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Maximum overlap population principle and direct calculation of bond order, valence and atomic charge
Journal of Molecular Structure (Theochem), 231 (1991) 47-56 Elsevier Science Publishers B.V., Amsterdam
47
MAXIMUM OVERLAP POPULATION PRINCIPLE AND DIRECT CALCULATION OF BOND ORDER, VALENCE AND ATOMIC CHARGE
CHANG-GUO ZHAN Department of Chemistry, Central China Normal University, Wuhan 430070 (People’s Rep. of China) (Received 5 October 1990)
ABSTRACT A maximum overlap population principle is introduced to give a very simple approximate scheme for calculating bond order, atomic valence and atomic charge directly from the eigenvectors of the A0 overlap matrix without performing the ordinary molecular orbital calculation. The calculated results are close to or on the whole coincident with those obtained from the canonical molecular orbitals constructed by use of the ab initio calculation, which shows that the calculation scheme suggested in this paper is feasible.
INTRODUCTION
As is well known, calculation methods for bond order and valence have been studied by many scientists, such as Coulson [ 11, Pariser and Parr [ 21, Pople [ 3 1, Mulliken [ 41, Wiberg [ 51, Armstrong et al. [ 61, Mayer [ 7-91, Jug [lo131 and many others [ 14-201. Bond order and valence have become strong tools for studying molecular structure and chemical reactivity. Generally, calculation of bond order and valence is on the basis of molecular orbital data. By use of the calculated molecular orbital coefficients, one can calculate the bond orders and valences defined by different authors. Mayer’s definitions [ 71 of bond order and valence are based on the Mulliken population analysis method [ 41. Jug’s maximum bond order [lo] between two atoms is defined as the maximum of the trace of elements of the two atomic orbital (A0 ) basis sets coupled through the density operator. These pioneering works inspired us to introduce a new idea for studying these problems. In the present paper we attempt to present a maximum overlap population principle in which the molecular orbitals used for the population analysis and for calculation of the bond order, valence and atomic charge are determined by a maximum overlap population criterion
0166-1236/91/$03.50
0 1991-
Elsevier Science Publishers B.V.
48
P”)= F<&P$! = F<&C$C,iSpv =maximum
(1)
where P$ is the overlap population [21] between @h and vth AOs with one electron occupying the ith molecular orbital. Clearly, Pci) is the total overlap population of the ith molecular orbital. The molecular orbitals determined by eqn. ( 1) are called “maximum overlap population orbitals”. By use of the maximum overlap population orbitals and numbers of electrons occupying them, the bond order, atomic valence and atomic charge may be calculated very easily. The main purpose of this paper is to suggest a very simple approximate scheme for calculating directly the bond order, atomic valence and atomic charge without performing the customary molecular orbital calculation. CONSTRUCTION OF THE MAXIMUM OVERLAP POPULATION ORBITALS
Let us denote a normalized non-orthogonal row matrix
A0 basis set in a molecule by a (1)
@=(k)lW”‘MnN The corresponding orthonormalized be expressed as
maximum overlap population orbitals can
(2) By use of the normalization condition of Ivi), P w can be expressed as pci,_~$cZicyisfiu - T
(3) Tc;icvispu
Generally, eqn. (1) can be satisfied by using the conditions dP(i)=O
j= 1, 2, .... n
dCji
(4)
From eqn. (4) we can obtain C C,iSjv -2P”‘CC”iSj, (vZj)
Y
=O
that is (S-I)C. ‘ =2Pci’sc. L
(5)
where S= a+@, I is a unit matrix, and Ci is a column matrix formed from the ith column of matrix C. Equation (5 ) can be expanded as f (S-I)C=SCD
(6)
49
where D is a diagonal matrix, and the ith diagonal element of D is Pci). Rewriting eqn. (6) as +S-i’z(S_I)S
- 1’2S1’2CzSif2CD
that is
(7)
f (I-S-1)S”2CsS1’2CD it follows that S”?! is a unitary matrix, an eigenvector matrix of matrix $(1-S-‘). It is very easy to prove that 4 (I-S - ’ ) commutes with overlap matrix S. Therefore, S1’2C is also an eigenvector matrix of S. Let the eigenvalue equation of S be SC’ = CA
(3)
where C’ is an eigenvector matrix of S, and A is the corresponding eigenvalue matrix. We can choose ZP2C as C’, i.e. S’DC- - C’
(9)
Equation (9) gives C=S - 1’2CI
(IO)
By use of eqn. (8)) matrix S - 1’2 can be expressed as S--‘2=CIA-1’2C)+
(11)
Substitution of eqn. (11) into eqn. (10) gives C=C’A-
1’2
(12)
Substitution of eqn. ( 12 ) into eqn. (2) gives
Y~IBC’A-“~
(13)
It follows that the maximum overlap population orbitals are just the orthonormalized orbitals obtained by making use of the canonical orthogonalization [ 221 of the A0 basis set. Their symmetry properties have been studied in the preceding paper [ 211. It should be pointed out that eqn. (8) has been obtained by many others [ 23-251, and has been employed to construct the maximum overlap atomic and molecular orbitals [ 23 1, (PC’, which are somewhat different from the maximum overlap population orbitals, @C’A-“2. DETERMINATION
OF THE ELECTRON OCCUPATION NUMBERS
Because Pci) is the total overlap population of the ith maximum overlap population orbital 1vi), the orbital energy of 1vi) is closely related to Pci). Generally, the bigger the value of Pfi), the better the bonding and therefore
50
the lower the orbital energy. It follows that electrons should occupy the orbitals with the bigger values of Pci) if the considered molecule is in its ground state. Now let us probe the relationship between P w and the eigenvalue 1 i of overlap matrix S. Substitution of eqn. (9) into eqn. (7) gives $(I-s-l)C’=C’D
(7a)
Equation (7a) can be rewritten as IC’ - S - ‘C’ - 2CD C’(I-2D)=S-‘C’
SC’=C’(I-20)-l
(14)
Comparison of eqns. (8) and (14) gives A= (I-20)-’ A--=1-20
(15)
or
D=f(I-A-‘)
(16)
Equation (16) shows that p (9
=$(I-l)
(17)
Li
It follows that the bigger the value of izi,the bigger the value of Pti) and therefore the lower the orbital energy. In brief, one can determine the numbers of electrons occupying the maximum overlap population orbitals by use of the eigenvalues lis of the overlap matrix S. From the electron occupation numbers and the coefficient matrix of the maximum overlap population orbitals one can obtain the density matrix P [ 161 which can be used for calculating bond order, valence and atomic charge. CALCULATION OF BOND ORDER AND ATOMIC VALENCE
In the present paper, we adopt Mayer’s definitions of bond order and valence [ 731 and consider only the closed shell systems. The sum of electrons occupying the all maximum overlap population orbitals is expressed as [ 141
N= ftr(PS)‘=
‘6
5 (PS).bWS)b,
Expanding the right-hand side of eqn. (18) we obtain
(18)
51
A
b
a
@#A)
(19)
@+A)
where BAn is the bond order of A-B bond BAB
= $
(~s)ab(~s)b,
$
(20)
The valence VAof atom A is defined as
VA=
c
BAB
(21)
(B#A)
Equations (20) and (21) show that so long as the elements of matrix PS are known, the bond order and atomic valence can be evaluated easily. The element Pajcan be expressed as Paj = CniC,iCz
= CniChiCjtllli
i
(22)
i
where ni is the electron occupation number of the ith maximum overlap population orbital. From eqn. (8) we obtain S=C’AC’+
(23)
that is (24) From eqns. (22) and (24) we can obtain (PS),b =
CPajsjb j
= ccc
(ni/ni)Cb;C~C~&pC&
jir
=CC(nin,/ni)CbiCb*,(CC7Cl,) i
P
=xX
j
(25)
t&J-p/~i)ChiCb*,&p
Equation (25) shows that PS appearing in this paper is a Hermitian matrix. Substitution of eqn. (25) into eqn. (20) gives BAB=$$~~W&C~~!~
(26)
52
Substitution of eqn. (26) into eqn. (21) gives
It follows that one can easily evaluate the bond order and atomic valence by using eqns. (26) and (27) so long as the electron occupation numbers nis and the eigenvectors of the overlap matrix S are known. CALCULATION OF ATOMIC CHARGE
Generally, the contributions of bond order BAB to the electron charges of atoms A and B are considered equal. In this way, the charge of atom A can be BAB ) /2 [ 141. Many other methods [ 26-30 ] for calcudefined as (BAA + c @+A) lation of atomic charge have also been proposed. In order to calculate atomic charge from the maximum overlap population orbitals, we introduce a weighted distribution factor f in which satisfies the conditions f&+f&j=l
w3)
and (29)
fiB=fiA
By use of the factor, eqn. (19) can be rewritten as N=iz]&A+
c
A
BAB(f
kB+f !&>I
CB:A) =?[~BAA+
&
(30) BABfhl
(B#A)
From eqn. (30) the charge of atom A can be defined as QA=@AA+
c
Bmfh
(31)
CB:A)
As a test, for practical calculation in the present paper we choose the weighted distribution factor f & as f
iBZIx Ii
+I;
(32)
where IA is the average value of the ionization energies of all valence electrons of atom A.
53 RESULTS FROM LdWDIN ORTHONORMALIZED
A0 BASIS SET
Generally, the matrix element (PS), is closely related to the A0 basis set employed, and so are the bond order, valence and atomic charge. Hence, the values of the bond order, valence and atomic charge obtained from the Lijwdin orthonormalized A0 basis set may be different from those obtained from the normalized non-orthogonal A0 basis set. In this section, we will demonstrate the interesting fact that, according to the maximum overlap population principle, the results obtained from the Lowdin orthonormalized A0 basis set are the same as those from the normalized non-orthogonal A0 basis set. From eqn. (13) we obtain y_cpc’A-‘/2c’+c’_~~c’
(33)
where @” is the orthonormalized A0 basis set obtained b!Yuse of the Lijwdin orthogonalization process [ 311, i.e. @” _@(-J’A-lDc’+ -
(34)
By use of cPOinstead of @, the matrix elements Pti, Sj, and (PS), become Paj =
(224
C&Chic; i
Sj, = (W+W)jb
respectively
=Sjb
(244
CTtiCbiC$ i
(254
and (PS),, =Pa* =
It follows from eqns. (25) and (25a) that the values of the matrix element (PS), obtained from the two A0 basis sets are the same. Furthermore, according to the calculation scheme expressed in the previous sections, eqns. (20), (21), (31) and (19) show, respectively, that the values of the bond order, valence and atomic charge obtained from the two A0 basis sets are the same so long as the values of (PS), from the two basis sets are the same. Therefore, according to the maximum overlap population principle the bond order, atomic valence and atomic charge can be evaluated by using eqns. (26), (27) and (31) , respectively, no matter whether Liiwdin orthonormalized or the normalized non-orthogonal A0 basis set is employed. NUMERICAL RESULTS
As an illustration, we consider only the valence AOs, and use Slater orbitals. The used orbital exponents and valence A0 ionization energies are the default
54
values in the EHMO program [32]. By use of eqns. (26), (27) and (31) the bond orders, atomic valence and atomic charges for many molecules have been obtained from the eigenvectors of the overlap matrix and the electron occupation numbers. Part of the calculation results are listed in Table 1 to enable comparison with other theoretical results obtained from the canonical molecular orbitals. As seen in Table 1, the values of the bond order and atomic valence obtained from the eigenvectors of the A0 overlap matrix are close to those from the canonical molecular orbitals constructed by use of the ab initio TABLE 1 Bond orders, atomic valences and net atomic charges of some small molecules’ Molecule
Bond
Bond order Calc.
Ab initiob
HZ N2 F* HF
HH NN FF HF
1.000 3.000 1.000 1.000
1.00 3.00 1.00 0.96
co
co
3.000
2.52
Hz0
HO
0.957
0.96
NH,
NH
0.963
0.96
HCN
HC CN
0.993
3.000
0.97 2.99
CHI
CH
0.976
0.99
C,Hd
CH cc CH cc CH co
0.964
0.98
2.000 0.990 2.989 0.966 2.042
2.01 0.98 3.00 0.94 2.03
0.988
0.87 0.82 0.91 0.92
&Hz H&O
CHsOH
co OH CH, CH,.
0.961 0.970 0.974
Atom
H N F H F C 0 0 H N H H C N C H C H C H H C 0 C 0 H H, H,.
Atomic valence
Net atomic charge
Calc.
Ab initiob
Calc.
1.000 3.000 1.000 1.000 1.000 3.000 3.000 1.915 0.998 2.888 0.997 1.000 3.993 3.007 3.905 0.998 3.947 1.000 3.986 1.000 0.999 3.974 2.105 3.935 1.991 0.998 1.000 1.000
1.00 3.00 1.00 0.96
0.000 0.000 0.000 0.262 -0.262 0.339 -0.339 -0.475 0.237 - 0.598 0.199 0.088 0.096 -0.184 - 0.525 0.131 -0.212 0.106 - 0.096 0.096 0.071 0.034 -0.177 -0.218 - 0.342 0.237 0.106 0.111
2.52 1.92 0.97 2.88 0.99 0.98 3.96 3.00 3.96 1.00 3.97 1.00 3.98 0.99 0.96 3.91 2.11 3.61 1.72 0.83 0.91 0.91
“All ab initio calculation results are based on the Mulliken population analysis. bThe ab initio calculation results of CHBOH are from ref. 33, and the others from ref. 14. ‘The ab initio calculation results are from ref. 33.
Ab initio”
-0.67 0.34
- 0.66 0.17
0.06 0.06 -0.19 -0.27 -0.68 0.37 0.18 0.22
55
calculation, and the atomic charges obtained from the eigenvectors are on the whole coincident with the ab initio calculation results. Therefore, the calculation scheme proposed in the present paper is indeed a reasonable one for calculation of the bond order, atomic valence and atomic charge. CONCLUSION
A maximum overlap population principle has been introduced to give a very simple approximate scheme for calculation of bond order, atomic valence and atomic charge. The maximum overlap population orbitals determined by the maximum overlap population criterion are the orthonormalized orbitals obtained by making use of the canonical orthogonalization of the A0 basis set. By use of the scheme one can obtain satisfactory calculation results directly from the eigenvectors of the A0 overlap matrix without performing the ordinary molecular orbital calculation. No matter whether the normalized nonorthogonal or Lijwdin orthonormalized A0 basis set is used, the values of the bond order, atomic valence and atomic charge are the same and can be evaluatedbyusingeqns. (26), (27) and (31).
REFERENCES 1 C.A. Coulson, Proc. R. Sot. London, Ser. A, 169 (1938) 413. 2 R. Pariser and R. Parr, J. Chem. Phys., 21 (1953) 466,762. 3 J.A. Pople, Trans. Faraday Sot., 49 (1953) 1375. 4 R.S. Mulliken, J. Chem. Phys., 23 (1955) 1833. 5 K.A. Wiberg, Tetrahedron, 24 (1968) 1083. 6 D.R. Armstrong, P.C. Perkins and J.J.P. Stewart, J. Chem. Sot., Dalton Trans., (1973) 838. 7 I. Mayer, Chem. Phys. Lett., 97 (1983) 270. 8 I. Mayer, Int. J. Quantum Chem., 29 (1986) 73. 9 I. Mayer, J. Mol. Struct. (Theochem), 55 (1989) 43. 10 K. Jug, J. Am. Chem. Sot., 99 (1977) 7800. 11 K. Jug, Croat. Chem. Acta, 57 (1984) 941. 12 K. Jug, J. Comput. Chem., 5 (1984) 555. 13 K. Jug, Mol. Phys., Chem., Biol., 3 (1989) 149. 14 A.B. Sannigrahi and T. Kar, J. Chem. Educ., 65 (1988) 674. 15 MS. Gopinathan and K. Jug, Theor. Chim. Acta, 63 (1983) 497. 16 O.G. Stradella, H.O. Villar and E.A. Castro, Theor. Chim. Acta, 70 (1986) 67. 17 S.G. Semenov, Valence in Progress, Mir, Moscow, 1980. 18 M.S. Gopinathan, P. Siddarth and C. Ravimohan, Theor. Chim. Acta, 70 (1986) 303. 19 P. Siddarth and MS. Gopinathan, Int. J. Quantum Chem., 37 (1990) 685. 20 C.-G. Zhan, Q.-L. Wang and F. Zheng, Theor. Chim. Acta, 78 (1990) 129. 21 C.-G. Zhan and F. Zheng, J. Mol. Struct. (Theochem), 226 (1991) 339. 22 A. Szabo and N.S. Ostlund, Modem Quantum Chemistry, Macmillan, New York, 1982, p. 144. 23 P.G. Lykos and H.N. Schmeising, J. Chem. Phys., 35 (1961) 288. 24 R.J. Bartlett and Y. ohm, Theor. Chim. Acta, 21 (1971) 215.
56 25 26 27 28 29 30 31 32 33
T. Morikawa and Y.J. I’Haya, Int. J. Quantum Chem., 13 (1978) 199. R.E. Christoffersen and K.A. Baker, Chem. Phys. Lett., 8 (1971) 4. M. Pollak and R. Rein, J. Chem. Phys., 47 (1967) 2045. E.R. Davidson, J. Chem. Phys., 46 (1967) 3320. P. Politzer and R.R. Harris, J. Am. Chem. Sot., 92 (1970) 6451. M. Yauze, R.F. Stewart and J-A. Pople, Acta Crystallogr., Sect. A, 34 (1978) 641. P.-O. Lijwdin, J. Chem. Phys., 18 (1950) 365. J. Howell, A. Rossi, D. Wallace, K. Haraki and R. Hoffmann, Quantum Chemistry Exchange Program, No. 344, Chemistry Department, Indiana University, Bloomington, IN. J. Baker, Theor. Chim. Acta, 68 (1985) 221.
47
MAXIMUM OVERLAP POPULATION PRINCIPLE AND DIRECT CALCULATION OF BOND ORDER, VALENCE AND ATOMIC CHARGE
CHANG-GUO ZHAN Department of Chemistry, Central China Normal University, Wuhan 430070 (People’s Rep. of China) (Received 5 October 1990)
ABSTRACT A maximum overlap population principle is introduced to give a very simple approximate scheme for calculating bond order, atomic valence and atomic charge directly from the eigenvectors of the A0 overlap matrix without performing the ordinary molecular orbital calculation. The calculated results are close to or on the whole coincident with those obtained from the canonical molecular orbitals constructed by use of the ab initio calculation, which shows that the calculation scheme suggested in this paper is feasible.
INTRODUCTION
As is well known, calculation methods for bond order and valence have been studied by many scientists, such as Coulson [ 11, Pariser and Parr [ 21, Pople [ 3 1, Mulliken [ 41, Wiberg [ 51, Armstrong et al. [ 61, Mayer [ 7-91, Jug [lo131 and many others [ 14-201. Bond order and valence have become strong tools for studying molecular structure and chemical reactivity. Generally, calculation of bond order and valence is on the basis of molecular orbital data. By use of the calculated molecular orbital coefficients, one can calculate the bond orders and valences defined by different authors. Mayer’s definitions [ 71 of bond order and valence are based on the Mulliken population analysis method [ 41. Jug’s maximum bond order [lo] between two atoms is defined as the maximum of the trace of elements of the two atomic orbital (A0 ) basis sets coupled through the density operator. These pioneering works inspired us to introduce a new idea for studying these problems. In the present paper we attempt to present a maximum overlap population principle in which the molecular orbitals used for the population analysis and for calculation of the bond order, valence and atomic charge are determined by a maximum overlap population criterion
0166-1236/91/$03.50
0 1991-
Elsevier Science Publishers B.V.
48
P”)= F<&P$! = F<&C$C,iSpv =maximum
(1)
where P$ is the overlap population [21] between @h and vth AOs with one electron occupying the ith molecular orbital. Clearly, Pci) is the total overlap population of the ith molecular orbital. The molecular orbitals determined by eqn. ( 1) are called “maximum overlap population orbitals”. By use of the maximum overlap population orbitals and numbers of electrons occupying them, the bond order, atomic valence and atomic charge may be calculated very easily. The main purpose of this paper is to suggest a very simple approximate scheme for calculating directly the bond order, atomic valence and atomic charge without performing the customary molecular orbital calculation. CONSTRUCTION OF THE MAXIMUM OVERLAP POPULATION ORBITALS
Let us denote a normalized non-orthogonal row matrix
A0 basis set in a molecule by a (1)
@=(k)lW”‘MnN The corresponding orthonormalized be expressed as
maximum overlap population orbitals can
(2) By use of the normalization condition of Ivi), P w can be expressed as pci,_~$cZicyisfiu - T
(3) Tc;icvispu
Generally, eqn. (1) can be satisfied by using the conditions dP(i)=O
j= 1, 2, .... n
dCji
(4)
From eqn. (4) we can obtain C C,iSjv -2P”‘CC”iSj, (vZj)
Y
=O
that is (S-I)C. ‘ =2Pci’sc. L
(5)
where S= a+@, I is a unit matrix, and Ci is a column matrix formed from the ith column of matrix C. Equation (5 ) can be expanded as f (S-I)C=SCD
(6)
49
where D is a diagonal matrix, and the ith diagonal element of D is Pci). Rewriting eqn. (6) as +S-i’z(S_I)S
- 1’2S1’2CzSif2CD
that is
(7)
f (I-S-1)S”2CsS1’2CD it follows that S”?! is a unitary matrix, an eigenvector matrix of matrix $(1-S-‘). It is very easy to prove that 4 (I-S - ’ ) commutes with overlap matrix S. Therefore, S1’2C is also an eigenvector matrix of S. Let the eigenvalue equation of S be SC’ = CA
(3)
where C’ is an eigenvector matrix of S, and A is the corresponding eigenvalue matrix. We can choose ZP2C as C’, i.e. S’DC- - C’
(9)
Equation (9) gives C=S - 1’2CI
(IO)
By use of eqn. (8)) matrix S - 1’2 can be expressed as S--‘2=CIA-1’2C)+
(11)
Substitution of eqn. (11) into eqn. (10) gives C=C’A-
1’2
(12)
Substitution of eqn. ( 12 ) into eqn. (2) gives
Y~IBC’A-“~
(13)
It follows that the maximum overlap population orbitals are just the orthonormalized orbitals obtained by making use of the canonical orthogonalization [ 221 of the A0 basis set. Their symmetry properties have been studied in the preceding paper [ 211. It should be pointed out that eqn. (8) has been obtained by many others [ 23-251, and has been employed to construct the maximum overlap atomic and molecular orbitals [ 23 1, (PC’, which are somewhat different from the maximum overlap population orbitals, @C’A-“2. DETERMINATION
OF THE ELECTRON OCCUPATION NUMBERS
Because Pci) is the total overlap population of the ith maximum overlap population orbital 1vi), the orbital energy of 1vi) is closely related to Pci). Generally, the bigger the value of Pfi), the better the bonding and therefore
50
the lower the orbital energy. It follows that electrons should occupy the orbitals with the bigger values of Pci) if the considered molecule is in its ground state. Now let us probe the relationship between P w and the eigenvalue 1 i of overlap matrix S. Substitution of eqn. (9) into eqn. (7) gives $(I-s-l)C’=C’D
(7a)
Equation (7a) can be rewritten as IC’ - S - ‘C’ - 2CD C’(I-2D)=S-‘C’
SC’=C’(I-20)-l
(14)
Comparison of eqns. (8) and (14) gives A= (I-20)-’ A--=1-20
(15)
or
D=f(I-A-‘)
(16)
Equation (16) shows that p (9
=$(I-l)
(17)
Li
It follows that the bigger the value of izi,the bigger the value of Pti) and therefore the lower the orbital energy. In brief, one can determine the numbers of electrons occupying the maximum overlap population orbitals by use of the eigenvalues lis of the overlap matrix S. From the electron occupation numbers and the coefficient matrix of the maximum overlap population orbitals one can obtain the density matrix P [ 161 which can be used for calculating bond order, valence and atomic charge. CALCULATION OF BOND ORDER AND ATOMIC VALENCE
In the present paper, we adopt Mayer’s definitions of bond order and valence [ 731 and consider only the closed shell systems. The sum of electrons occupying the all maximum overlap population orbitals is expressed as [ 141
N= ftr(PS)‘=
‘6
5 (PS).bWS)b,
Expanding the right-hand side of eqn. (18) we obtain
(18)
51
A
b
a
@#A)
(19)
@+A)
where BAn is the bond order of A-B bond BAB
= $
(~s)ab(~s)b,
$
(20)
The valence VAof atom A is defined as
VA=
c
BAB
(21)
(B#A)
Equations (20) and (21) show that so long as the elements of matrix PS are known, the bond order and atomic valence can be evaluated easily. The element Pajcan be expressed as Paj = CniC,iCz
= CniChiCjtllli
i
(22)
i
where ni is the electron occupation number of the ith maximum overlap population orbital. From eqn. (8) we obtain S=C’AC’+
(23)
that is (24) From eqns. (22) and (24) we can obtain (PS),b =
CPajsjb j
= ccc
(ni/ni)Cb;C~C~&pC&
jir
=CC(nin,/ni)CbiCb*,(CC7Cl,) i
P
=xX
j
(25)
t&J-p/~i)ChiCb*,&p
Equation (25) shows that PS appearing in this paper is a Hermitian matrix. Substitution of eqn. (25) into eqn. (20) gives BAB=$$~~W&C~~!~
(26)
52
Substitution of eqn. (26) into eqn. (21) gives
It follows that one can easily evaluate the bond order and atomic valence by using eqns. (26) and (27) so long as the electron occupation numbers nis and the eigenvectors of the overlap matrix S are known. CALCULATION OF ATOMIC CHARGE
Generally, the contributions of bond order BAB to the electron charges of atoms A and B are considered equal. In this way, the charge of atom A can be BAB ) /2 [ 141. Many other methods [ 26-30 ] for calcudefined as (BAA + c @+A) lation of atomic charge have also been proposed. In order to calculate atomic charge from the maximum overlap population orbitals, we introduce a weighted distribution factor f in which satisfies the conditions f&+f&j=l
w3)
and (29)
fiB=fiA
By use of the factor, eqn. (19) can be rewritten as N=iz]&A+
c
A
BAB(f
kB+f !&>I
CB:A) =?[~BAA+
&
(30) BABfhl
(B#A)
From eqn. (30) the charge of atom A can be defined as QA=@AA+
c
Bmfh
(31)
CB:A)
As a test, for practical calculation in the present paper we choose the weighted distribution factor f & as f
iBZIx Ii
+I;
(32)
where IA is the average value of the ionization energies of all valence electrons of atom A.
53 RESULTS FROM LdWDIN ORTHONORMALIZED
A0 BASIS SET
Generally, the matrix element (PS), is closely related to the A0 basis set employed, and so are the bond order, valence and atomic charge. Hence, the values of the bond order, valence and atomic charge obtained from the Lijwdin orthonormalized A0 basis set may be different from those obtained from the normalized non-orthogonal A0 basis set. In this section, we will demonstrate the interesting fact that, according to the maximum overlap population principle, the results obtained from the Lowdin orthonormalized A0 basis set are the same as those from the normalized non-orthogonal A0 basis set. From eqn. (13) we obtain y_cpc’A-‘/2c’+c’_~~c’
(33)
where @” is the orthonormalized A0 basis set obtained b!Yuse of the Lijwdin orthogonalization process [ 311, i.e. @” _@(-J’A-lDc’+ -
(34)
By use of cPOinstead of @, the matrix elements Pti, Sj, and (PS), become Paj =
(224
C&Chic; i
Sj, = (W+W)jb
respectively
=Sjb
(244
CTtiCbiC$ i
(254
and (PS),, =Pa* =
It follows from eqns. (25) and (25a) that the values of the matrix element (PS), obtained from the two A0 basis sets are the same. Furthermore, according to the calculation scheme expressed in the previous sections, eqns. (20), (21), (31) and (19) show, respectively, that the values of the bond order, valence and atomic charge obtained from the two A0 basis sets are the same so long as the values of (PS), from the two basis sets are the same. Therefore, according to the maximum overlap population principle the bond order, atomic valence and atomic charge can be evaluated by using eqns. (26), (27) and (31) , respectively, no matter whether Liiwdin orthonormalized or the normalized non-orthogonal A0 basis set is employed. NUMERICAL RESULTS
As an illustration, we consider only the valence AOs, and use Slater orbitals. The used orbital exponents and valence A0 ionization energies are the default
54
values in the EHMO program [32]. By use of eqns. (26), (27) and (31) the bond orders, atomic valence and atomic charges for many molecules have been obtained from the eigenvectors of the overlap matrix and the electron occupation numbers. Part of the calculation results are listed in Table 1 to enable comparison with other theoretical results obtained from the canonical molecular orbitals. As seen in Table 1, the values of the bond order and atomic valence obtained from the eigenvectors of the A0 overlap matrix are close to those from the canonical molecular orbitals constructed by use of the ab initio TABLE 1 Bond orders, atomic valences and net atomic charges of some small molecules’ Molecule
Bond
Bond order Calc.
Ab initiob
HZ N2 F* HF
HH NN FF HF
1.000 3.000 1.000 1.000
1.00 3.00 1.00 0.96
co
co
3.000
2.52
Hz0
HO
0.957
0.96
NH,
NH
0.963
0.96
HCN
HC CN
0.993
3.000
0.97 2.99
CHI
CH
0.976
0.99
C,Hd
CH cc CH cc CH co
0.964
0.98
2.000 0.990 2.989 0.966 2.042
2.01 0.98 3.00 0.94 2.03
0.988
0.87 0.82 0.91 0.92
&Hz H&O
CHsOH
co OH CH, CH,.
0.961 0.970 0.974
Atom
H N F H F C 0 0 H N H H C N C H C H C H H C 0 C 0 H H, H,.
Atomic valence
Net atomic charge
Calc.
Ab initiob
Calc.
1.000 3.000 1.000 1.000 1.000 3.000 3.000 1.915 0.998 2.888 0.997 1.000 3.993 3.007 3.905 0.998 3.947 1.000 3.986 1.000 0.999 3.974 2.105 3.935 1.991 0.998 1.000 1.000
1.00 3.00 1.00 0.96
0.000 0.000 0.000 0.262 -0.262 0.339 -0.339 -0.475 0.237 - 0.598 0.199 0.088 0.096 -0.184 - 0.525 0.131 -0.212 0.106 - 0.096 0.096 0.071 0.034 -0.177 -0.218 - 0.342 0.237 0.106 0.111
2.52 1.92 0.97 2.88 0.99 0.98 3.96 3.00 3.96 1.00 3.97 1.00 3.98 0.99 0.96 3.91 2.11 3.61 1.72 0.83 0.91 0.91
“All ab initio calculation results are based on the Mulliken population analysis. bThe ab initio calculation results of CHBOH are from ref. 33, and the others from ref. 14. ‘The ab initio calculation results are from ref. 33.
Ab initio”
-0.67 0.34
- 0.66 0.17
0.06 0.06 -0.19 -0.27 -0.68 0.37 0.18 0.22
55
calculation, and the atomic charges obtained from the eigenvectors are on the whole coincident with the ab initio calculation results. Therefore, the calculation scheme proposed in the present paper is indeed a reasonable one for calculation of the bond order, atomic valence and atomic charge. CONCLUSION
A maximum overlap population principle has been introduced to give a very simple approximate scheme for calculation of bond order, atomic valence and atomic charge. The maximum overlap population orbitals determined by the maximum overlap population criterion are the orthonormalized orbitals obtained by making use of the canonical orthogonalization of the A0 basis set. By use of the scheme one can obtain satisfactory calculation results directly from the eigenvectors of the A0 overlap matrix without performing the ordinary molecular orbital calculation. No matter whether the normalized nonorthogonal or Lijwdin orthonormalized A0 basis set is used, the values of the bond order, atomic valence and atomic charge are the same and can be evaluatedbyusingeqns. (26), (27) and (31).
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