Ab initio calculation of natural hybrid orbitals

THEO CHEM ELSEVIER

Journal of Molecular Structure (Theochem) 313 (1994) 231-236

Ab initio calculation of natural hybrid orbitals Shi-Yong Ye a'b, Chang-Guo Zhan a'*, Jian Wan u, Chang-Jun Zhang u aDepartment o['Chemistry, Central China Normal University, Wuhan 430070, People's Republie of China bDepartment Of Chemist~3', Anhui Normal University, Wuhu, Anhui 241000, People'.s" Republic Of China

Received 6 September 1993: revised 27 November 1993;accepted 6 December 1993

Abstract

The natural hybrid orbital (NHO) procedure is tested at the ab initio level by use of the density matrix in a L6wdin orthogonalized atomic orbital basis. The direct NHO calculation based on the whole density matrix also includes the hybridization of the inner atomic orbitals, and the NHO calculation employing the valence orbital part of the density matrix considers only the hybridization of the valence atomic orbitals. The numerical results obtained by using the NHO calculations based on the ab initio calculation with an STO-3G basis show that the components of the s atomic orbitals in the NHOs obtained from the two kinds of NHO calculations are very close to each other, and that the two kinds of NHOs have excellent correlation with the nuclear spin spin coupling constants.

1. Calculation scheme

As is well known, the concept of orbital hybridization was first proposed by Pauling [1]. Later, many others have presented systematic methods for constructing hybrid atomic orbitals [2 32]. These methods have been employed to elucidate molecular structures and to connect the s-character of hybrid atomic orbitals with many physicochemical properties, especially with the nuclear spin spin coupling constants, bond stretching frequencies and thermodynamic proton acidities of a large number of molecules [17,20,29-32]• In 1980, Foster and Weinhold [19] proposed a method, called the "natural hybrid orbital" (NHO) * Corresponding author. This project was supported by the National Natural Science Foundation of China and the Excellent Young University Teacher's Foundation of State Education Commission of China.

method, for extracting a unique set of directed orthonormal hybrid atomic orbitals from the information contained in the first-order density matrix for a given molecule, thereby constructing its "Lewis structure" in an a priori manner. Although closely related to the scheme introduced by McWeeny and Del Re [11] for extracting hybrid atomic orbitals, the N H O procedure proposed by Foster and Weinhold [19] requires no simultaneous minimization of the SCF energy expression. According to the N H O method, the first-order density matrix is partitioned into atomic subblocks [19] PAA

PAB

PAL

PBA

PBB

PBL

(1)

p =

0166-1280/94/$07.00 ~d::1994 Elsevier Science B.V. All rights reserved SSDI 0166-1280(94)03705-P

PLA

PLB

PLL

S.-Y. Ye et al./J. Mol. Struct. (Theochem) 313 (1994j 231 236

232

Table 1 Calculated coefficients of s orbitals in the natural hybrid orbitals and the occupation numbers No.

Molecule

Bond a

Center

Valence orbitals

All orbitals Cls

t'2s

Hi

l'2s

Hi

1

CH 4

CH

C

0.0599

0.4964

2.0000

0.5000

2.0000

2

CH3F

CF

C F C

0.0648 0.0438 0.0614

0.4550 0.3596 0.5091

2.0000

0.4590 0.3598 0.5130

2.0000

3

CHF~

C F C

0.0658 0.0471 0.0639

0.4791 0.3962 0.5427

2.0000

0.4833 0.3967 0.5470

2.0000

4

CH2F 2

C F C

0.0653 0.0453 0.0620

0.4662 0.3758 0.5241

2.0000

0.47(/2 0.3762 0.5282

2.0000

0.0457 0.0397 0.0457 0.0397 0.0704 0.0468 0.0714

0.4195 0.3850 0.4195 0.3850 0.5203 0.3902 0.6023

2.0000

2.0000

CH CF CH CF CH

2.0000

2.0000

2.0000

2.0000

2.0000

CH

C O C O C F C

2.0000

0.4222 0.3857 0.4222 (I.3857 0.5245 0.3907 0.6067

6

CH-CH

CC((y) CC(Tr) CH

C C C

0.0675 0.0000 0.0806

0.7305 0.0000 0.6733

2.000(/ 2.0000 1.9988

0.7351 0.0000 0.6780

2.0000 2.0000 1.9987

7

CH 2 - C H 2

CC(o') CC(Tr) CH

C C C

0.0660 0.0000 0.0668

0.6091 0.0000 0.5549

2.0000 2.0000 1.9937

0.6127 0.0000 0.5589

2.0000 2.0000 1.9937

8

CH3CH 3

CC CH

C C

0.0629 0.0598

0.4882 0.4990

2.0000 1.9962

0.4919 0.5027

2.0000 1.9962

9

Cyclopropane

CC CH

C C

0.0558 0.0650

0.4358 0.5503

1.9961 1.9957

0.4389 0.5544

1.996(I 1.9957

10

Cyclobutane

CC CH

C C

0.0614 0.0617

0.4802 0.5117

1.9957 1.9949

0.4839 0.5156

1.9956 1.9949

11

H20

OH

O

0.0485

0.4376

2.0000

0.4384

2.0000

12

NH 3

NH

N

0.0545

0.4916

2.000(/

/).4938

2.00(1(/

13

H;C=CF 2

*CC(c9

*C C *C,C C F *C

0.0631 0.0678 0.0000 0.0697 0.0481 0.0684

0.5858 0.6654 0.0000 0.5125 0.4067 0.5607

2.0000

2.0000

1.9856

0.5773 0.6551 0.0000 0.5168 0.407 [ 0.5649

5

HCO(F)

CO(aTr I )

2.0000

CO(trot2 ) CF

*CC(Tr) CF *CH

2.0000 2.0000

2.0000 1.9975

2.0000 2.0000 2.0000

2.0000 1.9975 1.9856

14

HF

HF

F

0.0437

0.3906

2.0000

0.3910

2.0000

15

HC-N

CN(~r)

C N C,N C

0.0642 0.0551 0.0000 0.0832

0.7052 0.6225 0.0000 0.7011

2.0000

0.6988 0.6235 0.0000 0.7057

2.0000

C O O C C

0.0641 0.0509 0.0496 0.0617 0.0601

0.4640 0.4203 0.4463 0.5125 0.5035

2.0000

0.4679 0.4214 0.4473 0.5164 0.5072

2.0000

CN(Tr) CH 16

CH3OH

CO OH CH(s) CH(a)

2.0000 2.0000

1.9930 1.9975 2.0000

2.0000 2.0000

1.9929 1.9975 2.0000

S.-Y. Ye et al./J. Mol. Struct. (Theochem) 313 (1994) 231-236

233

Table l (continued) No.

17

Molecule

H2C=O

Bond a

CO(a) CO(~) CH

18

CH3NH 2

CN NH CH(s) CH(a)

19

H2C=*C=CH 2

*CC(a) *CC(~) CH

20

CH~CmN

*CC *CN(a) *CN(~) CH

21

CH3OCH 3

CO CH(s) CH(a)

22

H*C~CF

*CC(o~I) *CC(o~2) *CC(on3) *CH CF

23

*CH3C~+CH

+CC(o) +CC(~) *CC *CH +CH

24

CH~CHO

*CO(a) *CO(~) C*C *CH CH(s) CH(a)

Center

All orbitals

Valence orbitals

Cls

C2s

ni

C2s

ni

C O C,O C

0.0648 0.0560 0.0000 0.0696

0.5734 0.5355 0.0000 0.5733

2.0000

0.5770 0.5364 0.0000 0.5775

2.0000

C N N C C

0.0632 0.0603 0.0564 0.0596 0.0607

0.4714 0.4980 0.5081 0.4999 0.5065

2.0000

C *C *C,C C

0.0742 0.0648 0.0000 0.0672

0.7032 0.6006 0.0000 0.5595

1.9997

C *C *C N *C,N C

0.0616 0.0846 0.064t 0.0552 0.0000 0.0602

0.4898 0.6990 0.7074 0.6192 0.0000 0.4985

2.0000

C O C C

0.0638 0.0529 0.0618 0.0598

0.4651 0.4392 0.5144 0.5021

1.9927

*C C *C C *C C *C C F

0.0590 0.0613 0.0008 0.0016 0.0350 0.0354 0.0817 0.0817 0.0494

0.6190 0.6796 0.0009 0.0176 0.3677 0.3931 0.6858 0.6100 0.4249

C +C +C,C *C C *C +C

0.0677 0.0681 0.0000 0.0624 0.0822 0.0600 0.0808

*C O *C,O *C C *C C C

0.0650 0.0559 0.0000 0.0716 0.0624 0.0683 0.0593 0.0594

a s means the C - H bond is in the plane of the molecular geometry. a means the C H bond is out of the plane of the molecular geometry.

2.0000 2.0000

2.0000 2.0000

0.4752 0.5005 0.5106 0.5035 0.5102

2.0000

0.7071 0.6039 0.0000 0.5636

1.9997

0.4933 0.7035 0.7107 0.6202 0.0000 0.5022

2.0000

0.4690 0.4403 0.5183 0.5057

1.9926

2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9977 1.9994

0.6562 0.7067 0.0043 0.0375 0.3040 0.3477 0,6907 0,6150 0.4254

2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9976 1.9993

0.7350 0.7306 0.0000 0.5009 0.6699 0.4948 0.6747

1.9999

0.7382 0.7337 0.0000 0.5045 0.6746 0.4985 0.6794

1.9999

0.5726 0.5307 0.0000 0.5805 0.4933 0.5667 0.4973 0.4975

1.9990

1.9957 2.0000 1.9971

1.9827 1.9827

1.9999 1.9932 1.9827

1.9966 1.9994

1.9911 1.9987 1.9845 1.9989

1.9950 2.0000 1.9980 1.9939 1.9859

0.5763 0.5315 0.0000 0.5847 0.4969 0.5710 0.5009 0.50ll

1.9957 2.0000 1.9971

1.9827 1.9827

1.9999 1.9932 1.9827

1.9966 1.9994

1.9911 1.9987 1.9846 1.9988 1.9990 1.9950 2.0000 1.9980 1.9939 1.9860

234

S.-Y. Ye et al.,J. Mol. Struct. (Theochem) 313 (1994) 231 236

Correspondingly, the overlap matrix in the atomic orbital basis set is also partitioned into atomic subblocks. As the first step of the N H O procedure, one can obtain a kind of bond orbital from the solution of the following equations:

PAAhl A) = n}A)SAAhlA)

(2)

/3(AL)hi(Ak) = HIA)s(AL)h}AL)

(3)

By solving Eq. (2), one expects an eigenvalue nl A) ~ 2 for each doubly occupied lone-pair orbital of atom A. By use of the eigenvalues and eigenvectors for the lone-pair orbitals of atoms A and L, matrix p(AL) can be formed from

L PEA PLL J

(4)

where

/SAA : - P A A -

lone-pair Z rllA)hlA)hlA)i

(5)

By diagonalizing the matrix p(AL), one can get an eigenvalue nl A) ~ 2 for each doubly occupied covalent bond between atoms A and L. The eigenvector t~fAL)

//-i(AL) : []~)A)] L)

According to the usual concept of orbital hybridization, all the orthonormal hybrid orbitals of an atom should be expressed in linear combinations of a set of orthonormal atomic orbitals. That is to say, the coefficient matrix of the hybrid orbitals for each atom should be self-adjoint. However, in ab initio molecular orbital calculations, the initial atomic orbitals adopted for each atom are nonorthogonal. In order to be in accordance with the customary concept of hybridization and for ease of calculation of the components of the atomic orbitals in the constructed hybrid orbitals, we can use a minimal basis set in the ab initio molecular orbital calculation and employ the density matrix in a LOwdin orthogonalized atomic orbital basis to perform the N H O procedure. If we use P0 and S0 to represent the standard density matrix and overlap matrix in the non-orthogonalized atomic orbital basis for the molecule, respectively, then the density matrix P used in this work can be written as

P : s /eposU

(v)

In addition, the overlap matrix S used here is a unit matrix, and Eqs. (2) and (3) become the standard eigenvalue problems.

(6) 2. Calculation results and discussion

associated with the eigenvalue nl A) ~ 2 is chosen to represent the bond orbital between a directed hybrid ~ a / o n center A and the corresponding hybrid ha(L/ on center L. In this way, one can get all the initial hybrids for each center. Finally, since the hybrids found in this manner are generally nonorthogonal, these initial hybrids for each center are symmetrically orthogonalized in order to obtain the final NHOs for each center. It follows that the NHOs are determined completely by the density matrix, and that the best NHOs should depend on ab initio calculation of the density matrix. The N H O procedure has been performed by use of the density matrix obtained from some semiempirical molecular orbital calculations. In this short report, we attempt to perform the N H O procedure by use of the density matrix obtained from the ab initio calculation.

If/3(AL) in Eq. (3) is the submatrix of P over the basis of all LOwdin orthogonatized atomic orbitals of atoms A and L, then the hybrids/~}A) of atom A and /~L) of atom L are linear combinations of all the atomic orbitals of atoms A and L, respectively. In this case, the hybridization includes not only the valence atomic orbitals, but also the inner atomic orbitals. However, because of the larger difference between the orbital energies of inner and valence orbitals, the components of the inner orbitals in the NHOs should be small. As examples, we have adopted the STO-3G basis to perform the whole ab initio N H O procedure for some typical molecules. Part of the results obtained are listed in Table 1. We can see from Table 1 that the components of the inner orbitals (ls) in the obtained NHOs are less than 0.72%.

235

S.-Y. Ye et al./J. Mol. Struct. (Theochem) 313 (1994) 231-236

If we consider only the hybridization of the valence atomic orbitals, then p(AL) in Eq. (3) is the valence orbital part of the submatrix over the basis of all L6wdin orthogonalized atomic orbitals of atoms A and L. The N H O calculation results for the valence atomic orbitals are also listed in Table 1 for comparison with those for all the atomic orbitats. Table 1 reveals that the total components of the s atomic orbitals in the NHOs for the two kinds of calculations are very close to each other. This means that c~s(all) +

c2,(all)

~ c~(valence)

(8)

For example, for the carbon atom in methane, c~s(all) + C~s(all) = 0.0036 + 0.2464 = 0.2500 and c~(valence) = 0.2500. For the nitrogen atom in ammonia, c~(all) +c2s(all) = 0.0030 + 0.2417 = 0.2447 and c~(valence)=0.2438. Correspondingly, as indicated in the table, the occupation numbers of the bond orbitals obtained from the two kinds of calculations are also very close to each other and very near or equal to the expected value of two. We have also examined the correlativity of the NHOs obtained from the two kinds of calculations with C - H and C C spin-coupling constants through analyzing the same molecules as those analyzed by use of the maximum bond order hybrid orbital procedure [31]. By applying the least-squares process for hydrocarbons with the experimental Jci~ values taken from ref. 31, we get the following good linear relationships: arCH =

5.892[ls% + 2s%] - 20.12

(R = 0.9958; SD = 4.88 Hz)

(9)

arch = 5.89712s% (valence)] - 20.31 (R = 0.9959; SD = 4.83Hz)

(10)

Similar relationships have also been obtained for Jcc values. For the single C C bonds with the experimental Jcc values from ref. 31, we get

For the multiple C C bonds with the experimental Jcc values from the previous paper we obtain Jcc = 6.710[ls% + 2S%]c 1[ls% + 2S%]c2 -21.56(R=0.9986;

SD=2.67Hz)

(13)

Jcc = 6.65812s%(valence)]cl [2s%(valence)]c2 -21.83 (R=0.9986; SD=2.72Hz)

(14)

These relationships are quite satisfactory. It follows that the two kinds of N H O calculations at the ab initio level are qualified for calculation of the nuclear spin spin coupling constants, and that the results obtained from the two kinds of calculations are also very close to each other.

3. S u m m a r y and conclusion

The N H O procedure has been tested at the ab initio level. The density matrix needed for the N H O calculation is the one in the L6wdin orthogonalized atomic orbital basis. The direct N H O calculation based on the whole density matrix includes the hybridization of not only the valence atomic orbitals, but also the inner atomic orbitals. The N H O calculation based on the valence orbital part of the density matrix considers only the hybridization of the valence atomic orbitals. We have examined the N H O results based on the ab initio calculation with the STO-3G basis and can conclude that the total components of the s atomic orbitals in the NHOs obtained from the two kinds of N H O calculations are very close to each other, and that the two kinds of NHOs all have excellent correlation with the nuclear spin spin coupling constants. In order to be in accordance with the usual idea of hybridization, we prefer to consider only the hybridization of the valence atomic orbitals in practical N H O calculations.

J c c = 3.511[ls% + 2S%]c I [ls% + 2S%]c2 + 13.83

(R = 0.9603; SD = 2.53 Hz)

(11)

Jc(: = 3.52512s%(valence)]cJ [2s%(valence)]c2 -~ 13.77 (R = 0.9603; SD = 2.53Hz)

(12)

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236

S.-Y. Ye et al./J. Mol. Struct. (Theochem) 313 (1994) 231 236

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