INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XVI, 5 7 6 4 (1979)

Ab initio Molecular Fragment Calculations with Pseudopotentials

R. GASPAR, JR.

R. GASPAR

Department of Biophysics, Medical University of Debrecen, H-4012 Debrecen, Hungary

Institute of Theoretical Physics, Kossuth Lajos Unioersity, H-4010 Debrecen, Hungary

Abstract

Pseudopotential theory is introduced into the ab initio FSGO molecular fragment method. Pseudopotential molecular fragments using pseudopotentials of double-zeta quality are characterized, and a method of their assembly into larger molecules is presented. Core-valence electron separation is achieved at both levels of the molecular calculations. Heteroatom incorporation into the method is also considered. Applications of the pseudo-FSGO molecular fragment method to hydrocarbons and simple molecular systems containing heteroatoms are discussed. Results are compared to those of the original FSGO method and experiment.

Introduction

Application of theoretical ideas of quantum mechanics to large molecular systems is one of the basic problems of quantum biology. The application of semiempirical methods led to the solution of many problems in this field; however, they provide a first-level approximation only [l-31. Recently the use of ab initio methods became quite usual, and the introduction of new mathematical and computational ideas along with the developments in computer techniques increased the size of the molecules that can be treated by these methods [4-61. A special method has been proposed for the treatment of large molecular systems, namely, the ab initio molecular fragment method of Christoffersen et al. [7]. This method is based on the floating spherical Gaussian orbitals FSGO method of Frost by introducing the new idea of fragmentation of molecular systems into it.

Pseudopotentials have been in use for a long time in connection with the calculation of atomic and molecular systems in order to describe the cores of many-electron atoms in them [8]. Recently detailed investigations have been carried out on the theoretical background of the pseudopotential approach and many ab initio and semiempirical pseudopotentials have been discussed [9]. All these investigations show a common feature, namely, that they reduce the all-electron problem to the problem of the valence electrons.

Starting with some empirical data and a simple model system (one electron in a model field, etc.) and fitting the calculated and experimental data, a core-electron effective potential field can be defined. Moreover, the orbitals and energies of more complicated atomic systems can also be determined in this field. In principle this approach may compete with more sophisticated all-electron theoretical

@ 1979 John Wiley & Sons, Inc. 0020-7608/79/0016-0057$01.00

58 GASPAR AND GASPAR

methods. However, the main problem in connection with the pseudopotential approach is the consistent determination of the parameters.

The first step toward this goal has been described elsewhere [lo]. The Gaussian form of the pseudopotential has been proposed for use in molecular calculations and a variational method with a double-zeta basis set has been used for the determination of the pseudopotential parameters. The pseudopotential has been incorporated into the FSGO method and the pSeUdO-FSGO method has been established [lo, 111. In contrast with earlier investigations in this field this method does not introduce any new empirical parameter or does not make any adjustment of the previously determined pseudopotential parameters during the molecular calculations.

The present work makes an attempt to combine the pSeUdO-FSGO method with the FSGO fragment method, and further to increase the power of the latter method in treating large molecules and also to simplify its computational requirements. By this means the ab initio nature of the molecular calculations will be preserved and the computational requirements of the resulting pseudo-FSGO fragment method are reduced to approximately that of an all-valence electrons SCF calculation.

Method

The potential field acting on one of the valence electrons in an atom or ion may be described in the following form:

where the actual analytical form of the radial factor of the non-Coulombic part of the pseudopotential is

In the above expressions 2 is the net charge of the ion core consisting of the core electrons and the nucleus of the atom, A, and a, are the pseudopotential parameters, Pl is the angular momentum projection operator, and r, I have their usual meanings. This form of the radial factor in the pseudopotential is appro- priate for the correct representation of it as has been shown earlier [lo]. The form described by Eq. (2) is very appropriate from the computational point of view especially when it is used together with Gaussian wave functions. The variational method with double zeta basis set is used for the determination of the parameters in (2) and Table I contains them for all the elements used in the present investigation,

The above-defined pseudopotential may be introduced into the FSGO method of Frost in order to replace the atomic cores in the molecule [lo-121. In the FSGO

method, normalized FSGO are used as basis orbitals, and both the position and the size of the FSGO and the location of the nuclei are varied until the energy of the molecule is minimized. In the original FSGO method a substantial fraction of the

AB INITIO MOLECULAR FRAGMENT CALCULATIONS 59

TABLE I. Double-zeta parameters of the pseudopotential V,(r) = A( e-"f'2 for 1 = 0 and 1=1.

Element Ao ff0 Ai f f 1

C 31.6899 8.0076 -15.2716 20.6316 N 39.3905 9.6892 -16.603 1 29.1564 0 54.9425 13.3 183 - 17.0904 37.8607

basis orbitals is used to accommodate only the core electrons of the atoms. The effect of the cores can be replaced by suitable pseudopotentials and no core orbitals are required.

Furthermore, molecular fragments, small molecules that appear as entities in larger molecules, can be defined within the framework of the resulting pseudo- FSGO method [ 1 3 ] . N, valence electrons in a pseudopotential molecular fragment containing n atoms can be described by the valence pseudo-Hamiltonian

where Vy is the pseudopotential representing the core of atom k. The normalized FSGO have the following forms:

(4) G i ( r ) = (1/2.rrpi ) exp{-p;2(r-Ri)2}

where pi is the orbital radius and Ri is the vector determining the location of the center of the FSGO. Both pi and Ri of a molecular fragment are varied until the valence electronic energy E,, associated with the pseudo-Hamiltonian XPs is minimized. The nuclear geometry of the fragment remains unchanged during the energy minimization. Furthermore, Eva,, the total valence energy of the fragment, is determined

2 314

where Z, and Z , are the charges of the atomic cores and Rk, is the internuclear distance. The goodness of the resulting molecular fragments can only be tested by incorporating them into larger molecular systems. Optimized parameters are presented in Table I1 for various pseudopotential fragments. Fragments CH, (tetrahedral) and CH3 (planar) bear special importance in hydrocarbon descrip- tion, while fragments NH3 (tetrahedral), NH4+ (tetrahedral), and H20 (bent) may be used in the description of amines, ammonium salts, ethers and alcohols, respectively [ 131.

For large molecule formation from the pseudopotential molecular fragments a slightly modified version of the pseudo-FsGo procedure can be applied. In this procedure the atomic orbitals are replaced by the pseudopotential fragment FSGO.

Moreover, those hydrogen atoms are removed from the fragments, which served

60 GASPAR AND GASPAR

TABLE 11. Optimized molecular fragment parameters

Distance from Molecular Fragment Orbital radius ( p ) the heavy atom parameter

CH4 (tetrahedral) PCH = 1.67351407 1.1 8969120 RcH = 2.05982176 Eva, =-7.12590014

CH3 (planar, sp’) pCH = 1.50273080 1.08381477 RCH 1.78562447 P,, = 1.95286313 50.1 Eval = -6.56341222

NH3 (tetrahedral) h H = 1.52372425 0.90386007 RNH= 1.91242167 m p = 1.54312657 0.23487709 Eva, = -10.23277637

NH4+ (tetrahedral) h H = 1.49793482 0.77676604 R N H = 1.95401452 E,,1=-10.61511856

H 2 0 (bent) POH= 1.33774152 0.67599354 R O H = 1.81039250 fi.p. = 1,33102223 0.05775029 Eva, = -15.01920823

only to polarize the environment for the determination of the nonlinear parameters of the molecular fragments. The FSGO are retained in these regions and new chemical bonds between the fragments are formed by taking the linear combination of the molecular fragment FSGO. The above-described procedure of large molecule formation is schematically represented in Figure 1 by forming methanol from a CH4 (tetrahedral) and a H20 (bent) fragment.

Results and Discussion

In order to prove the applicability of the pseudopotential molecular frag- ments, detailed in the previous section, prototype molecules have to be con- structed from them. Geometrical and energy predictions of the method should be

Figure 1 . Formation of methanol from CH4 (tetrahedral) and H20 (bent) pseudo- potential molecular fragments. ( X ) Approximate position of the FSGO in the molecule. Cores of the C and 0 atoms are represented in the present method by

pseudopotentials.

A B INITIO MOLECULAR FRAGMENT CALCULATIONS 61

TABLE 111. Distance and angle predictions for some hydrocarbons."

Parameter Experimental Calculated Deviation' Molecule predicted value value 0 0 1

Ethane C-C distance 2.90 2.80 3.4 Ethylene C=C distance 2.47 2.57 4.0 Propane CCC angle 112.4 112.2 0.2

a See Ref. 13 for detailed results. Measured from the experimental value of the parameter.

examined. In Table I11 distance and angle predictions gained by the pseudo- potential fragment method for some hydrocarbons are displayed and compared to those gained by experiment. Detailed description of the results of hydrocarbon pseudopotential molecular calculations are given elsewhere [ 131. As it has been shown, the total valence electronic energy of hydrocarbons is reproduced by the present method with an average accuracy of 11.3%. Geometrical predictions of the method also seem quite reasonable, especially if one takes into account the relatively primitive basis orbital set applied in the description of the pseudo- potential molecular fragments.

Table IV comprises the results gained by the pseudopotential fragment method for molecules containing heteroatoms. The simplest possible molecular

TABLE IV. Geometry and energy predictions for molecules containing heteroatoms." -

Calculated value by

Experimental pseudopotential Deviation" Molecule Parameter predicted value' fragment method ( O h )

Methanol Total valence energy -24.1466 -21.1994 12.2

Methylamine Total valence energy - 18.6456 -16.3843 12.1

C-0 distance 2.6853 2.7344 1.83

C-N distance 2.7968 2.7773 0.7

Hydroxylamine Total valence energy -27.7901 -24.3031 12.6 N-0 distance 2.7591 2.7522 0.25

Hydrazine Total valence energy -22.2730 -19.4787 12.5 N-N distance 2.7780 2.7925 0.52

Hydroxyl- Total valence energy -28.4387 -24.7358 13.0 ammonium ion N - 0 distance 2.7401 2.7790 1.4

a All parameters are given in a.u. For experimental values of parameters see Ref. 15. Estimated value of total valence energies using experimental ionization potentials and bond

Measured from the experimental value of the parameter. energies.

62 GASPAR AND GASPAR

2 I

I r m -

-2115 W

- > 7 2 - 2 1 20- c

systems have been selected in order to check the reliability of the heteroatom- containing fragments when they are combined with the CH, (tetrahedral) and CH3 (planar) fragments that have already been checked and with each other too.

The procedure of molecule formation for methylamine, hydroxylamine, hydrazine, and hydroxyl-ammonium ion is similar to that of methanol, schemat- ically shown in Figure 1. The total valence energy of the investigated heteroatom- containing molecules is calculated by the present method, with the average accuracy of 12.5 % . Predictions of internuclear distances within these molecules are even better than similar distance predictions of hydrocarbons. A represent- ative example is shown in Figure 2 for the determination of the C-0 distance in

b

/*’ \R /* *-*..-*-*’

\. , 1 I

20 25 30 35

methanol. The internuclear distance belonging to the minimum of the total valence energy of the molecule is determined by fitting a parabolic formula to the theoretically determined points of the curve.

The main goal of the present investigation is the introduction of the pseudo- potential theory within the framework of the ab initio FSGO fragment method of Christoffersen [14]. The suitable form of the pseudopotential is given and applicability is proved through various examples of molecular systems. Pseudo- potential molecular fragments are established and the method for large molecule formation from them is described. The description of the pseudopotential frag- ment is based on a relatively primitive basis set and a net gain in the number of FSGO is achieved if one compares the present method to the original FSGO fragment method. This fact is of special importance if one considers that the molecular fragments are further involved in a pSeUdO-FSGO type of calculational procedure during their assembly into larger molecules. Furthermore, the core-

AB INITIO MOLECULAR FRAGMENT CALCULATIONS 63

valence electron separation results in conceptual advantages too. Well-known ideas of chemistry like hybridization, the existence of different types of chemical bonds, etc., can directly be applied in the selection of the appropriate molecular fragments and in the construction of large molecules from them. Moreover, during the molecular calculations the valence electrons of the molecule are treated explicitly only and the description of the cores of the atoms in the molecule is done by pseudopotentials. It must be noted that the total valence electronic energy of the molecule, which is always determined by the present method, is much smaller in magnitude than the total electronic energy of the molecule. The separate determination of the total valence electronic energy means that the valence FSGO are not required to compensate for the incorrect description of the core orbitals by the core FSGO as may be the situation in the original FSGO method. We regard our good predictions for the geometrical properties of the investigated molecules due to this latter fact.

At the present level of approximation electron correlation of the valence electrons was excluded and this provides a basis for further improvement apart from the possibility of using more accurate pseudopotentials. The present method shares the favorable characteristics of the original FSGO fragment method, namely, the accuracy of the calculations increases as the molecules become larger. This advantage of the method is due to the fact that during the combination of the molecular fragments into molecules additional orbitals are added to the basis set. The result of this is the increasing percentage of virtual orbitals with the increasing size of the molecule.

We would like to emphasize that the present method has no semiempirical character at the molecular level of the calculations. The pseudopotential parameters are determined first at the atomic level according to a procedure described earlier, and after their determination no adjustment of them happens during the molecular calculations and all matrix elements of the pseudo-Hamil- tonian are calculated exactly.

Bibliography

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[3] R. Daudel and C. Sandorfy, Semiempirical Wave-Mechanical Calcularions on Polyaromic

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64 GASPAR AND GASPAR

[12] A . A. Frost, J. Chem. Phys. 47, 3707 (1967); 47, 3714 (1967); J. Am. Chem. SOC. 89, 3064

[13] R. Gaspar, Jr. and R. Gaspar, Int. J. Quantum Chem. (to be published). [14] R. E. Christoffersen, Adv. Quantum Chem. 6, 333 (1972). [15] L. E. Sutton (Ed.), Tables of Interatomic Distances and Configuration in Molecules and Ions (The

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Received September 17, 1978 Accepted for publication January 18, 1979