Gaussian basis sets for polarizability calculations

CHEMICAL PHYSICS LETTERS

Volume 55, number 3

GAU!WAN

BASZS SETS FOR POLAREZABiLITY

1 May L978

CALCULATIONS

PhilEp A, CHRISTIANSEN and E-A. MCCULLOUGH Jr. De.wrttnent of Chemistry and Biochemistry, Logan. Utah 84322, USA Received 31 lammy

UMC 03, Utah State University,

1976

A simple set of rules for choosing gaussian basis sets for molecular pola+ability calculations is proposed. The rules have been applied in coupled Hartree-Fock calculations on several frrst row diatomics and have been found to give polarizabilities accurate to within 2%. Because of their simolicitv and lack of laborious exponent optimization, the rules should prove especially useful for polyatomic mobcde cak&tio~

A number of attempts to compute coupled HartreeFock polarizabilities for first row diatomic molecules using the LCAO SCF procedure have appeared in the literature during the recent past [ l-6]. The wide variations in the polarizabilities reported by different workers for the same molecule have left substantial doubt as to the adequacy of some of the basis sets employed_ Recently, the partial-wave SCF procedure was used to compute nearly exact parallel pola~b~ties for a few of these molecules [7]. The results, when taken in conjunction with the previous work of others, revealed the following. Fh-st, a basis set capable of yielding near Hartree-Fock accuracy for a number of properties may still result in a very inaccurate azz, and vice-versa. In particular, the total energy is a very poor indicator of the quality of (y,. Second, azz is sensitive to seemingly minor variations in the basis set. Third, those LCAO calcuiations which do give reliable polarizabilities nearly always include basis functions more diffuse than those usuaily found in energy-optiruiied basis sets. Thus, basis set selection appears to be of critical importance in LCAO polar&ability calculations. Although selecting the types (s, p, d, etc.) of basis function needed for polarizability calculations is straightforward IS], selecting the exponents is not. Rules for choosing &xponents using a second-order perturbation optimization applied either to individual basis functions [9,10] or to the entire basis set [ I,3,4]

have been used with considerab’fe success. Unfortunately, these mles are not always straightforward in their application and may require extensive experimentation with the basis set, an expensive process for large molecules. However, all these procedures have the common effect of introducing diffuse functions into the basis. The need for such functions is appare-zt from plots of the first-order orbital corrections c&used by the external field [7]. These corrections are very diffuse and probably lie outside the region spanned by the usual er*er~-opt~ed basis sets. We have investigated the possibility that a set of simple rules for constructing reasonably small basis sets which effectively span this space and give reasonably accurate polarizabilities can be formulated.

2. Basis set setection procedure Our starting points are the Dunning (4s3p/2s) [I I] and (5s4p/3s) [ 121 contractions of the Huzinaga (9~5~14s) aI?d (lOs6p15s) primitive gaussian basis sets [t3]. These basis sets were chosen because of their widespread use and because they are both accurate enough and small enough for a wide variety of molecular calculations. To help account for molecular bond formation we added to the basis sets for B, C, N, 0, and F single d functions contracted from two primitive gaussians according to Dunning’s rules [ 14]. The eMective Slater exponent for the d function is crudeIy 439

CHEMICAL

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PHYSICS LETTERS

chosen to correspond to average d exponents from energy-optimized Slater basis calculations I I 51. (The same exponent was used for both do and dir.) A single p function contracted from two primitives was added to the hydrogen basis sets. The p exponents were chosen in the same marmer as the d exponents. The (4~3~ ld/?,sl p) basis sets generated in this way give total energies which deviate from the Hartree-Fock limit by (5-20) X 10-3 hartree for first row moleCl&S.

Our rule for adding diffuse functions is based on the observation that the exponents in the original sets approximate geometric progressions with ratios in the range of 3 to 4 between successive exponents. We mereiy extended these sets by one free prhmitive per class of function (s, p, d) with the exponents roughly chosen to continue the progressions downward. Thus a (4s3pld/Mp) basis set becomes (Ss4p2dl3sZp). We emphasize that no attempt was made to optimize the diffuse function exponents. To iIIustrate this mle, table I gives the exponents in our BH basis set derived from Dunning’s (4s3p/2s) set, The solid lines separate contracted functions in the original basis whiIe the dashed lines indicate the added diffuse functions. interestingIy, Morrison and Hay independently arrived at a nearly identical basis set selection procedure in their calculations on Nz and COZ [ i6f _They differ from us mainly in the choice of d functions. Table 1 BH (5s 4p 2df3s 2p) basis set evpanents Boron P

d

s

P

2788.4 1 419.039

11.3413 2.4360

0.71187 0.20F09

17.24296 2.59816

1.610 0.3745

-

i.3057

-

0.3245 0.1022 ----....0.0341 440

We employed the fmite-field method for caiculating aZZ [I 71. Field strengths were chosen to g&e induced dipole moments of *O-G2 au, which is large enough to yield adequate significant figures, yet small enough so that higher order terms in the field are usually negligible. We first carried out extensive calculations on BH to test the adequacy of our basis sets and the effect of basis set variation. Tabie 2 presents our results. The essentially exact partial-wave values have been included for comparative purposes, One sees that the two smallest basis sets without diffuse functions give the same value for cyzt which is in error by more than 20%. The large Dunning set augmented with two d functions on boron and two p functions on hydrogen (roughly energy-optimized), improves 01, somewhat, but the error is still 16%. This behavior is typical of that observed in other calcufations without diffuse functions [2,5,6], and it demonstrates that the flexibility of even large energyoptimized basis sets may not be sufficient to guarantee accurate pola~~bilities. In contrast, our smallest basis set with diffuse functions gives (uZZcorrect to within a fraction of one percent. Note that the energy is only marginally improved by the addition of diffuse functions. Replacing the high exponent primitive p function on N with a Table 2 BH polarizabiiity versus basis set

S

3.4062

3, Pollarizabiity calculatiorls

Hydrogen

96.4683 28.0694 9.3760

0.6836 --___ 0.0670 0.2134

0.0701 _---0-0234

0.58563 -

0.15912 -----0.0530

1 May 1978

---0.10

without diffuse functions (4~3~12s) C4s3pld/2slp) 1%4p2d]3s2p) with diffuse functions (ls4p2df3s2p) a) f%4p2d/3s2p) b) (6sSp2dj4sZp) uj (6dp2d/4s3p) c) (6sSp2df4s3p)d) partiakwave

E Chartree)

a,, Cbohr’ )

-25.1167 -25.1274 -25.1301

17.3 17.3 18.8

-25.1277 -25.1279 -25.1299 -25.f301 -25.1301 -25.1314

22.3 22.2 19.4 21.8 22.3 22.3

a) Primitive high exponent p + diffuse p on H. b, Contracted high exponent p + diffuse p on H. c) Split P contraction from b + diffuse p on H. d) Contracted p from b + 2 diffuse p on Ii.

Volume 55, number3

CHEMICAL PHYSICS LETTERS

contracted function, which lowers the diffuse p function exponent by a factor of about two, leaves aZZ essentially unchanged_ Although our aZz value is very good, comparison of the individuaI orbital components with the partialwave values indicates that this is largely due to cancellation of errors in the components, which may be in error by 20%. Such cancellation does not appear to be unique to induced dipoIe moments: it also seems to occur with permanent moments as wefl. This cancellation is very sensitive to the balance of the basis set, as became readily apparent when we tried to improve the origiual basis by going to Dunning’s larger set. Using this set with two primitive p func-

tions on H, we get a value for aZXwhich is in error by 13%. This error can be traced almost entirely to the p set on H, which is now very unbalanced with respect to B. Zncreasing the fiexibility of the p set again gives very good results, as table 2 shows.

Our results for BH suggested that Dunning’s &all basis was an adequate starting poiut for a=,- calcufations. Therefore, we carried out calculations on FH and CO using the (4s3pfZs) Dunning set augmented according to our rules. Table 3 shows these results, along with those of Morrison and Hay [ 161 * for N2_ One sees that the values are uniformly excellent, the largest error being approximately 2%. f We thank Dr. ?. Jeffrey Hay for making the de&& of this calculation availableto us.

Table 3 Molecularpolarizabilities E FH

(Ss4p2d/3s2p) a) (.%4P2d/SsZp)b, partial-wave

CO CSs4p2d) partial-wave NZ (6s4p2d) partial-wave

(hartree)

r%, Cbohr3)

-100.0521 --190.0528 -100.0706

5.65 5.64 5.78

-112.7701 -112.7910 -108.9742 -108.9939

14.4 14.5 C)

14.8 ef 14.9

a) Primitivehi&~exponent p + diffusep on H. b) Contracted high exponent p + diffusep on H. c) Morrisonand Hay [ 161.

1 May 1978

4. Discussion Very accurate potarizabilities for both CO and FH have been previously computed using LCAO methods f 1,4], but the basis sets were chosen by the secondorder energy optimization rule. Furthermore, in the most recent of these calculations [4], a very Iarge uncontracted gaussian basis (1 ls6p3d/SsZp) was used, Aithough these methods clearly work very well, they are laborious and expensive, especiaiiy for polyatomits, Our results suggest that if accuracies of a few per cent are desired, then neithel the optimization nor such a large basis is realty necessary as Iong as the basis adequately spans the region of the first order corrections and is reasonably well balanced from atom to atom. Basis set balance problems were most apparent with BH. They should be less severe in systems with two heavy atoms bonded together (e.g., CO) and in systems like FH, which is so nearly one ceuter that the p basis on H has litt!e effect. Since we presently cannot compute perpendicular polarizabilities with the partial-wave program, we are unable to determine the reliability of our rules for such calctdations. However, we would expect accuracies similar to those obtainable for the parallel component since the diffuse functions should span the appropriate space. Of course, one would have to include functions of the proper symmetries, some of which may be unnecessary in the absence of the field. For instance d6 functions would be needed for diatomics with occupied B orbitats. For comparative purposes, we have restricted our LCAO calculations to diatomic molecuIes in as much as the partial-wave procedure is presently applicable only to such systems. However, it is our hope that basis sets chosen according to a rule such as we propose would prove adequate for calculations on polyatomics as well. This would greatly simplify the choice of basis sets for polyatomic poIa~zability calculations and thereby greatly reduce their cost. Electron correlation cannot, of course, be ignored in pola~zab~ity calculations; for atoms, the pdarizability correlation is around 10% [ 181. However, correlation does not decrease the need for diffuse basis functions since the basis must still be capable of describing the distortion due to the field. Indeed, Werner and Meyer [4] obtained excellent agreement 441

Volume 55, number 3

CHEMlCAL

PHYSICS LETTERS

with experiment in their correlated calculations using basis set exponents chosen according to the secondorder energy optimization rule. Their exponents are very similar to those that would be obtained from our much simpler rule. To summarize, it appears that reasonably accr&tte (within about 2% of the Hartree-Fock limit) polarizabilities can be calculated using Dunning’s (4~3~12s) basis plus one contracted high exponent d function on each heavy atom and one contracted high exponent p function on H if the basis is augmented with one primitive diffuse function of each type with exponents chosen roughly as we propose. From our results, PO optimization appears necessary.

Acknowledgement We wish to thank Dr. P. Jeffrey Hay for several helpful discussions and Professor William E. PaIke for providing us with a copy of POLYATOM. This research was partially supported by the National Science

Foundation, Grant CHE76-08768 and by computer funds provided by Utah State University.

442

1 May 1978

References R.M. Stevens and W.N. Lipscomb, f. C&m. ?hys- 41 (1964) 184. f2] AD- McLean and M- Yoshimie, J. Chem- Phys. 46 (1967) 3682; 47 (1967) 32.96. [3] V.P. Gutschick and V. McKay, f. Ckm. Phys. 58 (1973) 2397. [4] H.J. Werner and W. Meyer, Mol. Phys. 31(1976) 855. [S] B. Day and A.D. Buck&ham. MoL Phys 32 (1976) 343. fl]

(61 P. Swanstrom, W-P. Kraemer and G.H.F. Diercksen, Theoret. Chhn. Acta 44 (1977) 109.

[7] P-A. Christiansen and E.A. McCuUough Jr., Chem. Phys. Letters S 1 (1977) 468. [S] R.E. Sitter and R.P. Hurst, Phys. Rev. A5 (1972) 5. [9] F.P. Bklingsley 11 and M. Krauss, Phys. Rev. A6 (1972) 855. [ 101 P-J- Fortune and P-R- Certain, J. Chem. Phys 61(1974) 2620. fllj T-H- Dunning, J. Chem. Phyr 53 (1970) 2823. fl2] T-H- Dunning, J- Chem. Phys 55 (1971) 716. [ 131 S. Huzinaga, J. Chem. Phys. 42 (1965) f293. [ 141 T.H. Dunning, J- Chem. Phys_ 55 (1971) 3958,

[15] P.E. Cade and W. Huo, J. Chem. Phys. 47 (1967) 614; A-D- McLean and M. Yoshhine, Tables of Linear Molecube Wavefunctions (IBM Corp., San Jose, 1967). [ 161 M. Morrison and P. Jeffrey Hay, J- Phys- BlO (1977) 1647. [ 171 H- Cohen, J. Chem- Phys. 43 (1965) 3558; H- Cohen and C.C.J. Roothaan, J. Chem. Phys. 43 (1965) S34. [ 181 R.R. Teachout and R.T Pack, Atomic Data 3 (1971) 195.