NDDO/MC: A new semiempiricalSCFMO method for transition metal complexes

INTERNATIONAL JOURNAL OF QUANTUM

NDDO/MC:

CHEMISTRY, VOL. 44, 565-585 (1992)

A New Semiempirical SCF MO Method for Transition Metal Complexes

MICHAEL

J,

FILATOV, IGOR L. ZILBERBERG, GEORGE M. ZHIDOMIROV

AND

Institute of Catalysis, Novosibirsk, 630090 Russia

Abstract A new semiempirical SCF MO procedure available for prediction of the transition-metal complexes' binding energy and molecular geometry is developed. The features of the method are (i) an explicit account of the orthogonality of the basis set; (ii) use of a new formula for the resonance integral; and (iii) an effective account of the Coulomb correlation of electrons in the calculation of the two-electron integrals based on approach of a model Coulomb hole function. The parameterization for H, C, N, 0, Co, and Ni atoms is presented. The results of NDDO/MC (NDDO for Metal Compounds) calculations of molecular geometries and binding energies for a number of organic compounds and more than 30 cobalt and nickel complexes are compared with the available experimental and ab initio data. The average absolute errors for the binding energies of organic molecules and metal complexes are 8.8 and 5.0 kcal/mol, respectively. © 1992 John Wiley & Sons, Inc.

Introduction

A quantum chemical calculation of the potential energy surfaces of molecular systems is an important element of theoretical investigation of chemical reactions. High-quality ab initio methods are powerful tools for treating this problem and enable us to calculate energetic and structural properties of the small molecules containing light atoms [1] and transition-metal atoms [2,3] with high precision. However, there are some tasks that are connected with the calculation of potential energy surfaces of large organometallic compounds. For instance, calculations of the potential energy surfaces of transition-metal clusters and complexes with large organic ligands are particularly interesting. In our opinion, application of semiempirical zero differential overlap (zoo) methods to investigate such systems could be very fruitfuL Thus, development of semiempirical zoo methods for the quantitative determination of transition-metal compound energies and geometries remains a challenging task for quantum chemical methodology [4]. This problem becomes more interesting due to the appearance of powerful personal computers, since corresponding programs could be easily realized on PCs. Earlier, we developed the semiempirical SCF MO method CNOO-S2 [5-7], with the help of which several reactions with palladium complexes have been investigated [7]. Features of this method are an approximate account of the orthogonality of the basis atomic orbitals and the use of a modified expression for the resonance integral [5]. The latter gives a physically correct treatment of the interactions be© 1992 John Wiley & Sons, Inc.

CCC 0020-7608/92/040565-21

566

FILATOV, ZILBERBERG,

AND ZHIDOMIROV

tween diffuse (s or p) metal valence orbitals and ligand orbitals. However, the assumption is less accurate, when applied to calculation of systems with the open shells. Such systems are of great interest for organometallic chemistry. In this work, a new semiempirical SCF MO method based on the NDDO level of approximation is presented. From experience in the development of different semiempirical methods, it is the NDDO assumption that permits us to calculate energetic and structural properties of molecules with the best accuracy [4,8]. As in the CNDO-S2, an approximate account of orthogonality is done in the new method and a modified formula for the resonance integral, which is an orthogonal basis analog of the so-called weighted formula in EHT [9], is used. An essential advantage of the semiempirical methods is the possibility of evaluation of the one-center Coulomb and exchange integrals from the data on atomic spectra that enables some allowance of the correlation effects in atoms [4,10]. The two-center Coulomb integrals in such schemes are evaluated by the semiempirical formulae, thus also taking correlation effects into account [4,10]. Contrary to the different versions of the MNDO scheme, in our method for evaluation of the Coulomb integrals, an approximation of a model Coulomb hole function [11] is used. We choose the so-called q function [q(r12 - r0)] as a model of the Coulomb hole. A similar approximation has been used by Clementi et al. for the calculation of the correlation energy in atoms [12] and molecules [13]. The present scheme has been programmed in standard FORTRAN-77 and installed on the AT/386-compatible computer. The parameterization for H, C, N, 0, Co, and Ni atoms has been developed and calculations for a number of organic molecules and nickel complexes have been performed. The results of the calculations are compared with the literature data. CNDO

Method and Parameterization

In the are [14]

NDDO

approximation, the matrix elements of the Hartree-Fock operator

FAmnA

=

+

HAmnA

S

PAs'

{(mV | la) -

1/2

(ml | va)}

A,sEA

+

S S B?A

FAmnB

=

Ppt' (mv|pt)

(1)

p,tEB

HAmnB +

S S

PAs

.

(ml

| va),

(2)

AEA sEB

where m, v, l, a, p, and t denote the atomic orbitals; A and B, atoms; Pmn, the elements of density matrix; and H mn, the elements of the core-Hamiltonian. Using the "different orbitals for different spins" approximation, i.e., without assuming that the spatial orbitals are occupied by two electrons with opposite spins, one

NDDO/MC: NEW SEMIEMPIRICAL

SCF MO METHOD

567

has for the matrix elements

S

aFAmnA = HAmnA +

- Pals · (mA IVs)}

(mV | As)

{Pls·

l,sEA

+ S S Ppr

(mV Irt)

•

(3)

B?A p·rEB

S S Pals

aFAmnB= HAmnB +

lEA

and corresponding expressions ments of density matrices for

· (mA | Vs),

(4)

sEB

for 13FAmnAand 13FAmnB·Here, Pamn and pen are the eleand b-spin electrons, respectively, and

a-

Pmn

Pamn + pen·

=

To account for the orthogonality of the basis atomic orbitals, we use the approximate formulae for the matrix elements F mn over the symmetrically orthogonalized basis set [15]· For the interelectronic integrals, the following approximation is adopted [16]:

A(mAVB

IsctD)

= dAB · BCD

•

(mV Ist) ·

Using the approximate Löwdin orthogonalization ~S2) for the core-Hamiltonian, one has

lHmm

=

Hmm + sS?mSms · {Hms

+ 1/4 ·

S S2ml

-

1/2

[5, 16, 17] (up to the terms

Ssm(Hmm

· (Hmm - Hll)

+ Hss)}

+ 0(S3)

(5)

l?m A

Hms

=

1 Hms - -Ssm(Hmm 2

+ Hss)

The first two terms in Eq· (6) are the well-known which is usually approximated by

Mmn

=

-bmn·

Smn'

2

+ o(S ) · Mulliken

(6)

function M mn [18],

(7)

where b mn is the resonance parameter depending on the nature of the m and vorbitals, and S mn is the overlap integral. Therefore, neglecting the last term in

568

FILATOV,

ZILBERBERG,

AND

ZHIDOMIROV

Eg. (5) and substituting Eg. (7) into Egs. (5) and (6), we obtain* lHAmnA = HAmnA + TAB

=

S dAB . TAmnB;

dAB = |dA

B?A

(T0AB)ms

R . lT0AB . R-1;

=

dms·

+ dBI/2

S (lH0AB)ml

(8)

. (S0AB)ml'

lEB

m E A lHAB

=

1 R . lHAB o . R- •,

SxAB x

= (T, p,

(9)

(lHAB) 0 =

ms

=

-b

ms

. SAB ms .

Sx

AS,

•

+ sxB1/2

ISA

d;

SA = S sB =

(10) 1.0.

Here CH0AB)ms and (S0AB)ms are the resonance and overlap integrals taken in the local diatomic coordinate system, and R is the transformation matrix rotating the atomic orbitals from the local diatomic coordinates to molecular ones. We chose the orthogonality correction term in the form given by Eg. (10) in order to preserve a rotational invariance of the core-Hamiltonian [16]. In Eg. (8), dA and dB are the empirical parameters depending on the nature of the A and B atoms. The empirical parameter d has been introduced to adjust the shortcomings of the approximations used in Eg. (8). The empirical parameters SA and sxB (x = (T, p, d) in Eg. (9) have been introduced to account for the different character of (T, p, and d resonance interactions. A modified expression for the resonance parameter bmn entering in Eg. (10) is used [5]: bmn

=

+ In)2) . Im . In/(Im

bAB . (2 - (Im - In)2/(Im bAB

=

IbA

+ bBI/2,

+ In); (11)

where I,. is the ionization potential of the AO m, and bA and bB are the empirical parameters depending on the nature of the A and B atoms. This expression gives a physically correct treatment of the interaction between diffuse metal valence s or p orbitals and localized ligand orbitals [5,19] and is an orthogonal basis analog of the so-called weighted formula for the off-diagonal matrix elements in the extended Hückel calculations [9]. In Egs. (8) and (10), different values of the parameters bA and dA (x = sp, d) for different shells (sp or d shell) are used for transition-metal atoms. The one-center core-Hamiltonian elements HAmnA are written as HAmnA = dmn· Umm

+

S ,

B?A

(12)

where Umm is the one-center one-electron part, and is the two-center core-electron attraction term, describing the interaction of a charge distribution

* Notation

Ix| means

that absolute

value of x has to be taken.

NDDO/MC: NEW SEMIEMPIRICAL

569

SCF MO METHOD

on atom A and the core of atom B. For the calculation of the two-center coreelectron attraction energy, we use the approximation of neglect of penetration integrals [20]:



=

nsB· (mAnAlSBSB)

S (mAnAlpkBpkB)

+ npB·

k=x,y,z

,S,

+ndB·

(mAnAldkBdkB)·

(13)

k=z2, Xz, y'z, x2- y2, Xy'

Here, ntB represents fixed occupation numbers of orbitals of the neutral atom; and SB, p kB, and d rare s, p, and d orbitals centered on atom B· The energy of core-core repulsion is calculated by the formula ENAB

=

VNAB + (l/RAB

- VNAB) · exp( -aAB

. RAB);

aAB

=

laA + aB|/2 (14)

VNAB

=

nsA · +

S

npA·



k=x,y, z

S

+ ndA .

k =z2, Xz, y'z, x2-

· y,2,

(15)

xy

Here, RAB is the interatomic distance, and aA and aB are the empirical parameters· This formula is analogous to that used in the MINDO/3 method [21]· As was mentioned in the Introduction, when we calculated the interelectronic interaction integrals, we used an approximation of the model Coulomb hole function in order to account for the correlation in the motion of electrons [11]. In line with Clementi's proposals [12,13], we chose the so-called q-function [q(rI2 - ro)] as the function modeling Coulomb hole. Therefore, the interelectronic interaction potential in our case is V(rI2)

=

q(r12 - rO)/r12,

(16)

where ro is the effective radius of the Coulomb hole. For evaluation of the Coulomb integrals, we used an approach based on the properties of Fourier transforms [22]. Using the Fourier transform of the potential (16) and formulae from [22], one can obtain for the Coulomb integral

cNN'LL'MM'

(Pa,Pb;

R)

=

J [NLM]a1

. V(rI2)

. [N'L'M']b2 . dtl · dt2

(here [NLM] is the basic charge distribution defined by Roothaan [23];pa and Pb, parameters of charge distributions; and R, the distance between centers of charge

570

FILATOV, ZILBERBERG,

AND ZHIDOMIROV

distributions) the following expression:

where (X + 1), = (X + r)!/X!; L< denotes the lesser of Land L'; [X], the largest integer in X; Ck(LM; L'M'), the Condon-Shortley coefficients as given by Slater [24]; and jk(x), the spherical Bessel functions. A one-dimensional definite integral entering in Eq. (17), ()io 8f(k)

. cos(kr0)

· jl(kR)

· dk

=

I/R . ()i8f(k/R)

· cos(kr0/R)

· j1(k) · dk,

0

(18)

can be easily evaluated numerically in the following way: Let us divide the integration area into two intervals [0, 1] and [1, 00]:

()io

8

f(x)

· cos(Cx) . jl(x) . dx

=

()i1f(X) . cos(Cx) · jl(x) . dx 0

+ ()81f(x) Then, we approximate functionf(x) on the Pk(X) [25]: N f(x) ˜ Sf(xk) . Pk(X) = k=0

· cos(Cx) . jl(x) . dx·

(19)

first interval by Legendre polynomials N Sf(xk) k=0

N · SDkl

· Xl,

l=0

where Dkl are the coefficients of polynomials· For evaluation of the second integral in Eq· (17), we use the approximation described in [26]: N

f(x)

1

N

˜ Sk=0f(Xk) . (1

+ xk)2 . Sl=0Akl . (1 + X)l+2'

Using these approximations and expressing the spherical Bessel function in terms of sines and cosines, evaluation of integral (18) is reduced to evaluation of the following simple integrals: S1N(A)

=

()i1xN. sin(Ax)dx,

C1N(A)

=

o

S2N(A)

=

()81 sin(Ax)/(1 +

()01xN. cos(Ax)dx 0

x)N · dx,

C2N(A)

=

()81 cos(Ax)/(1

+ x)N dx.

NDDO/MC: NEW SEMIEMPIRICAL

These integrals can be easily evaluated by the use of recurrence S1N(A)

=

C1N(A)

S2N(A) C2N(A)

=

l/(N

- N/A · S1N-l(A)

sin(A)/A =

A/(N

-

relations:

+ N/A · C1N-l(A),

-cos(A)/A

=

571

SCF MO METHOD

- 1) · C2N-1(A),

1) - A/(N

- 1) · S2N-1(A).

Here, A is a constant depending on Rand ro· The values of "Coulomb hole radii" r0 were determined by the fitting of the calculated one-center Coulomb integrals to the values evaluated from atomic spectra [27]· Different values of the Coulomb hole radii fa (x = sp, d) for different shells (sp or d shell) are used for transition-metal atoms· The atomic orbital exponents zx (x = s,p,d) are multiplied by scaling factors ax (x = s,p,d) in order to avoid dependence of the optimal ro values on the choice of basis functions· The values of scaling factors ax were fitted simultaneously with the r0 values (see Table II). The two-center Coulomb integrals for the A-B pair of atoms are calculated with the r0AB value determined by the formula 2 ·

AB

r0

=

A

r0A r0B B

ro + ro

To demonstrate what a dependence of so-obtained integrals on RAB looks like, we plotted some of them [(SASA | SBSB) and (dsA dsA I dsB dsB) for the Ni-Ni pair] in Figure 1·

Figure 1. The Coulomb integrals (a) (SASA|SBSB) and (b) (dsAdsA nickel-nickel pair as compared with (c) l/RAB.

|dsBdsB)

for the

572

FILATOV, ZILBERBERG,

AND ZHIDOMIROV

The orbital exponents for sand p orbitals of the H, C, N, and O atoms were taken from [28] and those for s, p, and d orbitals of the cobalt and nickel were taken from [29]. The one-center one-electron energies UIS and Upp for H, C, N, and 0 atoms were treated as empirical parameters. Their values were determined together with the values of aA, bAsp, oJ', and sAp by the fitting of the calculated energetic and structural properties to the observed ones for a chosen "basis set" of molecules. This was done by minimization of the sum of the squares of the weighted errors in the calculated binding energies and geometry. To reduce the computational efforts, the calculations were carried out at the experimental geometries and the gradients of energy with respect to the internal coordinates were taken as reference functions. A nonlinear least-squares minimization procedure proposed by Powell [30] was used to fit the empirical parameters. Standard molecules are listed in Table I (binding energies and gradients on bond lengths and valence angles were used as reference functions). For the metal atoms, the one-center one-electron energies U,s, Upp, and Udd were determined by the fitting of calculated energies of the atomic configurations to that calculated with the Oleari's parameters [27]. The other empirical parameters were determined for metal atoms in the same way as for light atoms. For the M-H pairs, parameters aMH, bspMH, OspMH, bdMH, and OdMH were treated as twocenter parameters and their values were determined separately (see Table I). For the metal-metal bonds, the aMM and bspMM parameters are used instead of a and bsP. The values of these parameters were determined by the fitting of binding energies and bond lengths of the M2 molecules. Table II contains the values of all parameters for H, C, N, 0, Co, and Ni atoms used in NDDO/MC calculations. The quantum chemical calculations for the cobalt and nickel compounds have been carried out by the unrestricted Hartree-Fock (UHF) method with annihilation of the components with spin S + 2k + 1 (k = 0,1,2, ... ) by the halfprojection operator [31]. We chose this procedure for at least two reasons: First, it is not more laborious than the other spin-purification methods [32,33], and, second, it is based on a very attractive physical idea [31], which could be particularly appropriate in the case of SDW solutions of the UHF method [34] for closedshell systems (with na = nb). A geometry optimization has been performed by the Oavidon-Fletcher-Powell [35] procedure with numerically formed gradients. TABLEI.

Standard molecules used for parameterization.

Atom

Set of molecules

Hand C N 0 Co-H Co Ni-H Ni

H" CH4, C,H6, C,H4, C,H" cyclo-C4H8, C6H6 N" NH3, HCN, N,H4, CH,NH" HONO 0" H20, H,CO, H,O" CH30H, HCOOH CoH, CoH, (bent), CoH, (linear) Co" CoCH3, Co(H,O)+, CoCO, CoO NiH, NiH, (bent), NiH, (linear) Ni" NiCH3, Ni(H,O)+, NiCO, NiO

NDDO/MC: NEW SEMIEMPIRICAL TABLE

II.

Atomic parameters used in calculations were fitted with the use of standard H

Parameter

Uss Upp Udd I, II' Id

z,

zp zd as ap ad sp

To

d

TO

a {3'p IdsI' {3d

(eV) (eV) (eV) (eV) (eV) (eV) (au) (au) (au) (au) (au) (au) (au) (au) (A -I)

13.32* -

13.585

1.0

1.0

0.867302 2.149987*

N

C

(only parameters marked with molecules listed in Table I).

0

52.15* 40.88* -

76.54* 58.80*

103.75* 78.84*

24.69

25.56

32.33

12.61

13.19

15.79

1.6438 1.3721 0.780184 0.780184

1.8761 1.5764 0.734861 0.776874

2.1925 1.9604 0.713961 0.699141

0.387244 1.020059*

0.164370 1.435863*

0.132076 1.953647*

0.330523*

0.372476*

0.550976*

0.858503*

0.153764*

0.120023*

0.190619*

0.361358*

1'dd

aMI! (A-I) {3'P MH asp

MH {3dMH ddM1! aMM (A-I) {3'P MM

-

sp

1.170411 * 2 2 0

Sd

n, np nd

-

1 0 0

Results of

-

1.435863* 2 3 0

NDDO/MC

573

SCF MO METHOD

-

1.065321 * 2 4 0

Co

an asterisk

Ni

84.60 67.53 121.32

97.34 78.69 141.57

8.28

8.67

3.84 12.90 1.4230 0.5380 2.8300 1.069127 2.382669 0.757987 1.088391 0.123435 0.222695*

3.97 13.93 1.4730 0.5500 2.9600 1.067137 2.397561 0.752102 1.001715 0.129833 0.256353*

0.809850*

0.687674*

-0.029792*

0.288414*

1.284957* 0.381784* 0.505324*

1.354686* 0.438457* 0.665100*

0.351527*

0.300100*

0.059688* 0.841299* 0.439968* 0.270237*

0.053332* 0.785950* 0.332671 * 0.291176*

0.497635* -0.764108* 1.000000 2 0 7

0.504049* 0.689377* 1.000000 2 0 8

Calculations

Table III lists molecular geometries and binding energies for the 69 organic molecules selected. The average absolute errors for these properties calculated by NDDO/MC and some modern semiempirical SCF MO methods are listed in Table IV. This comparison indicates that average differences among all four methods are quite small. In hydrocarbons, NDDO/MC predicts too long triple bonds and a short single bond near a double or triple bond. In molecules with Nand 0 heteroatoms, the NO bonds are too long. At the same time, bond lengths in H202 and N2H4 molecules predicted by NDDO/MC are better than those predicted by other methods (e.g.,MNDO predicts an 0-0 bond length of 1.295 and a N-N bond length of 1.397 [17]). As one can see from Table IV, NDDO/MC predicts XH bond lengths slightly better than do other methods. NDDO/MC strongly underestimates (=30 kcal/mol) binding energies in CO and HN03 molecules and overestimates

A

A

574 TABLE

FILATOV,

ZILBERBERG,

AND

ZHIDOMIROV

III. Results of the NDDO/MC calculations for compounds containing H, C, Nand (allexperimental data were taken from [17]). Geometry Molecule

(A, degree)

Binding energy (eV)

Expt.

NDDO/MC

0 atoms

NDDO/MC

H2

hh,

0.780

0.740

CH4

ch,

1.082

1.094

18.02

18.21

C2H6

cc, cc, cch,

1.476 1.089 112.8

1.536 1.091 110.9

30.71

30.86

C,H4

cc, ch, cch,

1.318 1.079 124.3

1.337 1.086 121.2

24.32

24.41

C2H,

cc, ch,

1.226 1.056

1.203 1.060

17.94

17.61

C,H8

cc, c1h, c2h, ccc, hc2h,

1.511 1.093 1.088 116.2 104.3

1.526 1.094 1.089 112.6 106.1

43.37

43.57

H,C=C=CH,

cc, ch, cch,

1.309 1.076 123.3

1.308 1.087 118.2

30.78

30.48

HC: :C-CH3

c1c2, c2c3, c3h, c1h, c2c3h,

1.234 1.429 1.086 1.057 110.2

1.206 1.459 1.105 1.056

31.02

30.56

C,H6

cc, ch, hch,

1.481 1.084 106.4

1.510 1.089 115.1

36.85

36.93

CH

ch,

1.101

1.128

4.07

3.65

CH2(s)

ch, hch,

1.104 97.9

1.110 102.4

8.29

CH2(t)

ch, hch, c1c2, c2c3, c1h, c3h, c1c2h, hc3h,

1.063 136.5 1.317 1.469 1.068 1.086 150.9 107.0

1.078 136.0 1.296 1.509 1.072 1.088 149.9 114.6

8.73

8.35

29.99

29.52

c1c2, c2c3, ch, c2c1h,

1.428 1.242 1.086 111.0

1.467 1.213 1.115 110.7

44.11

43.57

cyclo-C3H4

CH3-C:

:C-CH3

4.67

Expt. 4.76

NDDO/MC: NEW SEMIEMPIRICAL TABLE

III.

SCF MO METHOD

(Continued).

Geometry (A, degree) Molecule

Binding energy (eV)

Expt.

NDDO/MC

NDDO/MC

Expt.

37.19

37.31

CH2=CH-CH3

c1c2, c2c3, c1h, c2h, c3h, ccc,

1.325 1.462 1.078 1.085 1.088 119.1

CH(CH3)3

cc, c1h, c2h, c2c1h, c1c2h,

1.519 1.097 1.089 106.6 113.6

(CH3)2C=CH2

c1c2, c2c3, c1h, c3h, c1c2c3, hc1h, c2c3h,

1.333 1.473 1.077 1.087 121.9 109.2 113.4

1.330 1.507 1.088 1.095 122.4

1.233 1.382 1.056

1.205 1.376 1.046

31.08

CH: :C-C:

:CH

c1c2, c2c3, ch,

575

1.336 1.501 1.081 1.090 1.098 124.3

56.06

50.00

110.7 30.15

cyclo-C4H6

c1c2, c2c3, c3c4, c1h, c3h, c1c2h,

1.336 1.481 1.545 1.078 1.090 135.4

1.342 1.517 1.566 1.083 1.094 133.5

43.65

cyclo-C4H8

cc, ch, hch,

1.528 1.091 103.0

1.548 1.133 108.1

50.00

49.53

C6H6

cc, ch,

1.388 1.081

1.397 1.084

58.83

59.05

1.327 1.447 1.079 1.085 127.7 114.1 124.6

1.341 1.463 1.083 1.083 123.3 120.0 119.8

43.65

43.92

CH2=CH-CH=CH, trans

c1c2, c2c3, c1h, c2h, c1c2c3, c2c3h, c2c1h,

CH2=CH-CH=CH2 cis

c1c2, c2c3, c1h,

1.330 1.445 1.079

43.60

576

FILATOV,

ZILBERBERG, TABLE

AND

III.

(Continued).

Geometry Molecule

ZHIDOMIROV

(A, degree)

Binding energy (eV)

Expt.

NDDO/MC

NDDO/MC

Expt.

e2h, c1e2e3, e2e3h, e2c1h,

1.085 131.5 111.3 124.5

c1e2, e2e3, c1h, e2h, ele2e3, e2e3h,

1.462 1.334 1.088 1.086 129.5 117.9

ele2, e2e3, c1h, c2h, elc2e3, c2c3h,

1.462 1.334 1.088 1.086 132.2 115.9

C(CH3)4

cc, ch, cch,

1.525 1.088 113.9

1.539 1.120 110.0

68.74

eyclo-C5H10

cc, eh, c1c2, e2c3, c3c4, c1h, c3h, c4h, e2c3e4, c3c4c5,

1.528 1.091 1.454 1.343 1.483 1.078 1.077 1.089 109.4 103.3

1.546 1.114 1.469 1.342 1.509

63.28

109.2 102.8

cyclo-C6H12

ec, ch, ecc, hch, eccc,

1.528 1.093 113.3 103.5 49.0

1.536 1.121 111.4 107.5 54.9

75.89

76.60

NH(t)

nh,

1.045

1.038

3.94

3.81

NH2

nh, hnh,

1.046 99.2

1.024 103.3

8.07

8.08

NH3

nh, hnh,

1.041 101.8

1.014 107.1

12.51

12.93

N1

nn,

1.136

1.094

9.76

9.85

HN3

n1h,

1.042

0.975

14.91

14.45

CH3-CH=CH-CH3 trans

1.508 1.347

50.02

123.8

CH3-CH=CH-CH3 cis

cyclo-CsH6

49.99

127.8

63.51

50.98

NDDO/MC: NEW SEMIEMPIRICAL TABLE

SCF MO METHOD

(Continued).

III.

Geometry (Å, degree) Molecule

Binding energy (eV)

Expt.

NDDO/MC

nIn2, n2n3, n2n1h,

577

NDDO/MC

Expt.

1.281 1.173 118.8

1.237 1.133 114.8

1.178 1.068

1.153 1.066

14.18

13.55

1.157 1.458 1.104 109.5

27.18

26.53

13.86

12.66

HCN

cn, ch,

CH3CN

cn, cc, ch, cch,

1.186 1.449 1.085 111.0

N2H2 trans

nn, nh, nnh,

1.225 1.054 105.8

N2H4

nn, nh, hnh, hnnh,

1.471 1.048 99.1 180.0

1.449 1.022 106.0 90.0

19.37

18.44

CH3NH2

cn, nh, chi, ch2, hnh,

1.425 1.035 1.089 1.101 104.2

1.474 1.011 1.093 1.093 105.5

25.10

25.23

(CH3)2NH

cn, nh, chi, ch2,

1.440 1.036 1.090 1.098

1.466 1.022 1.091 1.091

37.62

37.66

(CH3)3N

cn, ch, cnc,

1.451 1.101 114.0

1.451 1.090 110.9

49.80

C2H4NH

cc, cn, nh, chi, ch2,

1.458 1.479 1.042 1.084 1.086

1.481 1.475 1.016 1.084 1.083

31.05

31.32

CH2N2

nn, cn, ch, hch,

1.181 1.302 1.073 117.9

20.11

19.27

cyclo-C4H4NH

nc1, c1c2, c2c3, nh, c1h, c2h,

45.93

45.44

1.403 1.367 1.420 1.024 1.077 1.073

1.12 1.32 1.08 127.0 1.370 1.382 1.417 0.996 1.076 1.077

578

FILATOV, ZILBERBERG, TABLEIII.

AND ZHIDOMIROV

(Continued).

Geometry (Å, degree) Molecule

Binding energy (eV)

Expt.

NDDO/MC

NDDO/MC

Expt.

cnc, nc1c2, c1c2h,

107.4 108.4 125.6

125.5

HNO

no, nh, hno,

1.200 1.080 112.2

1.239 1.020 114.4

9.27

9.01

HNO, trans

1.206 1.451 0.988 117.3 109.1 1.257 1.244 1.494 0.987 134.3 114.6

1.163 1.433 0.954 110.7 102.1 1.211 1.199 1.406 0.964 130.3 113.9

13.38

13.52

HN03

no1, no2, o2h, 01n02, n02h, no1, n02, n03, o3h, 01n02, 01n03,

15.29

16.76

NO

no,

1.170

1.151

6.72

6.61

N,O

nn, no,

1.173 1.263

1.129 1.188

11.42

11.73

no,

1.224 147.4

1.193 134.4

10.03

9.85

ono,

NH,CHO

cn, co, ch, nh1, nh2,

1.351 1.185 1.105 1.019 1.024

1.376 1.193 1.102 1.002 1.014

24.89

24.63

C,H,N

nc1, c1c2, c2c3, c1h, c2h, c3h,

1.325 1.398 1.392 1.084 1.079 1.081

1.338 1.394 1.392 1.086 1.082 1.081

53.62

OH

oh,

0.997

0.971

4.55

4.63

H2O

oh, hoh,

0.992 103.1

0.957 104.5

9.43

10.07

H2O,

00, oh, ooh, hooh,

1.485 0.999 103.3 135.6

1.467 0.965 98.5 120.0

12.33

11.62

HCOOH

co1, co2,

1.194 1.328

1.202 1.343

21.20

21.69

NO,

NDDO/MC: NEW SEMIEMPIRICAL TABLE

SCF MO METHOD

(Continued).

III.

Binding energy (eV)

Geometry (Å, degree) Molecule

Expt.

NDDO/MC

ch, 02h, olch, 01c02,

579

1.104 0.983 125.1 132.8

1.097 0.972 124.1 124.9

NDDO/MC

Expt.

O,(t)

00,

1.261

1.208

5.58

5.21

CO

co,

1.154

1.128

9.95

11.24

H2CO

co, ch, hch,

1.186 1.096 102.3

1.208 1.116 116.5

16.15

16.25

CH30H

co, oh, chI, ch2,

1.388 0.991 1.092 1.100

1.425 0.945 1.094 1.094

21.98

22.18

C2HsOH

co, cc, oh, c1h, c2h,

1.389 1.529 0.989 1.104 1.088

03 (rhf)

00, 000,

03 (hpuhf)

00, 000,

1.339 123.2 1.386

117.6

34.61

1.278

5.86

6.34

7.11

6.34

116.8 1.278

116.8

CO2

co,

1.182

1.162

16.41

16.84

CH3CHO

c10, c1h, c1c2, c2h,

1.184 1.101 1.481 1.089

1.216 1.114 1.501 1.086

29.26

29.28

H3C-O-CH3

co, chI, ch2,

1.401 1.092 1.099

1.410 1.091 1.100

34.43

34.53

OCo,

123.0

111.7

C2H4O

cc, co, ch, hch, cc(h-h),

1.466 1.458 1.088 106.9 155.4

1.470 1.435 1.084 116.3 158.1

27.52

28.16

CH,CO

co, cc, ch, hch,

1.178 1.318 1.070 110.9

1.161 1.314 1.083 122.6

23.11

23.18

CH3COOH

col,

1.196

1.214

34.15

580

FILATOV,

ZILBERBERG,

AND

TABLE III.

(Continued). (A, degree)

Geometry Molecule

NDDO/MC

C4H,O

TABLE IV.

ZHIDOMIROV

Average

Expt.

co2, o2h, cc, ch,

1.333 0.978 1.532 1.086

1.364 0.970 1.520

oc1, c1c2, c2c3, c1h, c2h,

1.388 1.376 1.429 1.079 1.073

1.362 1.361 1.431 1.077 1.075

errors

for molecules

NDDO/MC Length XY (Å) Length XH (A) Angle XYZ (deg.) Angle XYH, HXH (deg.) Binding energy (kcal)

Binding

0.024(87) 0.013(77) 4.6(17) 4.4(34) 8.8(56)

Data on SINDol, MINDO/3, and MNDOmethods

containing SIND01 0.028(181) 0.015(105) 2.5(53) 2.6(76) 8.3( 132)

energy

NDDO/MC

(eV) Expt.

42.35

42.34

H, C, N, and 0 atoms. MINDO/3 0.022(91 ) 0.017(67) 6.8(22) 4.1(43) 11.3(73)

MNDO 0.030(142) 0.017(89) 2.8( 43) 3.0(66) 9.0(118)

were taken from [17].

binding energies in molecules with single and double NN bonds. In hydrocarbons, binding energies of molecules with triple bonds are again overestimated. To see if our method overemphasizes closed (ring) structures, we have compared open and cyclic conformers of the 03 molecule and the C3H5' radical:

In both cases, open structures are the most stable. The energy differences are 15.0 kcal/mol for the ozone/trioxirane and 16.6 kcal/mol for the allyl radical! cyclopropenyl radical pairs. For the 03(C2v) molecule, the SDW solution of the UHF method is the lowest in energy. The half-projection lowers the energy of this state by 0.2435 eV (see Table III). The results of the NDDO/MC calculations for the 36 cobalt and nickel compounds are summarized in Table V. The average absolute errors for bond dissociation energies and bond lengths are 5.0 kcal/mol and 0.064 A, respectively. For some molecules, we have calculated not only the ground states but also low-lying excited states (see Table VI). Table VI lists d-orbital populations, net charges,

NDDO/MC: NEW SEMIEMPIRICAL SCF MO METHOD

581

582

FILATOV, ZILBERBERG,

AND ZHIDOMIROV

NDDO/MC: NEW SEMIEMPIRICAL

SCF MO METHOD

583

584

FILATOV,

TABLE VI. Configurations, cobalt and nickel complexes;

ZILBERBERG,

AND

ZHIDOMIROV

d-orbital's populations, and charges of the metal atoms for some ab initio results are taken from the references given in footnotes to Table V. Ab initio

NDDO/MC

Molecule CoH CoCH, CoCH3+ Co(H,O)+ CO(H,O)2+ NiH NiCH, NiCH3+ Ni(H,O)+ Ni(H,O)2+ NiCO NiCO+ NiCO+ NiN, NiN2+ NiN2+

Configuration s I.0 s 1.6 '1T3.0

(state)

d3.1

S0.6a1101e4.02e'·0

(3F) (3E)

s05a 11·'le'·02e,·0

(4E) s01a 1401a220b120bi0 (3B2) s0.1a g39b 12.g0b 22.g0b 13g.0 (3B,g) s 1.0s 1.7p4.0d3.0 (2D) s06a 110le4.02e4.0 (2AI) s0.4a11.81e402e'·0 (3E) s0.1a

r01a

i.0b 12.0b22.0

(3A1)

s01a g3.'b21g.°b22g.0b 13g0 (2Ag) s0.5 s2.0 p3.0d4.0 (1S+) s

0.2 s1.0p4.0d4.0

s

0.3 s1.9p4.0d3.0

so,5a SO.1 SO.2

I.3p3.9

d4.0

s 1.0 p 4.0d4.0 s J.9 p 4.083.0

nd

Q

7.75

0.30

7.99 7.33 7.98

0.39 1.16 0.95

7.87

0.98

8.73

0.26

8.98

0.47

8.82 8.99 8.99

0.82 0.93 0.90

8.95

0.58

(2S +)

8.98

0.81

(2D) (1S+) (2S+)

8.93 9.21 8.98 8.93

0.74 0.32 0.92 0.94

(2D)

State (configuration) 'F 'E 4E

nd

Q

7.60

'B2

'B,g 2D 'AI

7.39

0.26

7.21 7.84 7.66

0.98 0.88 0.70

8.65 8.96

0.34

'E 'AI 'B,g S0Ss1.5 p'"~d4.0(1S+) S0.1s1.0 p'"~ d4.0(2S+)

8.24 8.97 8.73

0.95 0.85 0.70

8.79

0.27

8.88

0.88

S"' s1.8p" d'0(2D) S0.8s1.4 p,5 d40( 1S+) S"·Is 1.11p'"~ d4.11(2S+)

8.72 8.87 8.91 8.76

0.87 0.22 0.92 0.92

S"'sI'

p'"~ d'·°(2 D)

and configurations of the metal atom in some molecules obtained by NDDO/MC. These data are compared with the available ab initio MCPF [42-45] and MCSCF [39,41] results. As can be seen, the NDDO/MC calculations of the electronic structure are in reasonable agreement with ab initio ones. Analysis of the HPUHF wave functions calculated by NDDO/MC indicates that they correlate well with the spectroscopic states obtained by ab initio methods (see Table VI). As one can easily see, the evaluation of the Coulomb integrals in our method is more laborious than it is in some other semiempirical methods (e.g., MNDO). Therefore, one must question how much time these calculations take. On our personal computer (TANDON 386/20, MS-DOS 3.3, compiler NDP-FORTRAN-77 ver. 2.1), single SCF calculations of C6H6, Ni2, and Ni(H20)62+ require 57, 109, and 664 s, respectively. In summarizing our findings, we can conclude that NDDO/MC could serve as a quite efficient and inexpensive tool of a theoretical investigation of electronic and molecular structure of transition-metal compounds. Bibliography [1] W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, Ab Initio Molecular Orbital Theory (Wiley-Interscience, New York, 1985). [2] M. Blomberg, U. Brandemark, I. Panas, P. Siegbahn, and U. Vahlgren, in NATO ASI Ser., Ser. C 176 (Quantum Chem.): Challenge Transition Met. Coord. Chem., A. Veillard, Ed. (1986), p. 1. [3] C. W. Bauschlicher, Jr., S. P. Walch, and S. R. Langhoff, in NATO ASI Ser., Ser. C 176 (Quantum Chem.): Challenge Transition Met. Coord. Chem., A. Veillard, Ed. (1986), p. 15. [4] W. Thiel, Tetrahedron 44, 7393 (1988). [5] M. J. Filatov, O.v. Gritsenko, and G. M. Zhidomirov, Theor. Chim. Acta 72, 211 (1987). [6] M. J. Filatov, O.v. Gritsenko, and G. M. Zhidomirov, Zurn. Struct. Chim. (in Russian) 28, 3 (1987).

NDDO/MC:

NEW

SEMIEMPIRICAL

SCF MO METHOD

585

[7] M. J. Filatov, O.v. Gritsenko, and G. M. Zhidomirov, J. Mol. Catal. 54, 462 (1989). [8] C. Nieke and J. Reinhold, Theor. Chim. Acta 65, 99 (1984). [9] J. H. Ammeter, H. B. Burgi, J. C. Thibeault, and R. Hoffman, J. Am. Chern. Soc. 100,3686 (1978). [10] M. J. S. Dewar and W. Thiel, J. Am. Chern. Soc. 99, 4799 (1977). [II] O.v. Gritsenko, A. A. Bagaturyants, and V. B. Kazansky, Int. J. Quantum Chern. 29, 1799 (1986). [12] E. Clementi, IBM J. Res. Dev. 9, 2 (1965). [13] E. Clementi, S. Chin, G. Corongiu, J. H. Detrich, M. Dupuis, D. Folsom, G.C. Lie, D. Logan, and V. Sonnad, Int. J. Quantum Chern. 35, 3 (1989). [14] J. A. Pople, D. P. Santry, and G. A. Segal, J. Chern. Phys. 43, S129 (1965). [IS] P.-O. Löwdin, J. Chern. Phys. 18,365 (1950). [16] R. D. Brown and K. R. Roby, Theor. Chim. Acta 16,175 (1970). [17] D. N. Nanda and K. Jug, Theor. Chim. Acta 57,95 (1980). [18] R.S. Mulliken, J. Chim. Phys. 46, 497 (1949). [19] A. A. Bagatur'jants, O.V. Gritsenko, and G. M. Zhidomirov, Zhurn. Phis. Chim. (in Russian) 54, 2993 (1980). [20] M. Goeppert-Mayer and A. L. Sklar, J. Chern. Phys. 6, 645 (1938). [21] R. S. Bingham, M. J. S. Dewar, and D. H. Lo, J. Am. Chern. Soc. 97, 1285 (1975). [22] M. Geller, J. Chern. Phys. 41, 4006 (1964). [23] C.C.J. Roothaan, J. Chern. Phys. 19, 1445 (1951). [24] J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960), Vol. 1. [25] M. Abramovitz and I. A. Stegun, Eds., Handbook of Mathematical Functions, Natl. Bur. Stand., Appl. Math. Ser. 55 (1964). [26] V. I. Krilov and L. G. Kruglikova, Spravotchnaya Kniga po Chislennomu Garmonitcheskomu Analizu (Nayka i Tekhnika, Minsk, 1971), p. 254. [27] (a) L. Oleari, L. Di Sipio, and G. De Michelis, Mol. Phys. 10,97 (1966); (b) L. Di Sipio, E. Tondello, G. De Michelis, and L. Oleari, Chern. Phys. Lett. 11,287 (1971). [28] J. A. Hashmall and S. Raynor, J. Am. Chern. Soc. 99, 4894 (1977). [29] M. Gouterman and M. Zerner, Theor. Chim. Acta 4, 44 (1966). [30] M. J. D. Powell, Comput. J. 7, 303 (1965). [31] Y. G. Smeyers and A. M. Brucena, Int. J. Quantum Chern. 14, 641 (1978). [32] A. D. Backon and M. C. Zerner, Theor. Chim. Acta 53, 21 (1979). [33] T. Amos and L.c. Snyder, J. Chern. Phys. 41,1773 (1964). [34] H. Fukutome, Progr. Theor. Phys. 47,1156 (1972); Ibid. 49,22 (1973); Ibid. 50, 1433 (1973). [35] R. Fletcher and M. J. D. Powell, Comput. J. 6, 163 (1963). [36] K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979). [37] T. H. Upton and W. A. Goddard, J. Am. Chern. Soc. 100,5659 (1978). [38] W. Weltner, Jr. and R. J. Van Zee, Annu. Rev. Phys. Chern. 35, 291 (1984). [39] S. R. Langhoff and C. w. Bauschlicher, Jr., Annu. Rev. Phys. Chern. 39,181 (1988). [40] M. R. A. Blomberg and P. E. M. Siegbahn, J. Chern. Phys. 78, 986 (1983). [41] A. K. Rappe and W. A. Goddard, J. Am. Chern. Soc. 99, 3966 (1977). [42] C.w. Bauschlicher, Jr., et. aI., J. Chern. Phys. 91, 2399 (1989). [43] M. R. A. Blomberg, U. Brandemark, and P. E. M. Siegbahn, J. Am. Chern. Soc. 105, 5557 (1983). [44] C.w. Bauschlicher, Jr., Chern. Phys. 129,431 (1989). [45] S. P. Walch and W. A. Goddard, J. Am. Chern. Soc. 100, 1338 (1978). [46] M. Rosi and C.w. Bauschlicher, Jr., J. Chern. Phys. 90, 7264 (1989). [47] P. E. M. Siegbahn, M. R. A. Blomberg, and C.w. Bauschlicher, Jr., J. Chern. Phys. 81, 1373 (1984). [48] M. Rosi, c.w. Bauschlicher, S. R. Langhoff, J. Phys. Chern. 94, 8656 (1990).

Received October 22, 1991 Revised manuscript received April 6, 1992 Accepted for publication April 8, 1992