Semiempirical implementation of strictly localized geminals for analysis of molecular electronic structure

A.M. Tokmachev and A.L. Tchougréeff
Karpov Institute of Physical Chemistry,
Vorontsovo pole 10, Moscow, 103064, RUSSIA

Abstract

Approximate electronic trial wave function taken as the antisymmetrized product of strictly localized geminals (APSLG) is implemented for semiempirical analysis of molecular electronic structure of ''organic'' compounds and for calculations of their heats of formation and potential energy surfaces. This resulted in an O(N)-scaling method. The use of the MINDO/3 form of the semiempirical Hamiltonian with the reparameterized b_AB values in combination with the APSLG form of the wave function yields the computational procedure BF'98. Calculations on the heats of formation and equilibrium geometries for a wide range of molecules show that the APSLG-MINDO/3 approach is more favorable than its self consistent field based counterpart. Also, the APSLG formalism allows to interpret molecular electronic wave function in chemically sensible terms.

Introduction.

Modern ab initio methods of quantum chemistry when applied to molecular objects of practical interest achieve acceptable results by extending the basis set of one-electron states, and by taking into account a great deal of electron correlation corrections of higher orders. These approaches require considerable computational resources which increase like N⁵¸N⁷ (where N is the dimension of the basis of one-electron states involved) when the system size grows. This reduces the applicability of ab initio methods to the real interesting systems (especially to their chemical transformations) []. Semiempirical methods employing the self consistent field (SCF or Hartree-Fock) approximation for the trial electronic wave function attain chemically reasonable results by using sophisticated parameterization schemes. Nevertheless, the required computational resources in this case grow as N³. Therefore, even the application of semiempirical methods to large systems may as well become problematic. That high requirements to the computational resources do not allow to apply the quantum-chemical methods to the massive calculations of potential energy surfaces (PES) of molecular systems of practical interest. These calculations may become necessary, for example, in the context of molecular dynamics studies on reactivity of large molecules. Thus it is important to develop quantum chemical methods with a weaker dependence on the size of the system than the cubic one. Preferably it should be O(N)-scaling methods i.e. those with a linear dependence of the required computation resources on the system (basis) size. Several attempts to construct such methods are reported in the literature. In Refs. fastcalc,Stewart it was proposed to eliminate the diagonalization of the matrices of the size N×N by ignoring matrix elements of the Fock matrix between the basis functions centered on distant atoms. This leads to the O(N)-scalability of the resulting procedure, but the elimination of the matrix elements of the Fockian was not counterbalanced. Another way to the methods with smaller computational costs has been proposed in Ref. Malrieu. It is constructed to get the local one-electron states directly from the SCF equations. The electron correlations can be then more effectively taken into account in the basis of the local one-electron states than it can be done in the basis of the delocalized canonical MO LCAO's.

The O(N)-scalability can be also achieved by employing the basis of one-electron states which are obtained without performing the Hartree-Fock step at all. The latter can be avoided by making use of some different form of the trial electronic wave function. Semiempirical quantum chemical methods can be constructed for arbitrary form of the electronic trial wave function; specific choice of its form is stipulated by the class of molecules the method is designed for and by a class of physical properties or phenomena it has to describe. For example, for the description of chemical transformations the trial wave function has to assure a correct asympthotic behavior when chemical bonds are cleaved or formed. For example, the trial wave function of the SCF MO LCAO approximation is generally known to have a wrong asympthotic behavior at large interatomic separations which is evident from the example of the hydrogen molecule: the electronic wave function in the SCF approximation (in the valence basis, of course) does not depend on the distance between the hydrogen atoms and thus the weights of ionic and covalent configurations in the ground state remain equal (and constant) for all internuclear separations. It leads to the physically absurd description of the system at large interatomic distances. This problem obviously persists for all the homolytic cleavages of s-bonds.

With the above example in mind we consider a theory of separate electron pairs with the trial wave function of the ground state of 2n-electron system taken as the antisymmetrized product of spin geminals [] as a starting point for constructing a semiempirical theory alternative to the SCF. The concept of two-electron bond wave functions (geminals) is due to V.A. Fock [] as the concept of single determinant wave function is. That construction takes into account different contributions to the bond wave functions - the covalent and ionic ones - with the amplitudes of these contributions determined by variation of the total energy. Such a wave function includes the intrabond correlations and allows to describe the limit of the homolytic cleavage of the s-bonds correctly.

In this context the well known PCILO approach [], which employs perturbation theory to account the electron correlations within the bonds, deserves special attention. The starting point of the PCILO approach is, nevertheless, the wave function of the SCF (one-electron) approximation. The bond wave functions are then constructed for each bond by maximizing the overlap of its hybrid orbitals (HO's). Such definition does not guarantee, however, that the obtained HO's are optimal from the point of view of the variational principle (provide a minimum to the total energy). Moreover, such scheme does not apply to many classes of compounds with electronic structures which can not be unambiguously represented by unique distribution of chemical bonds. The local and short-range character of the terms contributing to the Hamiltonian of this approach allows to achieve high speed of the electronic structure calculations.

In the PCILO approach the strong intrabond electron correlations are taken into account with use of perturbation theory, which can cause convergency problems. This is, however, not necessary, since the intrabond correlations can be alternatively taken into account variationally. For this end one must solve a sequence of quantum mechanical problems for each of two-electron bonds (electron pairs) in the subspace of one-electron functions ascribed to each bond. By this we arrive to a theory which gives the optimal local two electron wave function (geminal), representing a chemical bond. Such treatment of electron pairs in the molecules corresponds to the intuitive Lewis concept of two-electron bond []. In this framework the Lewis' picture appears as a result of optimization of total energy on a specific class of electronic trial wave functions.

Variational methods using strictly local orbitals are also known in the literature. These orbitals are localized on the bonds or on the electron lone pairs and do not contain tails on the other atoms of the molecule. Such MO's are called strictly localized (SLMO) []. For saturated organic molecules whose canonical MO's can be localized satisfactorily by standard methods [,,,,]. The orbitals relevant to the SLMO approach turn out to be close to the LMO's obtained by localization of the delocalized canonical MO LCAO which are solutions of the Hartree-Fock-Roothaan (HFR) equations. These a posteriori transformations from the atomic basis to the hybrid one are ambiguous. There exist several criteria for it, based on the geometry factors, when one either attempts to direct the HO's towards the other end of the bond Pauling or tries to maximize the overlap between the HO's, belonging to different ends of the bond []. However, it is more consistent to evaluate the HO's on the basis of the variational principle (i.e. from the energy minimum condition) as other parameters of the trial wave function.

An approach almost satisfying the criteria described above has been used in the work []. It has been implemented with use of the ab initio Hamiltonian. This approximation is called along with the type of the wave function of antisymmetrized product of strictly localized geminals (APSLG). This approximation is similar to other pair theories like the generalized method of valence bonds (GVB) [] and its descendants - those making use of perfect pairing [] and complete active space []. The difference between these methods lays in the way of determination of the one-electron states to be used for the bond function. The GVB approach describes electron pairs in terms of non-orthogonal orbitals. The product of two orbitals is then replaced by the wave function of the simple valence bond and the orbitals (one-electron states) of such wave function are optimized.

The ab initio APSLG approach [] was tested only for a limited series of simple molecules. Results obtained there do not allow to derive unambiguous conclusion about the validity of this approximation for larger molecules since even in the case of the CH₃F molecule the electronic energy of the APSLG approximation is significantly higher than that for the SCF method (in the same basis) []. The calculation of the potential curve for dissociation of one C-H bond in methane molecule shows that for the C-H separations close to those at the equilibrium the APSLG approximation results in a lower energy value than the GVB but for the dissociation limit the reverse is true. In this context we considered the possibility of semiempirical implementation of a quantum chemical method using the trial wave function of the APSLG approximation [].

Semiempirical implementation requires assessing a set of parameters to be used in calculations. We have used the well known MINDO/3 parameterization [] as a starting point for parameters evaluation. This set of parameters when used with the trial wave function of the HFR approximation for the valence electrons gives an adequate set of the energy characteristics and the equilibrium geometries for organic molecules. Furthermore, as it was noted in Ref. [], the one-center parameters of the MINDO/3 method are close to those estimates derived from the spectra of free atoms and ions [] and/or those made in the framework of the theory of the effective valence shell Hamiltonian []. This allows to conclude that the incorrect asympthotic behavior of the trial SCF wave function is compensated within the MINDO/3 approach by two-center parameters of the MINDO/3 method whereas the one-center parameters are quite universal and do not depend on the type of the trial electronic wave function. In this paper we construct the energy functional with use of the trial wave function of the APSLG type and with the MINDO/3 type of parametrization for atomic and two-center integrals. The variation of this functional gives the estimate of parameters of the wave function and the electronic (and total) energy of the system in question.

Theory.

Wave function and the Hamiltonian.

The wave function of electrons in the APSLG approximation has the form:

| F ñ =
Ő
m
g_m⁺|0 ñ ,
(0)
where the m-th geminal (i.e. the two-electron function of the m-th bond) is expanded through the operators creating electrons with the spin projection s respectively on the ''right'' and the ''left'' hybride orbital (HO) of the appropriate bond:

g_m⁺ = u_mr_ma⁺r_mb⁺+v_ml_ma⁺l_mb⁺+w_m( r_ma⁺l_mb⁺+l_ma⁺r_mb⁺) .
(0)
The normalization and orthogonality conditions for the geminals read:

á 0| g_mg_m^˘⁺|0 ñ = u_mu_m^˘+v_mv_m^˘+2w_mw_m^˘ = d_mm^˘.
(0)
Such functions describe singlet states of electron pairs. The quantities u_m and v_m are the amplitudes of the ionic contributions to the geminal of the m-th bond and w_m is proportional to the amplitude of the covalent contribution. If the HO's | r_m ñ and |l_m ñ are symmetrical the latter is the Heitler-London wave function. If all the amplitudes are accidentally equal then the function obtained is the spin restricted SCF wave function built on the HO's |r_m ñ and | l_m ñ . The electron lone pairs represent a special case of the geminals. For them only one ionic contribution to the geminal does not vanish: for the sake of definiteness we assume that the lone pair geminals have the u_m amplitudes equal to unity.

The electronic energy in the APSLG approximation is

E = á F| H| F ñ ,
(0)
where | F ñ is defined by Eqs. (1), ( 2).

The HO's | r_m ñ and | l_m ñ result from orthogonal transformations applied to the initial set of AO's for each ''heavy'' (non hydrogen) atom. Unlike most modern methods which use the HO's, satisfying some external localization criteria we determine the HO's variationally i.e. from applying the minimum condition to the energy functional. For each heavy atom (A) it is necessary to determine six independent angles which define the matrix h^A Î SO(4) of the orthogonal transformation in four dimensional space spanned by one s- and three p-orbitals per heavy atom []. The annihilation operators for the HO's are:

t_ms =
ĺ
i Î A
h_mi^Aa_is; t = r,l.
(0)
The Hamiltonian for a molecular system in the MINDO/3 approximation can be presented as a sum of one- and two-center contributions:

H =
ĺ
A
H_A+ 1
2

ĺ
A ą B
H_AB.
(0)
Molecular integrals for the MINDO/3 approximation are originally known in the basis of the valence s- and p-AO. As the geminals are expressed through the HO's, all the integrals have to be transformed to the HO basis, as well.

Suppose that | t_m ñ is any of the HO's (|r_m ñ or | l_m ñ ) belonging to the m-th geminal. At the same time the HO | t_m ñ belongs to the subspace spanned by the valence (s- and p-) orbitals centered on the atom A (this is denoted as t_m Î A). The parameter U_m^t representing the attraction of an electron on the HO | t_m ñ to the core of the atom A has the form:

U_m^t =
ĺ
i Î A
(h_mi^A)²U_i(A),
(0)
where the index i enumerates the s- and p-AO's belonging to the heavy atom A. The attraction of electron to other cores and repulsion of electrons on the orbitals of different atoms do not depend on the angular momentum (azimuthal) quantum number l. Thus the energy shifts of the one-electron states on the atom A due to the Coulomb interactions with the atomic cores are equal for the s- and p-AO's and therefore for all of their linear combinations. Thus they can be expressed through the Coulomb two-center integrals g_AB and the core charges Z_B as in the standard MINDO/3 method.

There are only a few integrals characterizing the repulsion of two electrons on one atom, essential for the energy estimate with use of the APSLG wave function. These are:

( \QATOPtm\QATOPtm | \QATOPtm\QATOPtm)^A =
ĺ
i Î A
(h_mi^A)⁴( ii | ii) ^A+

+2
ĺ
i < j Î A
(h_mi^Ah_mj^A)²[ ( ii | jj)^A+2( ij | ij) ^A] ;

(0)

( \QATOPtm₁\QATOPtm₁ | \QATOPt^˘m₂\QATOPt^˘m₂) ^A =
ĺ
i Î A
(h_m₁i^Ah_m₂i^A)²( ii | ii) ^A+

+
ĺ
i < j Î A
[(h_m₁i^Ah_m₂j^A)²+(h_m₁j^Ah_m₂i^A)²] ( ii | jj) ^A+

+4
ĺ
i < j Î A
h_m₁i^Ah_m₁j^Ah_m₂i^Ah_m₂j^A(ij | ij) ^A;

(0)

( \QATOPtm₁\QATOPt^˘m₂ | \QATOPt^˘m₂\QATOPtm₁) ^A =
ĺ
i Î A
(h_m₁i^Ah_m₂i^A)²( ii | ii) ^A+

+2
ĺ
i < j Î A
h_m₁i^Ah_m₁j^Ah_m₂i^Ah_m₂j^A( ii | jj) ^A+

+
ĺ
i < j Î A
[2h_m₁i^Ah_m₁j^Ah_m₂i^Ah_m₂j^A+(h_m₁i^Ah_m₂j^A)²+

+(h_m₁j^Ah_m₂i^A)²]( ij | ij) ^A.

(0)
The summation indici i and j in these sums refer to the s- and p-orbitals of the heavy atom A.

The resonance integral between the right (centered on the atom A) and the left (centered on the atom B) HO's is expressed through the resonance integrals in the AO basis:

b_m =
ĺ
i Î A

ĺ
j Î B
h_mi^Ah_mj^Bb_ij^AB.
(0)

To evaluate the electronic energy we use the second quantization technique. The fermion operators related to the HO's obey usual anticommutation relations. The contributions to the molecular Hamiltonian Eq. ( 6) in terms of these operators have the form:

H_A = ĺ
\Sb t_m Î A

s\endSb ć
č U_m^t-
ĺ
B ą A
g_ABZ_B ö
ř t_ms⁺t_ms-

- ĺ
\Sb t_m₁,t_m₂ Î A

m₁ < m₂\endSb
ĺ
s
b_m₁m₂^AA( t_m₁s⁺t_m₂s+h.c.) +

+ 1
2
ĺ
\Sb t_m₁,t_m₂^˘ Î A

t_m₃^˘˘,t_m₄^˘˘˘ Î A\endSb
ĺ
st
( \QATOPtm₁\QATOPt^˘m₂ | \QATOPt^˘˘m₃\QATOPt^˘˘˘m₄)^At_m₁s⁺t_m₃t^˘˘+t_m₄t^˘˘˘t_m₂s^˘,

(0)
where b_m₁m₂^AAis the resonance integral corresponding to electron transfer from the HO of the geminal m₁ to the HO of the geminal m₂ in the case if these HO's belong to the same (heavy) atom. Next

H_AB = - ĺ
\Sb t_m₁ Î A

t_m₂^˘ Î B\endSb
ĺ
s
b_m₁m₂^AB(t_m₁s⁺t_m₂s^˘+h.c) +

+g_AB ĺ
\Sb t_m₁ Î A

t_m₂^˘ Î B\endSb
ĺ
st
t_m₁s⁺t_m₂t^˘+t_m₂t^˘t_m₁s,

(0)
where b_m₁m₂^AB is the resonance integral corresponding to electron transfer between the HO's centered on different atoms. (Here and above h.c. stands for hermitian conjugation). If m₁ = m₂ the previous resonance integral transforms to the b_m integral for the m-th geminal (bond) Eq. (11)).

Electronic energy in the APSLG-MINDO/3
approximation.

Now let us consider the contributions to the energy from the one-center terms á F| H_A| F ñ . To do so we evaluate the averages of the operators with the APSLG wave function. The one-center part of the Hamiltonian contains the operators proportional to the number of electrons operator. The contribution from it to the energy can be represented as:

E₁ =
ĺ
A

ĺ
m
(U_m^t-
ĺ
B ą A
g_ABZ_B)
ĺ
s
P_m^ts,
(0)
where

P_m^ts = á 0| g_mt_ms⁺t_msg_m⁺| 0 ñ
(0)
and P_m^ts is expressed through the amplitudes of the m-th geminal:

P_m^rs = á 0| ( u_mr_m-s+w_ml_m-s) ( u_mr_m-s⁺+w_ml_m-s⁺) | 0 ñ = u_m²+w_m²,
(0)

P_m^ls = v_m²+w_m²
(0)
and does not depend on s. (Hereinafter we drop the s superscript in one-electron densities). Thus:

E₁ = 2
ĺ
A

ĺ
t_m Î A
(U_m^t-
ĺ
B ą A
g_ABZ_B)P_m^t.
(0)

The contribution from the Coulomb repulsion of electrons located on the same atom is:

E_coul⁽¹⁾ =
ĺ
A
[
ĺ
t_m Î A
( \QATOPtm\QATOPtm | \QATOPtm\QATOPtm) ^AR_m^t+

+2 ĺ
\Sb t_m₁,t_m₂^˘ Î A

m₁ < m₂\endSb ( 2( \QATOPtm₁\QATOPtm₁ | \QATOPt^˘m₂\QATOPt^˘m₂) ^A-( \QATOPtm₁ \QATOPtm₂ | \QATOPt^˘m₂\QATOPt^˘m₁)^A) P_m₁^tP_m₂^{t^˘}].

(0)
It is proportional to the averages of the form (see Eq. (12)):

á F| t_m₁s⁺t_m₂t^˘+t_m₃t^˘˘t_m₄s^˘˘˘|F ñ =

= d_m₁m₂d_m₃m₄d_m₁m₃( 1-d_st) R_m₁^t+

+[ d_m₁m₄d_m₂m₃( 1-d_m₁m₂)-d_m₁m₃d_m₂m₄( 1-d_m₁m₂) d_st] P_m₁^tP_m₂^{t^˘},

(0)
where the relevant elements of two-electron density matrix are:

R_m^r = á 0| g_mr_mb⁺r_ma⁺r_mar_mbg_m⁺| 0 ñ = u_m²
(0)
for the right HO and

R_m^l = á 0| g_ml_mb⁺l_ma⁺l_mal_mbg_m⁺| 0 ñ = v_m²
(0)
for the left HO of the m-th geminal.

The contribution to the energy from the resonance term of the Hamiltonian is proportional to the spin bond order between the HO's of the m-th geminal:

Q_m^s = á 0| g_mr_ms⁺l_msg_m⁺| 0 ñ = ( u_m+v_m) w_m.
(0)
The bond order is also s-independent for the singlet geminal. Total contribution to the energy from the resonance interaction is:

E_res = -4
ĺ
A < B

ĺ
m Î A,B
b_mQ_m.
(0)
The contribution from the Coulomb interaction of electrons located on different atoms (two-center) to the total energy is proportional to the following element of the two-electron density matrix:

á F| ^
n

t
m₁
^
n

t^˘
m₂
| F ñ =
ĺ
st
á F| t_m₁s⁺t_m₂t^˘+t_m₂t^˘t_m₁s| F ñ .
(0)
When we insert the APSLG wave function in this formula we get

á F| t_m₁s⁺t_m₂t^˘+t_m₂t^˘t_m₁s| F ñ =

= d_m₁m₂(1-d_st)(1-d_tt^˘) á 0| g_mr_ms⁺l_m-s⁺l_m-sr_msg_m⁺| 0 ñ +

+( 1-d_m₁m₂) P_m₁^tP_m₂^{t^˘}.

(0)

á 0| g_mr_ma⁺l_mb⁺l_mbr_mag_m⁺| 0 ñ = w_m².
(0)
So, the last (Coulomb two-center) contribution to the energy can be written as:

E_coul⁽²⁾ = 2
ĺ
A < B
g_AB é
ë 2 ĺ
\Sbt_m₁ Î A t_m₂^˘ Î B m₁ ą m₂\endSbP_m₁^tP_m₂^{t^˘}+
ĺ
m Î A,B
w_m² ů
ű .
(0)

Total electronic energy is a sum of the above four terms:

E = E₁+E_coul⁽¹⁾+E_res+E_coul⁽²⁾.
(0)
As we have mentioned above the quantities entering the operators H_A (Eq. (12)) and H_AB (Eq. (13)) depend on the matrix elements of the SO(4) matrices h^A, where A stands for a ''heavy'' atom. According to Ref. [] the elements of the group SO(4) are functions of six parameters. The electronic energy of the system is thus a function of 6L angles (L is the number of heavy atoms in the molecule), determining the matrices h^A of transformation from the AO basis to the HO basis on each of the heavy atoms A and of the 2M parameters defining the amplitudes of different contributions in the geminals (M is the number of bonds). Each sextuple of angles defines an SO(4) matrix for one atom. This matrix can be presented as a product of six sequential rotation matrices in different two-dimensional subspaces of the whole four-dimensional space of atomic one-electron states (the Jacobi matrices). Each rotation is defined by one angle. Therefore, the dependence of any matrix element of h^A on the angle a_n ^A (n=1¸6) from the A-th sextuple can be represented as:

h_mi^A = d_mi^{a_n ^A}+c_mi^{a_n ^A}cos( a_n ^A) +s_mi^{a_n ^A}sin( a_n^A) ,
(0)
i.e. each element of the hybridization matrix h^A of the atom A is the linear combination of products of the cosines and sines of the 6 angles a_m ^A (m=1¸6) and the derivative of this matrix element with respect to a_n ^A is

¶h_mi^A
¶a_n ^A
= -c_mi^{a_n ^A}sin( a_n ^A) +s_mi^{a_n ^A}cos( a_n ^A) .
(0)
To obtain the optimal matrices h^A we perform the gradient optimization of the energy with respect to the angles determining these matrices. The expression for derivatives of the electronic energy in a_n ^A is derived as follows. Let us represent the energy of the ground state for the APSLG-MINDO/3 method in the form, which explicitly includes all the contributions depending on the HO's of a ''heavy'' atom A.

E = const+
ĺ
A
[ ĺ
\Sb t_m Î A

i Î A\endSb ( X_{t_m}ⁱ(A)(h_mi^A)²+Y_{t_m}ⁱ(A)(h_mi^A)⁴)+

+ ĺ
\Sb t_m Î A

i,j Î A

i < j\endSb T_{t_m}^ij(A)(h_mi^Ah_mj^A)²+ ĺ
\Sb t_m,t_n^˘ Î A

m < n

i Î A\endSb U_{t_mt_n^˘}ⁱ(A)(h_mi^Ah_ni^A)²+

+ ĺ
\Sb t_m,t_n^˘ Î A

i,j Î A

m < n,i < j\endSb (V_{t_mt_n^˘}^ij(A)h_mi^Ah_mj^Ah_ni^Ah_nj^A+

+W_{t_mt_n^˘}^ij(A)[(h_mi^Ah_nj^A)²+(h_mj^Ah_ni^A)²] )]+

+
ĺ
m
ĺ
\Sb i Î A_mj Î B_m\endSbZ_m^ijh_mi^A_mh_mj^B_m,

(0)
where A_m and B_m are the right and left ends of the chemical bond m. Different quantities in this expression can be represented as:

Z_m^ij = -4Q_mb_ij^AB_m;

X_{t_m}ⁱ(A) = 2U_i(A)P_m^t;

Y_{t_m}ⁱ(A) = ( ii | ii) ^AR_m^t;

T_{t_m}^ij(A) = 2[ ( ii | jj) ^A+2( ij | ij)^A] R_m^t;

U_{t_mt_n^˘}ⁱ(A) = 2( ii | ii) ^AP_m^tP_n^{t^˘};

V_{t_mt_n^˘}^ij = 4[ 4( ij | ij) ^A-( ii | jj) ^A] P_m^tP_n^{t^˘};

W_{t_mt_n^˘}^ij = 2[ 2( ii | jj) ^A-( ij | ij) ^A] P_m^tP_n^{t^˘}.

(0)
The derivative of the electronic energy with respect to the angle a_n ^A reads:

¶E^el
¶a_n ^A
= ĺ
\Sb t_m Î A

i Î A\endSb ( 2X_{t_m}ⁱ(A)h_mi^A+4Y_{t_m}ⁱ(h_mi^A)³) ¶h_mi^A
¶a_n ^A
+

+2 ĺ
\Sbt_m Î A

i,j Î A

i < j\endSb T_{t_m}^ij(A) é
ę
ë (h_mi^A)²h_mj^A ¶h_mj^A
¶a_n ^A
+(h_mj^A)²h_mi^A ¶h_mi^A
¶a_n ^A
ů
ú
ű +

+2 ĺ
\Sbt_m,t_n^˘ Î A

m < n

i Î A\endSb U_{t_mt_n^˘}ⁱ(A) é
ę
ë (h_mi^A)²h_ni^A ¶h_ni^A
¶a_n ^A
+(h_ni^A)²h_mi^A ¶h_mi^A
¶a_n ^A
ů
ú
ű +

+ ĺ
\Sbt_m,t_n^˘ Î A

i,j Î A

m < n,i < j\endSb ( V_{t_mt_n^˘}^ij(A)h_mj^Ah_ni^Ah_nj^A+2W_{t_mt_n^˘}^ij(A)(h_nj^A)²h_mi^A) ¶h_mi^A
¶a_n ^A
+

+ ĺ
\Sbt_m,t_n^˘ Î A

i,j Î A

m < n,i < j\endSb ( V_{t_mt_n^˘}^ij(A)h_mi^Ah_ni^Ah_nj^A+2W_{t_mt_n^˘}^ij(A)(h_ni^A)²h_mj^A) ¶h_mj^A
¶a_n ^A
+

+ ĺ
\Sbt_m,t_n^˘ Î A

i,j Î A

m < n,i < j\endSb ( V_{t_mt_n^˘}^ij(A)h_mi^Ah_mj^Ah_nj^A+2W_{t_mt_n^˘}^ij(A)(h_mj^A)²h_ni^A) ¶h_ni^A
¶a_n ^A
+

+ ĺ
\Sbt_m,t_n^˘ Î A

i,j Î A

m < n,i < j\endSb ( V_{t_mt_n^˘}^ij(A)h_mi^Ah_mj^Ah_ni^A+2W_{t_mt_n^˘}^ij(A)(h_mi^A)²h_nj^A) ¶h_nj^A
¶a_n ^A
.

(0)
The optimal wave function of the system is obtained by varying the parameters [`(a)]^A for the matrices h^A defining the HO's expansion coefficients for the heavy atom A and those for the amplitudes u_m, v_m, and w_m for the bond geminals. The optimal amplitudes u_m, v_m, and w_m are obtained by diagonalizing the effective bond Hamiltonians for each bond geminal. Their exact form will be presented elsewhere []. Each iteration step includes optimization of the HO parameters and of the geminal amplitudes. For the set of the amplitudes u_m , v_m, and w_m the angles determining the matrices h^A are obtained. After that for the fixed coefficients of the HO's the expansion coefficients of the geminals are determined. The alternating optimizations and diagonalizations are consistent and the convergence is rapidly achieved in all variables. The number of the steps in the procedure of the optimization of the energy functional for a molecule grows linearly while its size increases.

Results.

The scheme of determination of the optimal parameters of the variational APSLG wave function described in the above Section has been implemented as a program package BF'98 [] designed for calculation of the electronic structure of organic molecules. To estimate the validity of the APSLG approach we, first, performed calculations with use of the BF'98 package which uses both the form of the MINDO/3 Hamiltonian and the precise numerical values of its parameters []. Some preliminary results obtained for the fixed experimental geometries were published in Ref. ZhFizKhim. The results of the calculations on molecular structures determined by the energy minimization with respect to all the geometry parameters are obtained in the present work and shown in the third and fifth columns of Table 1 respectively. The computed electronic energies and heats of formation (column 3) can be compared with the results of the standard MINDO/3 calculations (column 5) for the structures optimized with use of the standard SCF-MINDO/3 method. Comparison of the electronic energies allows to evaluate the relative importance of delocalization and correlation of electrons. For the dihydrogen molecule the APSLG wave function is that of the full configuration interaction in the basis of two s-AO. That is why the energy of dihydrogen molecule is lower for the APSLG function than for the SCF function for the same bond length with the same parameterization.

For more complex molecules the APSLG electronic energy with the standard MINDO/3 parameters can be either higher or lower than that of the SCF approximation. Similar picture was also obtained in the non-empirical calculations [] using the APSLG trial wave function. As for calculations on the energy of molecules with electron lone pairs the effects of the intrabond correlation outweight those of delocalization. Thus the energy loss owing to the restriction in the flexibility of one-electron states (HO's) as compared to the SCF-MINDO/3 method is smaller than the energy gain due to better description of intrabond electron correlations. At the same time the delocalization contribution is more important for the molecules with bonds between heavy atoms.

Comparison of the results of the APSLG-MINDO/3 method with the standard MINDO/3 parametrization (column 3) with the experimental ones (column 2) shows that its accuracy is not sufficient even for description of the simplest organic compounds. It is thus necessary to modify the parameters for the APSLG method. Indeed, the forms of the variational wave functions underlying, respectively, the SCF-MINDO/3 and the APSLG-MINDO/3 method are quite different. The latter by construction does take into account electron correlations. The SCF based semiempirical theories, on the other hand, usually assume that electron correlations can be absorbed by their parameters. However, the parameterization can not reproduce the qualitative features of the electronic structure which depend on correlation. For example, when the parameters are fixed the dissociation energy (the valley depth) of, say, H₂ molecule obtained by the SCF method is always larger than that obtained with use of variational function allowing for correlation. In the MINDO/3 method this discrepancy with the experiment is partially cured by adjusting the corresponding b_AB parameter. Additionally, a special functional form for the core repulsion is introduced and parameterized in the MINDO/3 method for heavier atoms as well. The APSLG-MINDO/3 method takes some fraction of correlation into account explicitly. Thus, the b_AB parameters must be readjusted.

To find the resonance parameters conforming to the APSLG form of the trial wave function we performed calculations on electronic structure of the test set of molecules containing hydrogen, carbon, nitrogen, and oxygen atoms. This set was taken from previous works devoted to the parametrization of semiempirical methods MNDO [], AM1 [] and PM3 []. These papers also contain a set of experimental data on the heats of formation of the molecules used. The parameters b_AB were adjusted to reproduce the heats of formation calculated by the APSLG-MINDO/3 method for the test set of molecules taken at their optimized structures to the experimental values. The criterium for the optimization was based on the reproducing the energy increments for the series of homologues rather than the heats of formation of all molecules. The resulting set of b_AB parameters of the APSLG-MINDO/3 method as compared to those of the SCF-MINDO/3 method is given in Table 2. This set slightly differs from that obtained in our previous work []. The similarity of the parameters of two methods using fundamentally different trial wave functions indicates some internal consistency of the MINDO/3 parametrization (see above). All subsequent calculations are made with use of the adjusted parameters b_AB given in Table 2. Table 1 contains the heats of formation calculated by the APSLG-MINDO/3 method (column 4) as compared to the experimental data (column 2) and the heats of formation calculated by the standard MINDO/3 method (column 5) at the optimized molecular geometries. In the previous work [] we have found that the total energies are approximately additive in the homologic series in agreement with experiment [,]. The analysis of the analogous data for optimized geometries shows that the additivity is well fulfilled and, moreover, the difference between them tends to the same value (-3613 kcal mol^-1) as in the case of the experimental molecular geometries. Data of table 1 show that the difference between the heats of formation calculated by the APSLG-MINDO/3 method for the closest homologues - saturated hydrocarbons (excluding the methane - ethane pair) is about 5 kcal/mol that is in a good agreement with the experimental data []. This difference is larger than 6 kcal/mol for the SCF-MINDO/3 method. It causes significant divergence between the experimental data on the heats of formation and the estimates by the SCF-MINDO/3 method increasing with the number of carbon atoms in the hydrocarbon molecule. Cyclic hydrocarbons do not manifest the additivity and should not since the strain energies for different cycles differ. The total energy increment in the APSLG-MINDO/3 method for homologous normal amines rapidly converges to that value, which is characteristic for the normal alkanes as well. This additivity i.e. the linear dependence of the system properties on its size allows to hope that the present semiempirical method based on the APSLG approximation can serve as a natural starting point for derivation of other additive schemes. The comparison with experiment for two computational methods (see Table 1) shows that the APSLG-MINDO/3 and the SCF-MINDO/3 methods reach similar accuracy when address the heats of formation of organic compounds. At the same time we can note that the mean deviation in the heats of formation given by the formulae

s = 1
N
N
ĺ
i = 1
| H_theor-H_exp|
(0)
is 6.6 kcal/mol for the APSLG-MINDO/3 method and 7.6 kcal/mol for the SCF-MINDO/3 method for the test set of molecules. When parameterizing the APSLG-MINDO/3 method we try to reproduce the heats of formation of non-branched compounds and to take into account the additivity of energy in the series of homologues. That is why we use the mean deviation of the form Eq. (35) and not the standard mean square deviation which ascribes larger weight for large deviations. Maximal deviation from the experimental data is observed for branched compounds for both the SCF and APSLG methods. The intrabond correlations in the APSLG-MINDO/3 method does not cure this defect which is common for both the MINDO/3 methods (or, may be, for the MINDO/3 parameterization itself).

The similarity of the calculated and experimental heats of formation can not be the ultimate criterium of the quality of the calculation. It is also essential to reach some consistency of the calculated minima positions on the potential energy surfaces with the experimental molecular geometries. Therefore we do not restrict ourselves by the electronic structure calculations at the selected points of the nuclear configuration space and have implemented the procedure of the gradient search for the local minima of the total energy. The expressions for the gradients of the energy with respect to the nuclear displacements were derived analytically. Using the adjusted values of the parameters b_AB, given in Table 2, we have calculated the optimal geometry structures for a series of simple molecules. In the case of the hydrogen molecule the length of the H-H bond is longer than the experimental value by 0.01 Å . The internuclear C-H separation in the methane molecule calculated by the APSLG-MINDO/3 method coincides with the experimental one with the accuracy of 0.002 Å (theoretical value 1.092 Å , experiment - 1.094 Å []). At the same time the calculated length of the C-C bond in the ethane molecule equals to 1.484 Å . This is smaller than the experimental value of 1.536 Å . However, for propane and the higher alkanes the discrepancy between the calculated and experimental bond lengths is slightly smaller (on the 0.01 Å ). It is important to notice, however, that the standard SCF-MINDO/3 method gives for the length of the C-C bond in ethane the value of 1.474 Å and, therefore, the APSLG-MINDO/3 approach slightly improves it. The APSLG-MINDO/3 optimized length of the C-C bond improves with the growth of the carbon chain. The optimized length of the C=C bond in the unsaturated ethylene molecule equals to 1.335 Å , which is very close to the experimental value of 1.339 Å . The SCF-MINDO/3 method gives the value 1.313 Å , which is not in a good agreement with the cited experimental value. More interesting example is the optimization of geometry structure of quadricyclane and norbornadiene molecules. Starting from the same geometry but different distributions of chemical bonds (connectivity scheme) we obtain after the geometry optimization two significantly different geometrical structures - one close to the experimentally observed structure of quadricyclane (lengths of four single bonds in the cyclobutane ring of the quadricyclane molecule equal to 1.54 Å each) and norbornadiene, respectively (lengths of two double bonds equal to 1.342 Å but the distance between the adjacent carbon atoms of two different double bonds equals to 2.46 Å ). Reproducing the geometrical structure of the cyclobutane molecule is a complex problem for molecular mechanics and quantum chemistry. The experimentally observed structure is non-planar with the torsion angle of 27 degrees. At the same time most of the computational procedures (for example, SCF-MINDO/3) lead to the planar structure. The analysis of the PES for the cyclobutane molecule by the APLSLG-MINDO/3 method shows that the global minimum on the PES corresponds to the planar configuration. At the same time the calculated PES has the local minimum for the non-planar configuration with the torsion angle of 22 degrees.

Discussion.

As it is mentioned in the Introduction, quantum chemistry of large molecules faces an important problem of constructing calculation procedures with the growth of computational costs linear in N (N characterizes the size of the system). Solution of this problem requires applying that or another approach to separation of electronic variables. The standard way of doing this assumed in quantum chemistry is the SCF (one-electron, Hartree-Fock) approximation for the wave function of the ground state of electrons. The requirements for computational resources of the SCF procedure grow as N³ and hence the latter can not to be considered as a basis for constructing methods linear in N. Moreover, the SCF approximation requires additional account of correlation to become useful for description of bond cleavage. Of course, it is not clear a priori what is preferable: to take into account one-electron transfers (resonance) between different AO's with maximal completeness and by this to get to delocalized form of the one-electron states and to incorrect description of the homolytic cleavage of the s-bonds or to take into account pair electron correlations within the bonds and to get correct behavior of the trial wave function under homolytic clevage of bonds and to be forced to consider the interbond one-electron transfers as corrections. At any rate second way preserves the prospects for the linear dependence of the computational costs on the system size and allows to recover traditional chemical concepts of bonds and lone pairs on the basis of variational calculation.

To demonstrate the computational capacities of the APSLG-MINDO/3 method we carried out calculations (for the fixed geometry) for a series of normal saturated hydrocarbons ranging from CH₄ to C₂₀H₄₂ by the APSLG-MINDO/3 and SCF-MINDO/3 methods. Figure 1 shows the dependence of the computation time (in seconds) on the number of carbon atoms in the homologue for the both methods. It can be easily seen that the dependence of computation time on the system size is essentially non-linear in the case of the SCF approximation and is practically linear for the geminal approach. The method APSLG-MINDO/3 is faster than the SCF-MINDO/3 already for the simplest hydrocarbons. In the case of normal hydrocarbon C₂₀H₄₂ (its semiempirical calculation uses 122 basis functions) the computation time for two methods differs 30 times in favor of the APSLG approach.

The APSLG-MINDO/3 approach can be also applied to unsaturated organic molecules (but not to aromatic ones). In this case two different geminals are assigned to each of the double bonds. In this context an old question about the character of each of two C-C bonds in ethylene [] (and other unsaturated hydrocarbons) can be considered. The APSLG-MINDO/3 method allows to construct two different APSLG wave functions for the ethylene molecule: one with two equivalent geminals for the double C=C bond which corresponds to equivalent (bent-type or ''banana'') bonds between the carbon atoms, and another one with two non-equivalent geminals which correspond to s- and p-bonds. The initial conditions of the energy minimization turn out to be very important. The type of double C=C bond (bent or s-p-separated) remains unchanged in the course of optimization. In the case of the bent-type bonds the optimal value of energy is higher than that for the s-p-separated bond, i.e. the energy calculation with use of the APSLG-MINDO/3 approach and optimized molecular geometry shows that the electronic structure with the nonequivalent s- and p-bonds is energetically favourable. However the difference is very small since the local minimum on the energy surface corresponding to the s-p double bond is only by less than 1 kcal mol^-1 deeper than that of the bent-type bond. The optimized bond length for the bent-type double bond is only slightly (by 0.0003 Å ) larger than that for the s-p-double bond. The question about the preferability of one or another type of double bond was also studied in [] by the full GVB. In the case of the ethylene molecule the conclusion about closeness of the energies for the two ways of bonding was drawn. The calculations with the frozen core have shown that the s-p-separation is slightly more preferable in agreement with our results.

Besides the correct dissociation limit the APSLG-MINDO/3 approximation has some computational advantages compared with usual MINDO/3 method. The method APSLG-MINDO/3 provides somewhat more precise values of energy. In the framework of the proposed approach it makes it possible to determine naturally such bond characteristics as ionic and covalent contributions and polarity of bond. We can rewrite the expression for each geminal as Surjan:

g_m⁺ = I_m
Ö2
é
ë
Ö

1+l_m

r_ma⁺r_mb⁺+
Ö

1-l_m

l_ma⁺l_mb⁺ ů
ű + C_m
Ö2
[ r_ma⁺l_mb⁺+l_ma⁺r_mb⁺] ,
(0)
where I_m is the amplitude of the ionic contribution to the geminal, C_m and l_m measure respectively the bond covalency and polarity and the normalization condition obviously leads I_m²+C_m² = 1. Obtained bond characteristics allow to determine charges on the atoms. Then for each electron pair (bond) we can determine the electron density at the right and left atoms of the m-th bond, respectively:

C_m²+I_m²( 1+l_m)
2
, C_m²+I_m²( 1-l_m)
2
.
(0)
For each atom the sum of the contributions Eq. (37) over all its geminals including electron lone pairs gives the electron density on the atom and consequently the effective atomic charge. Table 3 contains ionic and covalent contributions and polarities of the bonds and atomic charges for some simple organic compounds calculated for the optimized geometries. The covalent contribution prevails over the ionic one for all considered molecules. Even large differences in bond polarities only slightly affect the ratios of the ionic and covalent contributions. The values of bond polarities themselves qualitatively correspond to usual concepts of electronegativity of different atoms. The effective atomic charges can be compared with the charge distributions obtained by other methods. The APSLG-MINDO/3 method predicts a shift of electron density in methane from hydrogen atom to carbon atom meanwhile the SCF-MINDO/3 gives the reverse picture in contradiction with accepted notions. It should be noted that the C-N bond proves to be almost nonpolar (in the above sense). The parameters of the C-H bond in methane and the N-H bond in ammonia are very similar (see Table 3). Intrabond correlations (i.e. the deviations of the amplitudes u_m, v_m, and w_m of the two-electron states from the SCF value 0.5) are very significant for the wave functions of the ground states for polar species like HF or CH₃F due to large polarities of the H-F and C-F bonds.

Determination of hybridization for the HO's on the basis of variational principle is an important feature of the proposed approach. As it can be expected in methane the sp³-hybridization is recovered under the variational determination. The HO of the F-H bond in the HF molecule is almost pure p-orbital (more precisely the hybridization of this orbital can be represented as sp^39.0 which demonstrates an extreme unevenness of the hybridization scale). The HO's of electron lone pairs form two pure p -functions and one practically pure s-function. It is necessary to mention that the mixing (quantum mechanical interference) of the HO's of different electron lone pairs (not bonds) which are centered on the same atom does not change the electronic energy and degree of this mixing is defined by initial conditions of optimization. Despite the fact that the amplitudes of different two-electron states in the bond geminal expansions in methane and ammonia molecules are similar (see below), the HO's in ammonia differ significantly from the sp³ ones characteristic for methane. The analysis of the transformational matrix from the AO basis to the HO basis reveals the sp^7.14-hybridization for the N-H bonds and the sp^0.58-hybridization for the lone pair. In the framework of proposed method the form of the electron lone pairs is determined unequivocally on the basis of the variational principle.

The HO's of the carbon atoms in ethane slightly deviate from the sp³-form. The p-character is slightly more pronounced for the C-H bonds. The HO's of the C-C bond have the sp^2.26 -hybridization but the C-H bonds have the sp^3.33-hybridization. Directions of maximal density for the HO's of the C-C bonds in the cyclopropane molecule do not coincide with directions of the bonds themselves. As for the HF molecule the p-character of the HO participating in the chemical bond increases but the electron lone pair has essentially the s-character. The C-F bond in CH₃F has essentially covalent character and it is less polar than the H-F bond in HF. The properties of the C-H bond are not very much sensitive to its environment and are the same for CH₄, C₂H₆, and CH₃F. Main differences in the C-H bonds of C₂H₆ and CH₃F are due to different hybridization of orbitals. It is interesting that the HO's of fluorine in CH₃F are close to those in HF (sp^37.9 vs. sp^39.0).

Conclusions.

A semiempirical method of calculations on molecular electronic structure is developed in this work. This method uses the trial wave function of the antisymmetrized product of strictly localized geminals approximation. The calculations with the standard MINDO/3 parametrization have revealed that the APSLG wave function has quality comparable with that of the SCF version of the MINDO/3 approximation for the range of characteristic intramolecular interatomic distances and, moreover, have correct asymptotical behavior in the limit of cleaved bonds. It has been demonstrated that the slight modification of the pair resonance parameters b_AB makes it possible to obtain somewhat better agreement of the heats of formation and the equilibrium geometries calculated by the APSLG-MINDO/3 method with the experimental data than the SCF-MINDO/3 procedure permits. The APSLG-MINDO/3 method satisfies the O(N)-scalability condition and permits to calculate the electronic properties of very large organic molecules. In the framework of the present approach based on the variational wave function of the APSLG approximation it is possible to recover the intuitive chemical concepts like bonds, their ionic and covalent components, polarities, hybrid orbitals, and lone pairs which are not so transparent in the SCF-type approaches.

Helpful discussions with Dr. I.A. Misurkin and Dr. V.A. Tikhomirov are gratefully acknowledged.

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DH_f DH_f DH_f DH_f

Molecule MINDO/3 modified MINDO/3

(expt.) (APSLG) (APSLG) (SCF)

1 2 3 4 5

H₂ 0.0 -1.475 -0.101 0.131

CH₄ -17.8 -5.655 -8.430 -6.275

C₂H₆ -20.04 -8.254 -19.514 -19.849

C₃H₈ -25.00 -5.500 -25.185 -26.527

n-C₄H₁₀ -30.00 -2.226 -29.993 -32.655

iso-C₄H₁₀ -32.00 5.351 -22.379 -24.710

n-C₅H₁₂ -35.09 1.027 -35.225 -38.973

neo-C₅H₁₂ -40.15 22.811 -12.126 -14.632

cyclopropane 12.7 41.000 18.472 8.524

cyclobuthane 6.8 33.459 2.360 -8.088

cyclopenthane -18.3 12.694 -28.193 -27.795

cyclohexane -29.49 19.855 -29.166 -32.505

NH₃ -11.0 -21.245 -4.529 -9.125

CH₃NH₂ -5.5 4.583 -1.034 -4.615

C₂H₅NH₂ -11.3 2.007 -11.777 -15.708

n-C₃H₇NH₂ -16.8 5.270 -16.776 -21.874

iso-C₃H₇NH₂ -20.0 6.940 -14.718 -18.341

tert-BuNH₂ -28.9 22.597 -6.757 -13.215

(CH₃)₂NH -4.4 33.488 5.535 4.310

(CH₃)₃N -5.7 68.316 19.136 21.027

N₂H₄ 22.8 1.792 21.307 3.166

CH₃NHNH₂ 22.6 24.637 23.102 8.366

(CH₃)₂NNH₂ 20.1 56.978 33.039 22.242

CH₃NHNHCH₃ 22.0 50.517 27.588 15.019

H₂O -57.8 -60.451 -55.677 -53.611

CH₃OH -48.16 -33.099 -48.549 -50.573

C₂H₅OH -56.21 -36.503 -60.167 -64.292

1-C₃H₇OH -60.98 -33.325 -65.113 -70.417

2-C₃H₇OH -65.19 -34.383 -66.008 -69.117

tert-BuOH -74.7 -24.387 -63.509 -65.614

(CH₃)₂O -60.3 1.337 -33.905 -51.191

H₂O₂ -32.5 -38.670 -32.720 -29.265

Table 0: Heats of formation (kcal/mol) of test molecules obtained experimentally and calculated with use of SCF-MINDO/3 and APSLG-MINDO/3 methods.

A B b_AB (APSLG) b_AB (MINDO/3)

H H 0.243007 0.244770

H C 0.315839 0.315011

H N 0.353716 0.360776

H O 0.414559 0.417759

C C 0.427797 0.419907

C N 0.429386 0.410886

C O 0.486514 0.464514

N N 0.379342 0.377342

O O 0.657007 0.659407

Table 0: Parameters b_AB of APSLG-MINDO/3 and SCF-MINDO/3 methods (eV).

Ionic Covalent Bond

Molecule Bond contribution contribution polarity Charge Charge

A-B I² C² l A B

H₂ H-H 0.434 0.566 0 0 0

CH₄ C-H 0.415 0.585 -0.157 -0.260 0.065

C₂H₆ C-C 0.411 0.589 0 -0.126 -0.126

C-H 0.411 0.589 -0.102 -0.126 0.043

N₂H₄ N-N 0.363 0.637 0 -0.114 -0.114

N-H 0.415 0.585 -0.137 -0.114 0.057

NH₃ N-H 0.408 0.592 -0.111 -0.135 0.045

CH₃NH₂ C-N 0.398 0.602 0.051 -0.145 -0.142

C-H 0.411 0.589 -0.133 -0.145 0.055

N-H 0.415 0.585 -0.148 -0.142 0.061

HF F-H 0.470 0.530 -0.699 -0.329 0.329

CH₃F C-H 0.416 0.584 -0.147 0.032 0.061

C-F 0.386 0.614 0.556 0.032 -0.215

H₂O O-H 0.463 0.537 -0.523 -0.484 0.242

Table 0: Charges, ionic and covalent contributions and polarities of chemical bonds calculated by APSLG-MINDO/3 method.

File translated from T_EX by T_TH, version 2.67.
On 11 Apr 2000, 01:07.

	DH_f	DH_f	DH_f	DH_f
Molecule		MINDO/3	modified	MINDO/3
	(expt.)	(APSLG)	(APSLG)	(SCF)
1	2	3	4	5
H₂	0.0	-1.475	-0.101	0.131
CH₄	-17.8	-5.655	-8.430	-6.275
C₂H₆	-20.04	-8.254	-19.514	-19.849
C₃H₈	-25.00	-5.500	-25.185	-26.527
n-C₄H₁₀	-30.00	-2.226	-29.993	-32.655
iso-C₄H₁₀	-32.00	5.351	-22.379	-24.710
n-C₅H₁₂	-35.09	1.027	-35.225	-38.973
neo-C₅H₁₂	-40.15	22.811	-12.126	-14.632
cyclopropane	12.7	41.000	18.472	8.524
cyclobuthane	6.8	33.459	2.360	-8.088
cyclopenthane	-18.3	12.694	-28.193	-27.795
cyclohexane	-29.49	19.855	-29.166	-32.505
NH₃	-11.0	-21.245	-4.529	-9.125
CH₃NH₂	-5.5	4.583	-1.034	-4.615
C₂H₅NH₂	-11.3	2.007	-11.777	-15.708
n-C₃H₇NH₂	-16.8	5.270	-16.776	-21.874
iso-C₃H₇NH₂	-20.0	6.940	-14.718	-18.341
tert-BuNH₂	-28.9	22.597	-6.757	-13.215
(CH₃)₂NH	-4.4	33.488	5.535	4.310
(CH₃)₃N	-5.7	68.316	19.136	21.027
N₂H₄	22.8	1.792	21.307	3.166
CH₃NHNH₂	22.6	24.637	23.102	8.366
(CH₃)₂NNH₂	20.1	56.978	33.039	22.242
CH₃NHNHCH₃	22.0	50.517	27.588	15.019
H₂O	-57.8	-60.451	-55.677	-53.611
CH₃OH	-48.16	-33.099	-48.549	-50.573
C₂H₅OH	-56.21	-36.503	-60.167	-64.292
1-C₃H₇OH	-60.98	-33.325	-65.113	-70.417
2-C₃H₇OH	-65.19	-34.383	-66.008	-69.117
tert-BuOH	-74.7	-24.387	-63.509	-65.614
(CH₃)₂O	-60.3	1.337	-33.905	-51.191
H₂O₂	-32.5	-38.670	-32.720	-29.265

A	B	b_AB (APSLG)	b_AB (MINDO/3)
H	H	0.243007	0.244770
H	C	0.315839	0.315011
H	N	0.353716	0.360776
H	O	0.414559	0.417759
C	C	0.427797	0.419907
C	N	0.429386	0.410886
C	O	0.486514	0.464514
N	N	0.379342	0.377342
O	O	0.657007	0.659407

		Ionic	Covalent	Bond
Molecule	Bond	contribution	contribution	polarity	Charge	Charge
	A-B	I²	C²	l	A	B
H₂	H-H	0.434	0.566	0	0	0
CH₄	C-H	0.415	0.585	-0.157	-0.260	0.065
C₂H₆	C-C	0.411	0.589	0	-0.126	-0.126
	C-H	0.411	0.589	-0.102	-0.126	0.043
N₂H₄	N-N	0.363	0.637	0	-0.114	-0.114
	N-H	0.415	0.585	-0.137	-0.114	0.057
NH₃	N-H	0.408	0.592	-0.111	-0.135	0.045
CH₃NH₂	C-N	0.398	0.602	0.051	-0.145	-0.142
	C-H	0.411	0.589	-0.133	-0.145	0.055
	N-H	0.415	0.585	-0.148	-0.142	0.061
HF	F-H	0.470	0.530	-0.699	-0.329	0.329
CH₃F	C-H	0.416	0.584	-0.147	0.032	0.061
	C-F	0.386	0.614	0.556	0.032	-0.215
H₂O	O-H	0.463	0.537	-0.523	-0.484	0.242

A.M. Tokmachev and A.L. Tchougréeff Karpov Institute of Physical Chemistry, Vorontsovo pole 10, Moscow, 103064, RUSSIA

Semiempirical implementation of strictly localized geminals for analysis of molecular electronic structure

Abstract

Introduction.

Theory.

Wave function and the Hamiltonian.

Electronic energy in the APSLG-MINDO/3 approximation.

Results.

Discussion.

Conclusions.

References

A.M. Tokmachev and A.L. Tchougréeff
Karpov Institute of Physical Chemistry,
Vorontsovo pole 10, Moscow, 103064, RUSSIA

Electronic energy in the APSLG-MINDO/3
approximation.