Received ; accepted

Abstract

Previously we developed a computationally effective quantum chemical method based on the trial wave function in the form of antisymmetrized product of strictly localized geminals (SLG). Semiempirical implementations based on the MINDO/3, MNDO, AM1, and PM3 parameterization schemes have shown their preferability in description of molecular geometries, heats of formation and ionization potentials with respect to the SCF analogues. Significant progress is due to variational determination of localized one-electron states (hybrid orbitals). An important restriction is that the SLG scheme applies only to systems with well-defined two-center two-electron bonds. In the present communication we describe the extension of the SLG approach to molecular systems with essentially delocalized electrons. General electron group functions approach based on the combination of geminals and molecular orbitals (SLG/SCF) constructed from hybrid orbitals is used. It leads to computational effectiveness of the SLG/SCF method comparable with that of the SLG method. Free radicals and p-electron systems serve as tests of new approach.

hybridization, geminals, molecular orbitals, p-systems

31.90.

Introduction

Advances in applied quantum chemistry are mainly due to possibility of reliable electronic structure calculations for medium and large molecular systems. Unpleasant scaling properties of computational costs with growth of the object for calculation has led to understanding that special computation techniques should be developed for large molecules. There are two general strategies proposed in the literature: the first way is construction of methods with weak dependence of computational costs on the system's size, or the so called O(N)-methods (for reference see [,]); the second way is the so called hybrid schemes based on the combination of different level descriptions for different parts of the system - the reactive center is described by precise quantum chemical methods while the rest is described by molecular mechanics, i.e. by classical potential functions (for discussion on hybrid schemes see []). Both types of strategies are essentially based on local properties of wave functions.

Handling local elements of wave function is not a trivial problem. It requires separation of electron variables. In fact, the problem of junction between quantum and classical subsystems in hybrid methods is the most important one. Previously, we established that the trial wave function in the form of antisymmetrized product of strictly localized geminals (SLG) in combination with semiempirical implementation can be useful for solving problems arising in the field of quantum chemical modeling of large molecules. First of all, SLG-based methods possess pleasant scalability properties: computational costs of the SLG-MINDO/3 method grows linearly with the size of the system []. In the case of SLG-NDDO methods with three popular parameterization schemes (MNDO, AM1, and PM3) the quadratic contribution to the dependence of computational costs on the size of system becomes significant []. At the same time approximately linear scalability of computational costs can be achieved in the case of the SLG-NDDO method by using multipole expressions for two-center Coulomb interactions. It is noteworthy that the SLG methods give better agreement than the corresponding SCF analogues with the experiment for heats of formation, molecular geometries and ionization potentials. Moreover, the SLG wave function (unless the SCF one) possesses correct behaviour under cleavage of chemical bonds and constitutes natural basis for analysis of molecular structure in terms of chemical bonds and lone pairs. On the other hand, the SLG wave function can serve as a valuable tool for constructing junction in hybrid QM/MM schemes and its substantiation (see [] and references therein).

Though the SLG wave have proven to be appropriate for solving problems of quantum chemistry of large molecules, it has significant drawback: it is not universal and can be applied only to molecules with well defined two-center two-electron chemical bonds and lone pairs. In the present communication we describe method extending the SLG description to molecules where electron delocalization is significant. When delocalization is not over whole molecule and relatively small electron groups can be defined (for example, small p-electron systems) the resulting method would possess scalability properties close to those characteristic for the SLG method.

Method

To reach optimal scalability properties we use strictly local one-electron basis states. These states are hybrid orbitals (HOs) constructed for each non-hydrogen atom by orthogonal transformation (SO(4) matrix hA) of one s- and three p-orbitals:
tps+ =

i A 
hpiAais+,
(0)
where t denotes hybrid orbital. The orthogonality of transformation matrices ensures the strong orthogonality of carrier spaces for electron groups. Each hybrid orbital is uniquely assigned to particular electron group. Some electron groups are described by antisymmetrized product of molecular spin-orbitals constructed as:
bis s+ =

p {SCF} 
cis ptps+,
(0)
while others are described by geminals constructed as linear combinations of singlet configurations:
gm+ = umrma+rmb++vmlma+lmb++wm(rma+lmb++lma+rmb+),
(0)
where HOs rm and lm correspond to "right" and "left" ends of chemical bond and the normalization condition reads:
á 0| gmgm+| 0 ñ = um2+vm2+2wm2 = 1.
(0)
In the case of lone electron pair only one configuration survives with the amplitude equal to unity.

The wave function for the whole molecule (SLG/SCF) and is represented by antisymmetrized product of geminals and molecular orbitals:
| Y ñ =


is 
bis+



m 
gm+
| 0 ñ .
(0)
It is convenient to introduce the density matrices for separate groups. In the case of SCF groups the spin-dependent density matrices are defined as:
Ppqs =

is  
cis pcis q,
(0)
while in the case of geminals one- and two-electron density matrices are spin-independent and defined as:
Pmrr = um2+wm2, Pmll = vm2+wm2, Pmrl = Pmlr = (um+vm)wm,
Gmrr = um2Gmll = vm2Gmrl = Gmlr = wm2.
(0)

We consider the electronic Hamiltonian of the NDDO approximation transformed to the basis of hybrid orbitals. The expressions for molecular integrals in the basis of hybrid orbitals through the transformation matrices hA and molecular integrals in the basis of atomic orbitals can be found in Refs. [,]. The energy of the whole molecular system can be found as usually by averaging of the electronic Hamiltonian over electronic wave function and adding specific contribution representing core-core repulsion. The electronic energy in the present approximation is a sum of contributions of four types according to their physical meaning. The first type of contributions to the energy arises from attraction of electrons to the cores. For any atom A it can be written as:


pq {A,SCF} 
(UpqA+

B A 
Vpq,BA)

s 
Ppqs +
+2

tm {A,SLG} 
(UtmtmA+

B A 
Vtmtm,BA)Pmtt.
(0)
The second type of contributions to the energy corresponds to one-center electron-electron repulsion and can be written as:


ppqq {A,SCF} 
{(pp|qq)APppa Pqqb +
+ 1
2
[(pp|qq)A-(pq|qp)A]

s 
Ppps Pqqs }+
+

[(pq {A,SCF}) || (tm {A,SLG})] 
[2(pq|tmtm)A-(ptm|tmq)A]Pmtt

s 
Ppqs +
+

tmtn {A,SLG} 
[4(tmtm|tntn)A-2(tmtn|tntm)A]PmttPntt+
+

tm {A,SLG} 
(tmtm|tmtm)AGmtt.
(0)
The resonance (one-electron transfer) contributions to the energy are non-zero only for intragroup electron transfers:
-2

[(p {A,SCF}) || (q {B,SCF})] 
bpqAB

s 
Ppqs -4

[(tm {A,SLG}) || ([t\tilde]m {B,SLG})] 
btm[t\tilde]mABPmt[t\tilde].
(0)
The last type of contributions to the energy arises from two-center electron-electron repulsion. It takes the following form for any pair of atoms A and B:


[(pp {A,SCF}) || (qq {B,SCF})] 
(pp|qq)AB{(

s 
Ppps )(

t 
Pqqt )-

s 
PpqsPqps }+
+2

[(tm {A,SLG}) || (pq {B,SCF})] 
(tmtm|pq)ABPmtt

s 
Ppqs +
+2

[(pq {A,SCF}) || (tm {B,SLG})] 
(pq|tmtm)ABPmtt

s 
Ppqs -
-2

[(tm {A,SLG}) || ([t\tilde]m {B,SLG})] 


[(p {A,SCF}) || (q {B,SCF})] 
(tmp|q ~
t
 

m 
)ABPmt[t\tilde]

s 
Ppqs +
+2

[(tm {A,SLG}) || (tn {B,SLG})] 
(tmtm|tntn)AB[2(1-dmn)PmttPntt+dmnGmtt]-
-4

[(tm {A,SLG}) || ([t\tilde]m {B,SLG})] 


[(tn {A,SLG}) || ([t\tilde]n {B,SLG})] 
(tmtn| ~
t
 

n 
~
t
 

m 
)ABPmt[t\tilde]Pnt[t\tilde].
(0)

The parameters of electronic structure are obtained on the basis of the energy minimization procedure. Its implementation includes iterative optimizations of one-electron basis states and determinations of density matrices for groups. The optimization of the HOs is performed by gradient minimization of the energy with respect to sextuples of parameters defining SO(4) matrices hA. The optimal geminal amplitudes um, vm, and wm are determined by diagonalizations of 3×3 effective Hamiltonians. One-electron density matrices for groups described by molecular orbitals can be determined in the framework of one of three procedures: RHF, UHF, and ROHF ones. The corresponding Fock matrices are usual ones with modified one-electron matrix elements:
~
h
 

pq 
= dAB{UpqA+

B A 
Vpq,BA+
+

tm {A,SLG} 
[2(pq|tmtm)A-(ptm|tmq)A]Pmtt+
+2

tm {C A,SLG} 
(pq|tmtm)ACPmtt}-
-(1-dAB){bpqAB+

[(tm {A,SLG}) || ([t\tilde]m {B,SLG})] 
(tmp|q ~
t
 

m 
)ABPmt[t\tilde]}.
(0)
The averaging of two-electron operators producing Fock operator is not affected by the presence of other groups. It is clearly seen that the number of elementary steps in the optimization procedure is proportional to the size of the system if all the groups are chosen to be small. Implementation of geometry optimization procedure was based on the analytic gradients.

Results and Discussion

The method proposed was implemented for semiempirical Hamiltonians MINDO/3, MNDO, AM1, and PM3. The SLG/SCF method allows to study correlation and delocalization effects in molecules. The energy of delocalization of molecule can be defined as difference between molecular energies obtained in the framework of the SCF approach and in the framework of scheme based on the separation of molecule in groups each of them calculated by the SCF approach. The correlation energy can be calculated for any chemical bond: it is simply the difference between energies obtained for the geminal in the SLG and SCF approximations with all other electronic structure parameters fixed. Our numerical calculations have shown that the energies of electron delocalization grow with the size of molecular system and additivity in them is only approximate since, for example, the difference of these energies for ethane and methane is 18.7 kcal/mol while the same difference for propane and ethane is 19.3 kcal/mol. Another conclusion is that the delocalization energy for molecules with lone electron pairs is significantly smaller due to impossibility of electron transfer into lone electron pairs. The numerically close results for the delocalization energy were obtained using second order perturbation theory for electron transfers between different electron groups. It certifies that the scheme for determination of delocalization energies is internally consistent.

We also calculated the intrabond correlation energies for a series of compounds at their equilibrium geometry parameters. It is clear that the correlation energy becomes very large for large interatomic distances due to incorrect asymptotic behaviour of the SCF wave function. At the same time, the correlation energy for most chemical bonds at the equilibrium geometries are not very large and all close to 1 kcal/mol. Larger values are obtained for chemical bonds formed by heteroatoms. The correlation energies for similar bonds in similar environment are quite close. The most interesting situation is for multiple bonds: the correlation energy for s-bond is very small while the correlation energy for p-bonds is large (several kcal/mol).

To get quantitative coincidence with the experiment in estimation of physical properties of molecular systems the method should be carefully parameterized. It is clear that this problem is not trivial since the estimated electronic structure parameters depend on the particular separation of molecule on groups. Our idea is that unique parameters can be found for classes of molecules with well defined separation on groups. Here we consider molecules with p-electron systems. The best coincidence with the experiment can be achieved by total re-parameterization of semiempirical Hamiltonian. At the same time quite reasonable results can be obtained after only slight re-parameterization of the Hamiltonian. We note that the information about structure of wave function is mainly loaded onto the resonance parameters. We try to reproduce the experimental data on heats of formation and molecular geometries by adjusting only few resonance parameters for p-electron systems. The strategy of re-parameterization of only resonance parameters was quite successful in the case of the SLG-based methods [,]. The following values of resonance parameters were found to reproduce experimental data: in the case of MNDO parameterization bpp(C)=9.92 eV and bpp(N)=21.97 eV, while in the case of MINDO/3 parameterization bCCp=0.452, bNCp=0.371, and bNNp=0.400. It is clear that in the case of some systems like s-electron radicals with one orbital assigned to SCF subsystem the re-parameterization is not necessary at all since resonance interactions within SCF group are absent.

Let us consider results of calculations of some simple molecular systems using the SLG/SCF method in comparison with the SCF method. We calculated heats of formation for some test set of molecules using MNDO and MINDO/3 parameterization schemes. Our results show that in the case of MNDO parameterization the degree of coincidence with the experiment for the SCF and SLG/SCF methods depends on the molecule. The SCF scheme is somewhat better suited for molecules with multiple bonds, while the SLG/SCF scheme works better in the case of aromatic compounds. In the case of MINDO/3 parameterization the transition from the SCF to SLG/SCF trial wave function improves significantly the description of experimental data on heats of formation. Generally, the MNDO-based approaches are more suitable for description of heats of formation of molecules with p-electron systems. To compare the whole quality of the SLG/SCF and SCF schemes we considered 67 test molecules and calculated errors in determination of their heats of formation by both methods with MNDO parameterization. The analysis of errors have shown that they have approximately normal distribution. We estimated the parameters of these distributions for both methods. In the case of SCF-MNDO method the average over set of errors is approximately -3.3 kcal/mol while in the case of SLG/SCF-MNDO method this value is only 0.01 kcal/mol. These simple estimates point to the significant difference between the methods: SCF-MNDO method has systematic and large error in the calculated heats of formation, while the SLG/SCF-MNDO method is free from the systematic error and its errors lie more or less symmetrically around zero. Mean square deviation for the errors in the case of the SCF scheme is about 7.8 kcal/mol while the same quantity in the case of the SLG/SCF scheme is only 7.0 kcal/mol. It speaks about preferability of the SLG/SCF scheme.

The evaluation of the method requires also consideration of geometry parameters defined in its framework. Our calculations using the SLG wave function have shown that it leads to significant improvement (even qualitative) of description of lengths of chemical bonds formed by heteroatoms and torsion angles with respect to the SCF wave function. Our calculations based of p-electron systems based on the SLG/SCF and SCF wave functions show that the quality of optimal spatial structures obtained with different wave functions is similar. As in the case of heats of formation the SCF method better describes compounds with multiple bonds while the SLG/SCF method is better suited for aromatic compounds.

Acknowledgements

This work has been performed with financial support of RFBR through the grants 02-03-32087, 04-03-32146, and 04-03-32206.

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