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Extension of Gaussian-2 theory to molecules containing third-row atoms

Ga±Kr

Larry A. Curtiss

Chemical Technology/Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439

Mark P. McGrath

Research School of Chemistry, Australian National University, Canberra, ACT, 0200 Australia Jean-Philippe Blaudeau, Nancy E. Davis, and Robert C. Binning, Jr. Chemical Technology/Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439

Leo Radom

Research School of Chemistry, Australian National University, Canberra, ACT, 0200 Australia ~Received 3 April 1995; accepted 14 June 1995! Gaussian-2~G2!theory has been extended to molecules containing the third-row nontransition

elements Ga±Kr. Basis sets compatible with those used in G2 theory for molecules containing ®rst-

and second-row atoms have been derived. Spin±orbit corrections for atoms and molecules having spatially degenerate states~ 2 P, 3

Pfor atoms and

2

Pfor molecules in this work!are included

explicitly in the G2 energies. The average absolute deviation from experiment for 40 test reactions is 1.37 kcal/mol. In contrast to the situation for ®rst- and second-row molecules, inclusion of

spin±orbit effects is very important in attaining accurate energies for molecules containing third-row

atoms. Without spin±orbit effects, the average absolute deviation is 2.36 kcal/mol. ©1995

American Institute of Physics.

I. INTRODUCTION

The Gaussian-2~G2!theory

1 of molecular energies was introduced to provide a method for calculating accurate ther- mochemical data for molecules containing ®rst- and second- row atoms. Atomization energies, ionization energies, elec- tron af®nities, and proton af®nities were generally found to lie within 0.1 eV of established experimental values. G2 theory is a composite procedure based onab initiomolecular orbital theory and is a signi®cant improvement over its pre- decessor Gaussian-1~G1!theory. 2,3

Since its introduction,

G2 theory has been applied to numerous species for which experimental data are lacking or are uncertain. 4 In this paper, we extend G2 theory to molecules contain- ing the third-row nontransition elements Ga±Kr. There is substantial interest in the thermochemistry of compounds containing the atoms Ga±Kr.

5±7

For example, germanium

compounds play an important role in a variety of modern technologies such as in information-recording media, in optical-tracking devices, and as photoreceptors. The exten- sion of G2 theory to molecules containing Ga±Kr requires ~1!the development of new basis sets that are compatible with those of the ®rst- and second-row atoms,~2!the testing of the method on a suitable set of species with accurate ex- perimental data, and~3!an assessment of the importance of spin±orbit corrections which become signi®cant for third- row atoms and for some molecules containing third-row at- oms. G1 theory was previously extended to bromine- containing molecules by McGrath and Radom 8 and found to give an accuracy comparable to that of G1 theory for ®rst- and second-row molecules. 2,3

In some cases, molecular

spin±orbit corrections were found to be necessary to achieve an accuracy of 0.1 eV. In the development of basis sets for

Ga±Se and Kr, we have followed the methodology used inRef. 8 for the development of the bromine basis sets. G2

theory for Ga±Kr was then tested on a set of 40 reactions. In Sec. II, the theoretical procedures used for extension of G2 theory to the third-row atoms Ga±Kr are described. In Sec. III, the construction of the basis sets is described while in Sec. IV, G2 theory is evaluated for the set of test species.

II. THEORETICAL PROCEDURES

Gaussian-2 theory is based on standardab initiomolecu- lar orbital methods. 9,10

In the extension of G2 theory to the

third-row atoms Ga±Kr, the theoretical procedures used are the same as in G2 theory for ®rst- and second-row systems.

1±3

Equilibrium geometries are optimized using

second-order Mo"ller±Plesset perturbation theory~MP2!, and single-point energies are calculated at the second- and fourth-order Mo"ller±Plesset perturbation theory~MP4!and quadratic con®guration interaction@QCISD~T!# ~Ref. 11! levels of theory using basis sets described in Sec. III. A higher level correction~HLC!, based on the number of pairs of valence electrons, is used to account for residual basis set errors. The HLC used for third-row molecules is the same as that used for the ®rst- and second-row molecules.

1±3

Har- monic vibrational frequencies are calculated at the Hartree± Fock~HF!level to obtain zero-point vibrational energies.

These zero-point energies are scaled by 0.893.

The frozen-core approximation is used in single-point correlation energy calculations. For molecules containing ®rst-row atoms, the doubly-occupied orbital which effec- tively corresponds to the 1satomic orbital is excluded from the correlation treatment. For molecules containing second- row atoms, the doubly-occupied molecular orbitals which correspond approximately to the 1s,2s,2p x ,2p y , and 2p z atomic orbitals are de®ned as core orbitals and therefore ex-

6104 J. Chem. Phys.103(14), 8 October 1995 0021-9606/95/103(14)/6104/10/$6.00 © 1995 American Institute of PhysicsDownloaded¬17¬May¬2006¬to¬140.123.5.13.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp

cluded. The procedure for ®rst- and second-row atoms is thus the same as has been used previously in G2 theory. For mol- ecules containing third-row atoms, we include in the core the doubly-occupied molecular orbitals which correspond to the

1s,2s,2p

x ,2p y ,2p z ,3s,3p x ,3p y ,3p z ,3d z 2,3d x 2 2y 2, 3d xy ,3d yz , and 3d xz orbitals in the individual atoms. The effect of excluding the 3dorbitals from the correlation treat- ment was examined previously in the extension of G1 theory to bromine-containing systems and found to be small. 8 The

3dorbitals were also frozen in a study of dissociation ener-

gies and ionization energies of AH n ~A5Ge,As,Se,Br!mol- ecules calculated by a G1-like method which yielded results in good agreement with experiment. 12

In the single-point cal-

culations using the ``supplemented 6-311G'' basis sets for Ga±Kr described in the next section, there is always a high- lying virtual arising from the use of two basis functions for the 1sorbital. In the molecular situation, there will be one such orbital for each third-row atom. In the work reported here we have frozen these orbitals. However, the effect on the energy is negligible~;10 27
hartree in atoms and mol- ecules!, so whether or not these orbitals are frozen makes little difference. Basis function optimizations were carried out with the GAUSSIAN 90~Ref. 10!andTURBOMOLE~Ref. 13!programs.

III. CONSTRUCTION OF BASIS SETS

G2 theory involves approximating the QCISD~T!/6-311

1G(3df,2p)//MP2~full!/6-31G(d) energy using MP4 and

MP2 energies calculated with the 6-311G(d,p), 6-311

1G(d,p), and 6-311G(2df,p) basis sets, MP2 energies cal-

culated with the 6-3111G(3df,2p) basis, and QCISD~T! energies calculated with the 6-311G(d,p) basis. The 6-311G basis set and supplementary functions for the ®rst- and second-row atoms

14±17

were developed prior to G2 theory. In order to extend G2 theory to the third-row atoms Ga±Kr, it is necessary to derive a 6-311G-type basis set and supplemen- tary functions for Ga±Kr that are compatible with those used for H±Cl. For ®rst-row atoms, the 6-311G basis involves (11s,5p)

Gaussians contracted as@6311,311#resulting in 13

functions. 14

The 6-311G basis for second-row atoms has

been de®ned 3,17 as the basis developed by McLean and

Chandler.

16

It can be described as core double zeta, valence

double zeta ins, and valence triple zeta inp. It was derived by optimizing HF atomic energies. This basis involves (12s,9p) Gaussians contracted as@631111,52111#resulting in 21 functions. Beyond the ®rst row, ``6-311G'' is only a mnemonic for bases of similar quality that have become standard. For the third-rowp-block elements Ga±Kr, uncontracted (14s,11p,5d) Gaussian bases due to Dunning 18 and to

Huzinaga

19 which yield ground-state electronic energies

50±60 kcal/mol higher than the numerical Hartree±Fock

energies 20 have been available for some time. 21

The best seg-

mented contraction of the Dunning (14s,11p,5d) bases had been Dunning's [8s,6p,2d] double-zeta~core and valence! contraction for Br reported by del Condeet al., 22
in which optimization of both the contraction coef®cients and the

primitive Gaussian exponents was carried out to some extent,yielding a ground-state energy 30 kcal/mol higher than the

uncontracted 18

Hartree±Fock energy. Previously, we used

this [8s,6p,2d] Br basis 22
to derive the ``6-311G'' [8s,7p,2d] Br basis and its associated supplementary functions. 8

Since [8s,6p,2d] bases for Ga±Se and Kr have

not been previously reported, we derived such [8s,6p,2d] bases for these atoms as follows.

Using the [8s,6p,2d] Br basis

22
as a zeroth-order ap- proximation to our double-zeta Se basis, scale factors for all of thespdshells were optimized ®rst, followed by optimi- zation of the contraction coef®cients second, and optimiza- tion of the individual primitive exponents third. These three steps were repeated until the energy decrease became negli- gible, requiring typically 10 cycles. The resulting [8s,6p,2d] Se basis was then used as a zeroth-order ap- proximation to our double-zeta As basis, and so on, giving [8s,6p,2d] basis sets for Ga±Se. Having obtained good quality double-zeta bases for Ga±Se in this manner enables the straightforward derivation of ``6-311G'' [8s,7p,2d] bases and associated supplemen- tary functions for Ga±Se using procedures analogous to those described in Ref. 8 for Br. In summary, this entailed ®rst optimizing simultaneously the ®ve valence 4spGauss- ian exponents and all of the geometrical parameters for each of GaH, GeH 4 , AsH3 , and SeH 2 at the HF/6-311G level, the optimum 4spexponents combining with the core [6s,4p,2d] basis functions to give the 6-311G bases. Then, for each of these molecules, the exponent of a single polar- izationd-type Gaussian~®ve atomic functions!and the geo- metrical structure were optimized simultaneously at the HF/

6-311G (d,p) level, followed by optimization of a single

f-type Gaussian~seven atomic functions!at the MP4/6-

311G(2df,p) level. Finally, optimization of diffusesandp

exponents with the constraintsvp, along with the geometri- cal parameters, was undertaken for Ga 2 , GeH 32
, AsH 2 2 SeH 2 , and Br 2 at the HF/6-3111G(d,p) level. Note that in Ref. 8 the diffusesandpfunctions for Br were optimized separately. This procedure leads to tight valence-likesfunc- tions for Se, As, and Ge. Therefore, we have optimized all diffuse functions with the constraintsvp. The resulting dif- fuse functions for Br are slightly different from those in Ref.

8 on G1 theory, and caution should therefore be exercised in

comparisons of G2 total energies in the literature.

The ``6-311G'' [8s,7p,2d] Kr basis was optimized,

starting from the 6-311G Br basis. The ®nal ground-state energy was 9 kcal/mol higher than the uncontracted 18 Hartree±Fock value. Thedandfsupplementary function optimizations were carried out for KrH 1 in a manner com- pletely analogous to that described above. The diffusesand pGaussian exponents, a , were deduced by linear extrapola- tions ofa s anda p vsZ, whereZ534±36. The 6-311G basis set and supplementary functions for Ga±Kr are listed in the Appendix. The atomic energies ob- tained from these basis sets are listed in Table I along with energies from several other basis sets for comparison. The optimized [8s,7p,2d] bases for Ga±Se and Kr can be seen to yield ground-state energies only 7±10 kcal/mol higher than the uncontracted 18

Hartree±Fock values.

23

Before we can use our third-row 6-311G bases to calcu-6105Curtisset al.: Extension of Gaussian-2 theory

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late G2 energies, the question of how the polarizationd-type

Gaussian exponents

a d should be split when creating (2d) and (3d) sets of functions needs to be addressed. For the elements Li±Ar, the geometrical progressions ( a d /2,2a d and ( a d /4,a d ,4a d ) are used. 15~a!

That is, respective splitting

factors of 2 and 4 are standard. However, tests on the third- row systems GaBr, GeH 4 , AsH 3 ,As 2 , SeH 2 , BrH, and KrH 1 show that for MP2/6-311G(2df,p) and MP2/6-

311G(3df,2p), splitting factors of;1.5 for the (2d) set and

;2.8 for the (3d) set are energetically preferred, although the relevant energy surfaces are quite ``¯at'' around the points corresponding to minimum energy splitting factors. Therefore, without making too ®ne a distinction, we decided for the third-row bases to adapt standard splitting factors of 2 and 3, respectively, for creating (2d) and (3d) sets of basis functions. The structures required for G2 calculations on molecules containing ®rst- and second-row atoms are optimized at the

MP2~full!/6-31G(d) level.

1

Since there is no 6-31G(d) basis

for Ga±Kr, we adopt the 641(d) basis of Binning and

Curtiss.

24

This is a [6s,4p,1d] contraction of Dunning's

(14s,11p,5d) primitive set 18 with the addition of a single d-polarization function. This basis set has been tested on a series of third-row molecules and has given a good account of the geometries and vibrational frequencies. 24

In the exten-

sion of G1 theory to bromine, 8 the SV4P bromine basis set of

Andzelmet al.

25
was used. However, we have chosen here to use the 641(d) basis uniformly for all third-row elements. The geometries obtained for the bromine-containing mol- ecules using this basis are nearly the same as those obtained from the SV4P basis~see Table II!.

8,12~c!

In calculations of

``HF/6-31G(d)'' zero-point energies, the 641(d) basis was also used in combination with the 6-31G(d) basis. In all calculations with the 6-31G(d) basis, we use six second- order Cartesian Gaussians for all atoms. In the remainder of this paper, the notation 6-31G and 6-311G for the Ga±Kr atoms refer to the 641 and [8s,7p,2d] basis sets described above. The MP2~full!/6-31G(d) optimized geometries for the molecules used in evaluating G2 theory for the third-row Ga±Kr atoms are listed in Table II along with available ex- perimental values. The agreement with experiment is gener- ally good. For the 21 bond lengths involving atoms other than hydrogen, the average deviation of the MP2~full!/6-

31G(d) values from experiment is 0.023 Å. As expected, the

MP2~full!/6-31G(d) bond lengths are uniformly

overestimated. 24

IV. COMPARISON WITH ACCURATE EXPERIMENTAL

DATA G2 theory has been applied to the set of atoms, mol- ecules, and ions listed in Table III. This includes species chosen in Ref. 8 to test G1 theory on bromine compounds.

Table III contains G2 energies at 0 K~E

0 !for these systems and the individual energy components from which they are obtained. All of these components are de®ned in the same way as for the ®rst- and second-row species,

1±3

with the exception of the spin±orbit correction term@DE~SO!#. The DE~SO!includes energy lowering due to ®rst-order spin± orbit corrections. 26

The ®rst-order spin±orbit corrections for

the atomic species are taken to be the energy difference be- tween the spin±orbit±coupled ground state and the weighted J-averaged state. Also included in Table III are ®rst-order spin±orbit corrections forallmolecules which have such corrections, i.e., those that have spatially degenerate states.quotesdbs_dbs49.pdfusesText_49

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