Electrostatic molecular potentials: Mulliken approximation

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. IX, 193-214 (1975)

Electrostatic Molecular Potentials: Mulliken Approximation

RAMON CARBO AND MIGUEL MARTIN Secridn de Quimica Cua’nf ica, Departamento de Quimica Orgdnica,

Instituto Quimico de Sarriti, Barcelona-l7, Spain

Abstracts

The electrostatic potential is calculated, in the LCAO framework, through Mulliken’s approximation. An extremely simplified form of the potential expression is obtained, with a degree of accuracy comparable to a full ab initio treatment. Other related possible simplifications are also discussed.

Le potentiel tlectrostatique a t t t calculC, dans le cadre de la mtthode LCAO a l’aide de l’approximation de Mulliken. On obtient une forme trts simplifiee du potentiel avec un degri. de precision comparable B ce qu’on obtient dans un traitement ab initio. On discute aussi d’autres simplifications possibles.

Das elektrostatische Potential wird im Rahmen des Lao-Verfahrens mittels der Mulliken’schen Naherung berechnet. Eine sehr vereinfachte Form des Potentials wird erhalten, die aber eine mit der einer vollen ab-initio-Behandlung vergleichbare Genauigkeit giht. Andere mogliche Vereinfachungen werden auch diskutiert.

1. Introduction

The study of the intermolecular electrostatic interaction has recently obtained a high degree of attention [l-91. However, the application of simplifying elements as the use of point charge models [3] or the utilization of CNDO wave functions [ 5 ] has not implied a considerable reduction of computation time, The purpose of this work is to show that Mulliken’s approximation can be effectively used in order to obtain a considerable gain in time, preserving the accuracy of the results.

2. Electrostatic Potential

I n the scheme proposed by Bonaccorsi, Scrocco, and Tomasi [ l ] the interaction of a positive point charge with a molecule can be written, under the LCAO

framework as

= R(r ) - A(?)

where the first term corresponds to the nuclear repulsion and the second to the 193

0 1975 by John Wiley & Sons, Inc.

194 C A R B ~ AND MARTIN

electronic attraction. P,, is the element of the first-order density matrix, associated to the AO’S ,u and v ; ( P I l/r Iv) is the attraction integral between these orbitals, and a positive point charge.

The crucial point in the V(r) calculation, when the molecular wave function is known, lies on the integral evaluation. For each point, if n is the number of AO used, (n2 + n) /2 integrals are needed.

An easy way to obtain a considerable reduction of the integrals is the use of Mulliken’s approximation, which is used to express each integral as a function of the overlap, Shy , between the AO’S p and v [lo] :

(3)

Employing (2) in the attractive part of ( I ) , one obtains

which can be further simplified to

1 (4) 4 r ) = z: Dp,(Pl; IP)

P

taking into account the symmetry of the P and S matrices and using the following definition :

( 5 ) D,, = 2 p , v s , v V

The quantity D,, is the gross orbital population of the AO ,u [ 1 I], or the diag-

So with Equation (4), V ( r ) is transformed into onal element of the Chirgwin-Coulson density matrix [12].

and the necessary elements, in order to evaluate the potential, are reduced to the knowledge of the set {D,,} and a bicentric attraction integral programme. The ratio of the number of integrals between this approximation and the full treatment is 2 / (n + 1). Moreover, bicentric integrals of the type needed are easy obtained in any atomic basis. For example, a general expression over STO’S has been de- scribed [13, 141 and over GTO’S, the formulae [15], which give the attraction integrals can be greatly simplified.

To test Equation (6) the H,O and H,CO potentials have been calculated by means of ab initio orbital charges found in the literature [16]. The most in- teresting potential planes of both molecules shows a similar behaviour to some

ELECTROSTATIC POTENTIAL BY MULLIKEN APPROXIMATION 195

calculations made by other authors [3, 51. The calculated potential planes are shown in Figures 1, 2, and 3. By means of Mulliken’s approach the potential

Figure 1. Electrostatic potential map of H,O. Left-hand plane is the molecular plane, right-hand is a perpendicular plane bisecting the HOH angle. Contour

lines are expressed in Kcal/mol, as well as for the following figures.

minima are found in the adequate region of space, and do not show the defects associated to spherical charge distributions, like CNDO [5]. This is essentially due to the fact that Mulliken’s approximation preserves the anisotropic effect of the atomic orbitals with nonzero angular quantum number. In order to show the usefulness of the proposed technique, some other molecules have been calculated. A GTO basis set of double-zeta accuracy with data obtained from [16] has permitted us to calculate the potential surfaces of the following molecules: (1) allene (Figure 4)) (2) ketene (Figures 5 and 6), (3 ) diazomethane (Figures 7 and 8). Ab initio diazomethane potentials have been reported in the literature [ 191 ; the agreement of this calculation with the present is good. The values of the minima and its location show a similar trend with the present calculation, despite the differences in basis set and computational point of view. Values for ketene electrostatic

196 CARBO AND MARTIN

H,CO 'n, G T O

Figure 2. Electrostatic potential map of formaldehyde, representing the molecular plane.


H ,C 0 'A,

GTO

Figure 3. Electrostatic potential map of formaldehyde with a perpendicular plane bisecting the HCH angle.


H,C C C H,

Figure 4. Electrostatic potential map of allene. The molecular symmetry allows the drawing of a unique plane, which contains a CH, group (left-hand side) and

bisects the HCH angle of the remaining methylene group (right-hand side).

ELECTROSTATIC POTENTIAL B Y MULLIKEN APPROXIMATION 199

Figure 5. Electrostatic potential map of ketene, corresponding to the molecular plane.

2 00 GARB6 AND MARTIN

H&CO

Figure 6. Electrostatic potential map of ketene with a perpendicular plane bisecting the HCH angle.

ELECTROSTATIC POTENTIAL BY MULLIKEN APPROXIMATION 20 1

H2C NN

Figure 7. Electrostatic potential map of diazomethane, showing the molecular plane.

ab initio potentials have been available to us from Scrocco, Tomasi, and Petrongolo [20]. Also in this case the shape of both calculations coincide; the potential wells however are more positive in this work.

Data provided by Palke and Lipscomb [21] have been used to test an STO

minimal basis set in our framework. Figures 9 and 10 describe the methane electrostatic potential. Two planes are presented, one of which contains a CH

202 CARS0 AND MARTIN

bond, and the potential well; as expected, the minimum values are small compared with other molecules. In Figures 11 and 12, a new formaldehyde map is presented, to visualize the effect of the Ao-type and basis size. No significant changes occur, the maps being comparable to Figures 2 and 3. Comparison with ab initio calculations [5] also does not show qualitative differences.

ELECTROSTATIC POTENTIAL BY MULLIKEN APPROXIMATION

H2CO h, V, M B S-STO

203

Z

\ 1 2 4 5 6 7 X

H

Figure 9. Electrostatic potential map of formaldehyde using rnbs STO data, representing the molecular plane.

204 CARBO AXD MARTIN

H,CO 'n, V,

M BSSTO

Figure 10. Electrostatic potential map of formaldehyde with a perpendicular plane bisecting the HCH angle.

ELECTICOSTATIC POTENTIAL BY MULLIKEN APPROXIMATION 205

,z

CH4 'A, MBS-STO

Y j_l_

J (0.1.0)

Y

t---

Figures 11 and 12. Electrostatic potential map of methane. Two planes are drawn whose generating vectors are also given in the figures.

206 CARBC) AND MARTIN

Figure 12. See Figure 11.


The present formulation can also be applied to the CNDO wave functions and to the computational structure recommended by Giessner-Prettre and Pullman

If one uses Lowdin’s transformation [ 171 over the CNDO MO’S, for the evaluation [51.

of the density matrix, the orbital charges [5] may be defined by

where SFA and SE are the elements of the matrices S-Il2 and S+llz, respectively. PAD is an element of the CNDO density matrix. Formula (7) can be used in (6) instead of ( 5 ) . The use of Equation (7) in (6), however, has no real advantage over Mulliken treatment. CNDO wave functions fit better into a zero differential approximation. The features of this (more adequate) point of view are discussed elsewhere [22].

3. Electrostatic Interaction

The electrostatic interaction energy between two molecules A and B, can be written in the LCAO framework as [3] :

where (pv 1 26) corresponds to the Coulomb integral between AO’S p, v of A and A, G of B. The Mulliken’s approximation, applied with a similar criterion as before, gives

IeA v s B TI I e A J e B r I j

The Coulomb integral calculus is reduced this time to 4 / (nA + I ) ( n B + 1) of the full calculation, if n,, n, are the number of AO’S of molecules A and B, respectively. In order to test the approximation, the interaction of two water molecules has been studied and is shown in Figure 13. Literature data have been also used [16]. I t has been found that expression (9) can reproduce fairly well the shapes of the interaction curves of similar calculations [3]. The gain in computation time is considerable, compared with the full process. At the same time, the expressions for the Coulomb integrals between STO’S [13, 141 or GTO’S [15] are well known.


“ t I f ................. pqZz : ..... * ...............

Figure 13. Electrostatic interaction energy of two water molecules. The 0-0 distance is 2.76 A, and the used data is taken from a GTO basis set. Only a partial rotation is drawn because of the symmetric behaviour of the interaction.

4. Other Related Approaches

Ruedenberg [ 181 has pointed out an alternative version of Mulliken’s original Following his work, an integral (p I l I/’ IvJ) with the AO p centered at idea.

atom I and AO v at J, can be written as


where the first sum runs over all the molecular centers, and D,, is the corresponding element of the Chirgwin-Coulson density matrix [ 121 :

(12) D = *[PS + SP]

If in (1 1) the nondiagonal terms are neglected, the expression of A ( r ) transforms

If from expression (4) the remaining attraction integrals (pl 1/r Ip) are sub- into formula (4). The attraction integrals in ( I 1) are still bicentric integrals.

stituted by y l ( r ) , 'Vp E I , then

A ( T ) = 2 D I Y I ( S ) I

(13) is obtained, with

and it is found a CNDO-like or purely electrostatic potential form, depending on the evaluation of the y I ( y ) terms.

This two points show the possible use of a set of formulae decreasing in com- plexity and accuracy : full treatment, Ruedenberg approximation, Mulliken approximation, CNDO, and purely electrostatic. Mulliken's approximation can be viewed as an intermediate compromise between all of them. The same argu- ments can be applied to the electrostatic molecular interactions.

5. Rotational Invariance

One of the problems in Mulliken's approximation is the fact that rotational invariance is not preserved. This question has been discussed deeply by other authors [23]. But computational experience can prove that this effect is of minor importance, even in SCF calculations [24].

Out concern here is to find a compromise which, preserving the computational advantages of Equations (6) and (9), can keep as much as possible rotational invariance. In fact, attraction terms in both expressions can be seen as a sum of individual AO contributions, so the symbol

can be used for the AO p contribution, and one can write

if rotational invariance has to be held, being {A:,) the attractive contributions on other reference frame. If { TPy} are the elements of an orthogonal rotation matrix,


then, considering A;, as the elements of a diagonal matrix, one can write the transformation

then, the contribution due to atom I will be

Z: A,, = Z: 2TiaA:a = Z: ( 2T;a) ':a = 2 ,€I ,€I U t I a d ,GI a d

(18)

in this manner Equation (1 6) holds. With respect to the electronic repulsive terms, if one writes

(19) rPP I vvl = D,,D"V(PP I YV) and [,up I vvIo means the same expression on a rotated frame, invariance can be maintained if

this can be achieved, in a similar manner as the attraction terms treatment. Considering the transformation

which behaves, when summed over p, v as [ 171. In order to put in practice this simple idea, it is only necessary to have a

procedure to preserve the total gross atomic population on a given atom, through rotations, or

This can be obtained using a transformation scheme as [ 171, considering also that the terms 0:" with p, v E I and ( p # Y) vanish: thus

1 ~ p p ( p 1 IP) = I: Z: (TiaDfa) 1 I ~ U ) PEI ~ E I a d r

1 a d r

(23) = z D L ( a I - la)'

The last equality will be true if the transformation

ELECTROSTATIC POTENTIAL BY MULLIKEN APPROXIMATION 21 1

Y

Figure 14. Ammonia electrostatic potential map. Upper diagram shows a perpendicular plane to the Z axis, where the potential well is found. The lower dia-

gram shows the location of the minimum in the lone pair region.

212 CARS^ AND MARTIN

is also defined. Equation (24) can be admitted under the consideration that the off-diagonal terms (pl I/‘ IY) ; p, Y E I , (p # v) also vanish. A parallel property can be found for the electronic repulsive terms, provided that with condition (22) the integrals (pp’ I YY’) - 0, if p, p‘ E I ( p # p’) and Y, v‘ E J (Y # Y’). This last condition is a reliable approach, taking into account the distances ( 2 3 A) where the electrostatic treatment is valid.

The results previously presented prove that these conditions are sufficient to preserve invariance to a reasonable degree of accuracy. I t should be noted however, that in some cases, when the studied system may have a high degree of symmetry, potential maps may appear slightly distorted, but we feel that this is a minor inconvenience in view of the computational advantages of the present scheme.

Previous SCF computations show that molecules such as NH, have distorted charge distributions if rotational invariance is not maintained [24]. Mulliken’s approach however, does not show any effect on this molecule when electrostatic potential is calculated. Taking ab initio gross orbital populations from the literature [21], the NH, potential has been calculated, and the results are shown in Figure 14.

-10

-18 1 -2

Figure 15. Ammonia-water electrostatic interaction. The N-0 distance is 2.78 i\ and used data is taken from a GTO basis set. The double well potential corresponds

to the alignment of the OH bonds in the N - 0 line.


The potential map has the adequate symmetry and agrees with the chemical intuition. The same intention has induced calculations on the NH,-H,O interaction, as shown in Figure 15. Data from Reference [16] have been used. Also, in this case, the logical double-well potential curve has been found, corresponding to the alignment of both OH bonds of H,O with the NH, lone pair dircc tion.

6. Conclusions

Mulliken approximation appears as an adequate simplification for electrostatic potential calculations. The results are analogous to the a6 initio ones, when using a6 initio orbital populations. Computational time in this approach increases linearly for the potential or quadratically for the interactions when the OA basis size increases. The study of the potentials and interactions in the field of large molecules is thus easily feasible. Finally, the fact that in the Mulliken approximation only the orbital populations are needed allows the use of SCF

results with a minimal effort, since these quantities can be found, generally speaking in the major part of the published work or are a routine result in any SCF program.

Acknowledgements

The authors are deeply indebted to Drs. E. Scrocco, C. Petrongolo, and J. Tomasi, without whose kind support this work would not have been completed. Referees’ remarks are also acknowledged.

Bibliography

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Received February 8, 1974. Revised September 5, 1974.

Electrostatic molecular potentials: Mulliken approximation

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