Floating s and p-type gaussian orbitals

Volume 35, number 4

FLOATING

CHEMICAL

s AND p-TYPE GAUSSM

PHYSICS LETTERS

1 October

1975

OREITALS

D.F. BRAILSFORD, G.G. HALL, N. HEMMlNG and D. MARTIN Matl~errarics Depnrtrvent.

Clr~iversify of Xoftinglrorn. h’ortitrgl;arn NG7 2RD. UK

Kcceived 2 lune 197 5 Revised manuscript rcceivcd 21 July 1975

The advantages 0,: including a small number of p-type gaussian
[6] as a measure of the balance of the basis set. The correlations here arc excellent.

1. Introduction Since its introduction by Frost [ 11 and its adaptation by others (Preuss I:!], Blustin and Linnett [3], Christoffersen [4], Ford et al. [5]) the floating spherical gaussian orbital form of wavefunction has proved mory

to be a vivid and versatile

of theoretical

methods.

addition

not

alI molecules

have

models

of

this type which are numerically stable so constraints were introduced to prevent close functions from coalescing. This feature detracts from the purity of the appeal to optimization as de!ermining molecular structure and does not completely avoid the problems of numerkal sccuracy. This situation can be reme-

died by introducing p-type gaussians as the limit functions produced by the coalescence. The stability of the calculation is dramatically improved. Results are given hefi: for H20, H,O,, HF, F2, CzHl, CH30H, CHZCCHz and CHJCCH~, in their experimental

geometriem;, to illustrete

the effective-

-ness of this solution. The total energies represent about 95% of the molecular orbital energies. The extent to which

the orbital

of an exact Hartree-Fcck

energies

correlate

When two FSGO’s ark used to describe

to the ar-

In its original form the

minimum number of spherical gaussian was used to set up a determinantal wavefunction and, to compensate fir this small basis set, the positions and exponents of all functions wrc optimized. Unfortunately

2. Limit functions

with

those

calculatiori is often taken

the lone pairs

of the oxygen atom in a molecule they float towards one another. There is now considerable evidence (HyIton [l Z])that the optimum separation in circumstances like this is zero. Because the overlap matrix becomes singular as the functions coalesce (Ford and Hall [7]) the energy calculation becomes numerically unstable.

One way of avoiding this situation is to introduce a new basis function. Thus the two basis functions e1

=

e

-a(r+cv)2

g2

.I

=e -ar

z

,

are replaced by their sum and differences. In the limit the sum becomes ~2 but the difference becomes the p-type derivative function ?3 = Vg2 =remQ1.‘_

i3y symmetry, qZ and p3 have vanishing overlap integral so that the singularity iri the overlap matrix disappears. The introduction of a p-type gaussian dl-

rected along the line of centres of the previous functions will then give a much more stable molectilar model.

For a fluo&e

atom wiih 3 lone pairs two p-type 437

Vo!umc 3.5, number 4

CHEMICAL PHYSICS LETTERS

fuktions directed at righ.t angles .will be needed to span the set oi possible coalescences. The double bond is now most easily described in terms of u and TI funciions with the latter reprosented by p-type gaussians. There will be some advantage for most molecules in introducing the complete set of p functions (p,,p, ,p,) since this spans ali possibie directions that may be re.quired. In principle these p functions should be allowed to float freely through the molecule to optimize the energy. Practical calculation has shown, however, that they usuaiiy optimize on or very close to the nuclei so that it is more economical in terms of parameters to NIXthem on the nuclei. The FSCO’s that remain in the basis set can now be optimized more freeiy than before since the cause of their instability has been removed.

3. ~folecular models The introduction of p functions into the basis complicates the calculations in several respects. Additional procedures are required to label the functions, to classify the integrals and to evaluate the auxiliary functions F1(_r) and F2(x). Within the OPIT system these modifications have been Implenlented without major reprogramming (Bra&ford and Schnuelle [ 121). The rcstriction has been imposed that all p functions should be parallel to one of the coordinate axes. This eliminates the need for procedures to handle in tagrals of certain kinds. For multiply bonded and conjugated molecules the restriction is immaterial since it is already a desired feature of the model. To illusrra te the extended range of molecules which can now be easily modelIeS some selected examples have been caIculated. They faII into three classes: (0) Coll&zirlg 0 oforrzs. The dxygen atom is modclled using two spherical gausskns on the nucleus together with a set of p-type functions (p,, pY, p,). Its bonds tq other atoms are represented by spherical gaussians floating along the internuclear axes. Details of these functions are given in table 1. The location of bond s functions is given as a fraction of the bond lenght. The p functions are allowed to have different exponents. ,The smallest exponents arise when the p functioris lie in the direction of the bisecior of the bond.ing angle at the dxygen atom. The largest exponents

$38.

.’

Table 1 Parameters

1 October 1975

for models of 0 containing

Exponents, location

molecules

H202

H20

CH2OI-I

84.4549

83.9328

83.9069

12.35 11

12.2692

12.2652

0.4315 0.5601 3.3871

3.2355

sonOH

0.4714 0.7023 2.3964 0.5485

0.5705

0.5533

OS/OH

0.3253

0.5362

0.3967

Sl

on 0

on0 p on0

52

bisector lone pair

0.44 19 0.5445

correspond to p functions in the plane of the oxygen bonds but perpendicular to the previous ones. These functions are helping to describe the bonds whidh are also described by s functions in the bond directions. Because of this flexibility these s functions shrink closer to the oxygen nucleus and make some contribution to the description of the non-bonding electrons. The substantial differences in the exponents of the p functions shows that although the angular functions span spherica! symmetry the radial functions optimize so that there is no invariance around the atom. The exponents of the s functions are very similar from molecule to mokculc though small systematic trends can be observed across the table. (b) Corzrairlirlg F acorns. The fluorine a tom is also modelted using two s funcrions and a set of p functions on the nucleus. In addition there is one s function on the HF axis and one at the mid-point of F2. The parameter values are given in table 2. Table 2

Parameters for models of F containing Exponents,

molecules

HF

location

F2

_--

108.4639 l-5.9287

SI on E

s2 on

sonHF

1.0683 0.3931 0.8104

Fs/FX

0.0863

P on

pxz py PZ

.--

103.1813 15.1370 1.3756 0.3226 0.3852

0.5

The exponents of the p-functions are smallest when these are directed along the bonds I$,): The F atom in HF appears to be attracting its s electrons more strongly than the F atom in F2 but its lone-pair p elec-

Volume 3.5, number 4 Tnble 3 Porametcrs

CHEMICAL PHYSICS LEmERS

for models of CC conthin,o

molecules

-Exp’oncnts,

CH?=C=Cflz

loc;ltion 51

one

s2

one

I October 1975

CH=CH

CH3 -C=CH

--____ 45.8125

6.6288 0.3178

px. pu on C

s 0nC s onCEI WCC WCH

46.0934

6.6618 0.4171 0.4401

0.3768 0.4801 a; 0.6082

45.9374

6.6392 0.3829 0.4551 0.3865 0.5 0.5865

45X31? 6.6226

45.9501 45.9501 6.6407 6.6407 0.3824 0.3824 0.3831 0.4538

0.3638

(3.3844

0.4792 aJ0.5027 3) 0.5955 0.5896

a) Cs measured from central C. trons less strongly. This may be due to the much closer approach of the s function in the HF bond. (c) Containing nrrtltiple CC bonds. Double or triple CC bonds are now described in a ,more conventional way using pX and pmvftinctions normal to the internu-

Table 4 Compxison

Hz0

-72.3329

clear axis f5r the n bonds

H*O2

-143.x92 -109.6062 -94.2205 -184.2382 -110.9415 -73.6012

arId an s function

on the tis

for the cr bond. There are 1.~0 s functions on the C nucleus and one on each CH bond axis. The expo-

nents fcr the CSC in CH3 CCH have been constrained to be equal. The parameter values are given in table 3. The optimal parameter values for these moletiules are much closer to one another than for the earlier molecules. Nevertheless some trends can be observed such as the rise in the sI e:uponent with the fall in the number

4. Ener,v

of

H neighbours.

results calculated

4 toget:her

for all these molecules with those given by

Snyder and Bssch [8] using double zeta wavefunctions.From these results it is apparent that the use of p functions is slightly less successful for the molecules contsining F atoms than for the remainder since these have energies

This paper

U&OH HF Fz CHzCCH2 CHCH CH3CCH

Table 5 Approximate

close to 95% of the double

It has been argued by Mulliken

zeta value.

[6] that an approtimate wavefunction can produce reasonable values for molecular properties only if the basis set is balanced. One measure of this balance is the relation between the corresponding orbital energies and those of a Hartree-Fock wavefunction [9] _ For FSGQ wavefunctions it has been fburrd [lo] that the valence orbital enerdes correlate very weii with those of an ac-

-110.8995

Snyder and Basch

%

-76.0035 -150.7373

95.17 95.06

-115.0059 -100.0150 -198.6932 -115.8203 -76.7919 -

95.30 94.20 92.72 95.79 95.85

-

linear relation between ~_____

orbital ~ncrgics

a

b

Correlation coefticicnt

HZ02

1.01395 1.00311

0.27853 0.28355

CffjOH

0.99707

0.15333

{IT; Fz Ctl+xHz CHCH

1.01.235 l.C3123 0.39009 0.98669

0.36947 0.29860 0.10729

0.99984 0.99981 0.99979 0.99976 n.99954 0.99992

0.12825

0.99993

HZ

The total energies are shown in able

of totzl cncrgies (in hartrees) --.______

curate wavefunction. In tabIe 5 a similar correlation with the Snyder and Basch orbital energies fDZ ir: the

form

is reported

for each molecule. The correlation coefficients for the regressions are remarkably high and this shows that the Linear reIation is closely obeyed for each molecule. The ragres-

Volume 35, numlxr

4

CHEMICAL

PHYSICS

sion coefficient u remains close to unity. As is to be expected, it becomes greater than one for the poorer calculations as indicated by the percentages in table 4. The constant b increases with the inclusion of an 0 or F atom in the moIeculz.

‘1 October

LETTERS

1975

energies are not less than 0.9995. This demonstrates the balance of the basis set and suggests that other predicted properties will exhibit similar correlations.

Acknowledgement 5. Discussion The molecules considered above have been selected as ones for which the original Frost FSGO model was either numerically unstab!e or singular. The introduction of a limited numbx of p functions has eliminated this difficulty. None of these molecular modeis exhibited any tendency to numerical instability. Since a cusp function on each heavy nucleus has been used in addition. to the bond functions the results compare more directly with those of Ford et al. [5] than with Frost. III both calculations it is significant that the total energies are close to 95% of the Hartree-Fock results. The models of bonding that this treatment supports arc still recognisable as simple classical models. The nlajor distinction from the Frost model is that the multiple bond is described using CTand z orbit& and the multiple lone pairs are similarly resolved into s and p orbit&. The correlation coeftkients between the orbital energies of these model!; and those of the double zeta

The authors wish to thank the Science Research Council for financial support of N.H. 2nd D.M.

References [l] A.A. Frost, J. Chen. Phys. 47 (1967) 3707, 3714. I21 tl. Prcuss, Intern. J. Quantum Chcrn. 2 (1968) 651 J. Chem. Sot. Faraday i3j P.H. Blustin and J.W.-Linnctt, 70 (1974) 274,327. 141 R.E. Christoffersen,

II

Advan. Chcm. Phys. 6 (1972) 333. Intern. J. Quantum

[Sl B. Ford, G.G. Hall and J.C. Pxker,

Chem. 4 (1970) 533. J. Chcm. Phys. 36 (1962) 3428. [71 B. Ford nnd G.G. Hall, Computer Phys. Commun. 8 (1974) 337. [81 L.C. Snyder and H. Basch, Molecular WDVCfunctionsand propertics (Wiley, New York, 1972). 191 H.B. Jensen and P. Ros, Thcoret. Chim. Acta 27 (1972) 95. [lo1 R.E. Christoffersen, D. Span@, G.G. Hall and G.M. Mnsgiora. J. Am. Chcm. Sot. 95 (1973) 8526. [Ill J. Hy!ton, Ph.D. Thesis, University of Nottingham [61 R.S. hfullikcn,

(1973).

1121 D.F. Brailsford and C.W. Schnuelle, in preparation.