules, water-dimer and BH3-NH3 complexes. SM

ST

X-X (X=H or CH3) X-H…Y

BH3-NH3

16


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Table 2. The VB-EDA results by hc-DFVB with various basis sets (kcal/mol).

-0.85 -0.96 -0.95 -1.15 -1.07

∆Ecorr B3LYP M06 -1.98 -3.35 -2.12 -3.00 -1.19 -1.98 -1.62 -2.84 -0.96 -2.40

∆Eint B3LYP -6.38 -7.00 -5.75 -5.87 -4.41

-13.93 -16.03 -16.43 -16.10 -16.68

-25.11 -27.17 -27.54 -27.49 -28.56

-15.70 -15.05 -14.77 -15.33 -13.37

-14.76 -17.56 -17.27 -17.99 -19.09

-107.07 -103.27 -102.65 -102.92 -102.49

-106.13 -105.78 -105.15 -105.58 -106.21

13.10 14.25 17.54 16.58 25.00

-17.52 -16.01 -16.77 -17.29 -16.06

-35.21 -37.76 -39.61 -38.92 -47.81

-1.02 -0.20 0.77 -0.22 2.78

-0.79 -0.62 0.51 0.51 2.39

-40.65 -39.72 -38.07 -39.85 -36.09

-40.42 -40.14 -38.33 -39.12 -36.48

-19.47 -19.25 -17.90 -17.19 -17.59

-26.47 -24.61 -24.93 -21.08 -18.97

-31.85 -37.75 -38.30 -41.43 -45.07

5.32 4.77 4.47 2.54 3.02

6.13 8.10 10.31 9.65 10.95

-72.47 -76.84 -76.66 -77.16 -78.61

-71.66 -73.51 -71.02 -70.05 -70.68

∆Efro

∆ECT

∆Epol

(H2O)2

6-31G* 6-311G* 6-311G** cc-pVDZ cc-pVTZ

-2.37 -2.93 -2.65 -2.22 -1.67

-1.18 -0.99 -0.96 -0.88 -0.71

CH3-CH3


-52.33 -45.02 -43.91 -44.00 -43.88

BH3-NH3


C2H4-Mg2+


17


M06 -7.75 -7.88 -6.54 -7.09 -5.85


Page 18 of 28

Table 3. The SM and ST sets for the cation-π complexes in this paper. SM Φion

ST Φcov

C2H4-M+/2+

Φion

Φcov1

C6H6-M+/2+

18


Φcov2

Φcov3

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Table 4. The VB-EDA results of the cation-π interactions (kcal/mol). C2H4-Li+

∆Efro -8.29

∆ECT -2.46

∆Epol ∆Ecorr ∆Eint(this work) -13.48 0.53 -23.73

-19.7 -20.59 -20.9

∆Eint(ref) MP2/6-311++G** 49 DFT-SAPT//PBE0/aug-cc-pVDZ46 TPSS/ET TZ1diff 47

C2H4-Na+

-6.56

-2.13

-6.70

2.69

-12.70

-14.4 -14.4 -12.96

MP2/aV5Z 45 CCSD(T)//CBS 45 DFT-SAPT//PBE0/aug-cc-pVDZ 46

C2H4-K+

-4.55

-1.21

-3.90

3.43

-6.20

-9.25 -8.0 -6.8

MP2/aug-cc-pVDZ 50 PBE/aug-TZ2P 47 TPSS/ET TZ1diff 47

C2H4-Mg2+

-17.90 -24.93 -38.30

10.31

-71.02

-68.8 -71.64

MP2/6-311++G** 49 DFT-SAPT//PBE0/aug-cc-pVDZ 46

C2H4-Ca2+

-8.09

-6.55

-26.00

3.36

-37.29

-41.10

DFT-SAPT//PBE0/aug-cc-pVDZ 46

benzene-Li+

-11.55

-7.62

-22.31

4.16

-37.33

-36.70

DFT-SAPT//PBE0/aug-cc-pVDZ 46 experimental data56

-39.3±3.3 -35.8 benzene-Na+

-9.77

-3.34

-10.92

0.72

-23.24

-22.50 -23.2±1.4 -22.2

benzene-K+

benzene-Mg2+

-6.93

-2.10

-6.07

-15.62 -19.50 -81.04

-1.64

3.39

-16.37

-112.46

-17.9

CCSD(T)/6-311++G** 48 DFT-SAPT//PBE0/aug-cc-pVDZ 46 experimental data69 CCSD(T)/6-311++G** 48 TPSS/ET TZ1diff 47 experimental data56

-17.7±1.0 -16.5

CCSD(T)/6-311++G** 48

-114.41

DFT-SAPT// PBE0/aug-cc-pVDZ 46

19


The Journal of Physical Chemistry 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

benzene-Ca2+

-7.61

-10.65 -58.74

-6.59

-83.59

Page 20 of 28

-116.76

MP2/aug-cc-pVTZ 16

-68.33 -77.80

DFT-SAPT//PBE0/aug-cc-pVDZ 46 B3LYP/6-311++G**43

20


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Table 5. The polarizations in the cation-π interactions by VB-EDA (kcal/mol).

+

C2H4-Li C2H4-Na+ C2H4-K+ C2H4-Mg2+ C2H4-Ca2+ benzene-Li+ benzene-Na+ benzene-K+ benzene-Mg2+ benzene-Ca2+

∆Epol -13.48 -6.70 -3.90 -38.30 -25.98 -22.31 -10.92 -6.07 -81.04 -58.74

∆Epol(σ-core) -4.39 -2.50 -1.45 -20.49 -11.48 -11.08 -5.91 -3.44 -40.76 -25.40

∆Epol(val) -3.87 -2.19 -1.26 -23.35 -7.83 -7.98 -5.33 -3.21 -32.48 -20.50

21


Covalency index 0.261 0.281 0.256 0.595 0.354 0.376 0.361 0.352 0.447 0.405


Page 22 of 28

Table 6. The VB weights of the cation-π complexes.

+

C2H4-Li C2H4-Na+ C2H4-K+ C2H4-Mg2+ C2H4-Ca2+ benzene-Li+ benzene-Na+ benzene-K+ benzene-Mg2+ benzene-Ca2+

Φion 0.721 0.840 0.915 0.379 0.726 0.661 0.861 0.922 0.429 0.560

Φcov1 0.306 0.173 0.090 0.595 0.283 0.169 0.070 0.045 0.305 0.160

22


Φcov2

Φcov3

0.092 0.033 0.020 0.129 0.130

0.092 0.033 0.020 0.129 0.130

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Figure 1. The VB orbital representation for VB structures: (a) ethylene-M+/2+ and (b) benzene-M+/2+. .

px

py

pz

(a) +/2+

ethylene

M

(b) +/2+

M

benzene

23



Figure 2. The optimized geometries of the cation-π complexes by M06/6-311++G**

24


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TOC Graphic

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Valence Bond Based Energy Decomposition Analysis Scheme and Its

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