A multiple-scattering Xα study of silane and germane

Volume 20. number 5

1 July 1973

CHEhlICAL PHYSICS LETTERS

A MULTIPLE-SCATTERING

Xa

STUDY OF SILANE AND GERMANE

M.L. SINK and G.E. JUl?&S Batrelle Mmioriol Institute, Colurtrbus,Ohio 43201, UU

Received 30 April 1973 (MS Xo..)method are reported, Calculations for some of the group IV hydrides using the mu!tiple-scattering XCC and the results are compared with available experimental data and previous calculations, in those cases where such calculations exist. From these comparisons it appears that ionization potentials and transition energies can be adequately calculated by the his Xa method; however, in its present muffin-tin-restricted form the method gives binding energies and bond lengths that are in disagreement with experimen:.

I. Introduction The photoelectron spectra of the group IV hydrides, CH,, W-I,, GeH4 and SnH4 have been studied recently by Potts and Price [l] and Pullen et al. [5], while Hayes et al. [3] examined with high resolution the absorption spectra of SiiI4 and GeH4 in the energy range of 100-200 eV using synchrotron radiation. On the other hand, accurate Hartree-Fock (HF) calculations have been reported in the literature only for methane [4] and silane [5], since comparable ab initio-calculations for the larger hydride molecules are almost prohibitive, even ifwe neglect relativistic effects (which become increasingly more important as we go down the series). For these larger molecules, however, the rapidly converging multiple-scattering Xo method offers a real alternative. This method has

been applied recently by Danese [6] to the methane molecule, with results that are in good agreement with experiment [l, 71. They are also in good agreement with the HF calculations of Janoschek et al. [8] and cIso with the rest&s of ref. [4]. The mxessful application of the MS xol method to methane (and other tetrahedral complexes, such as SO:- and CIO, [9] and MnO, [ 101 for example) prompted us to investigate the applicability of the method to the heavier members of the series. As the details of the MS Xo ms-uiod can be found in t!!e review articles by Johnson [ 1 l] and Slater [ 121, and references therein, they will not be considered here.

474

In this paper, we present an MS Xo study of the silane and germane molecules. Since relativistic effects are important even for germane, we shall not apply the present nonrelativistic version of the MS Xor method to calculations of stannane. Ionization potentials (IP) and transition energies for the valence and highest-lying core states (Si 2s 2p; Ge 3s, 3p, 3d) are presented and compared with available experimental data and previous calculations, whenever such exist. In addition, the total electronic energies are calculated as functions of internuclear separation, and equilibrium values for these separations are established and compared with observed values. Also, the Rydberg nature of some of the excited-state orbita.Is is investigated directly from the MS Xo wavefunction;.

2. SiH, The ground-state

calculations

for srlane shown in

table 1 were carried out at R = Ml au, the experimental internuclear separation, using both the virialtheorem (avT) and the HF (cy& values for the exchange parameter of Si, as calculated by Schwas. [ 131 for ‘Ihe free atom. The intersphere value for a (c&l was taken to be a simple average of all o’s, while aOUT was chosen to be equal to ‘YH.The radii of the spheres surrounding the Si and H nuclei, Rsi and R,, were picked so as to minimize the non-muffm-

Volume 20, number 5

CHEhlICAL PHYSICS LETTERS

Table 1 Total statistical enerm and ground-state orbital energies (in Ry) of WI, for two different sets of MS Xc parameters (lengths are given in atcmic units) R

2.800 2.174 0.626 3.426 0.1275 1 = oHF”) 0.97804 0.92793 0.97804

RSi RH ROUT ?Si aH ‘YIN oOUT

2.800 2.174 0.626 3.426 0.72696 = oVTa) 0.97804 0.92782 0.97804

1 July 1973

Table 2 Variation of the total statistical energy E of to internuclear separation R

R (au)

E WY)

2.50 2.65 2.80 3.10 3.50 2.88 = R,

-581.696 -581.764 -581.792 -581.777 -581.684 -551.7918

s-&me

with

respect

;;

-130.828 -10.184 -1.070 -7.058 -0.668

-130.816 -10.176 -1.062 -7.050 -0.664

Table 3 Comparison of hlS Xa transition energies and ionization potentials (in eV) with experimental data and other HF calculations for sllane

E

-581.792

-581.759

Transition

&IF

-582.471”)

la1 2% 3%

tin contributions to the molecular potential, which are neglected in the present version of the MS XLY method. A procedure for choosing these sphere radii from the covalent radii 1141 of the constituent atoms has been discussed by Danese, in MS paper on methane [6]. The calculation using &YHF gives a value for the total statistical energy that is lower than the value obtained using &VT, and both of these values are above the total HF energy. In the transition-state calculations to be discussed below, therefore, we have used the aHF value for the exchange parameter of Si. (We make no explicit comparison between MS Xo and HF orbital energies because the two are different by defitzition [ 11, 121.) The variation of the total statistical energy E was studied as a function of internuclear separation R. of the E(Rj

HF

Experimental

2a, -) 3%

a) Ref. [13] ; b) ref. [5].

Five points

MS Xa

curve

were

calculated

by

R and keeping the ratio R,/R,i fixed (and equal to its value from table 1). These points are

varying

given in table 2. Using these data, a value of 2.88 au was obtained for the equilibrium internuclear separation. This value deviates only by 3% from the experimental value of 2.80 au [2, IS]. The corresponding deviation for CH4 is about 2% (see ref. [6]). The calculated dissociation energy for SiH, is only 1.13 eV, which disagrees with the experimental value of 13.3 eV [16] and the HF value of 10.97 eV [S] .

4ar le

146.1

rp

149.6

ltz43tz 4% le IP

3a, + 3% 4% le IP

%“% 4% le

IP

103.6 104.5 105.3 108.3 12.79 14.08 14.43 17.7

149

114.96 [S]

aj

107.53)

19.87 [5] 19.36 1231

18.17b) cl

11.89 1241 13.21 [5] 11.94 [23]

12.tnb) 12.36 d) 12.20c)

7.21 8.27 8.77 11.97

a) Absorption, ref. [ 31. b) Photoelectron spectroscopy, ref. [I]. c) This transition was not observed in the photoeIectron work of ref. [Z], even rhough 21.22 eV photons were used in the measurements. d) Photoelectron spectroscopy, ref. 121. e) ESCA, ref. [22].

In the transition-state calculations we have used the experimental internuclear separation. The results of these calctilations are shown in table 3 together with the corresponding experimental results and results of the HF ca!culation of ref. [5] _The 3nl and 1t2 HF ionization potentials calculated from ground-

Volume 20, number 5

CHEMICALPHYSICSLETI’ERS

state energies are not in as good agreement with experiment as is the calculated HF IP for the 2t, valence state. This is probably due to the neglect of core relaxation in the HF calculation; thus, individual calculations on the ions would probably improve the HF results. On the other hand, the MS x0! caiculations which take relaxaticn into account compare quite favorably with experiment. For the valence levels the IP’s agree closely with the photoelectron data of Potts and Price [ 1j and also with Pullen et al. [2]. The 2al dnd 1t2 IP’s agree well with the values obtained by Hayes et al. [33 from their absorption work. The ordering of the 1t2 transitions, however, does not agree with the ordering given in ref. [3]. The MS X& ordering is 3t2, 4al, le in order of increasing energy, whereas the ordering of ref. [3j is given as 4al, 3t2, le. It should be noted here that the level assignments of ref. [3] were based on: the results of a one-center SiH4 calculation. A close analysis of the distribution of electronic charge in the various spheres for the various excitedstate o@itals shows that the 4al orbital is very diffuse and can be identified as a Rydberg state. This is also true for the 3a, orbital of methane, i.e., the.electronic charge of this orbital is concentrated in the region outside ROUT. (We have established this with a test calculation on methane, using Danese’s parameters. The relative distribution of charge for the 3a, excited state of CH, for the It, + 3a, transition was found tb be: OUT(0.881), C(O.OSS>,H(O.OOO),IN(0.032).) The relative distribution of charge in the various muffin-tin spheres of SiH4 for th: 1t2 transitions is shown in table 4. From this table, the le and 3t2 orbitals show a mixture of diffuse and valence character. In addition, an examination of the partial-wave content of the wavefunctions of these excited-state molecular orbitals reveals that the 4al state is mainly s-like, the le state is d-like, and the 3t2 state is Table 4

Relative distribution of charge (normalized to unity) in the various muffin-tin spheres of the silane molecule for transitions from the It, orbital to excited state orbit& Final

Si

state

OUT

3t2 4a1 ,le

0.447

0.144

0.012

0.839

0.120

0.001

0.625

0.094

0.000

0.358 0.036 0.279

1 July 1973

mostly d-like with some mixture of p character. (Here, the partial waves are referred to the center of the molecule.)

3. GeH, Our calculations for germane used only the HF value for the Ge exchange parzmeter, a,-& = 0.70684 [I3]. The values for aOUT and QH were taken to be the same as in silane, and the average value for cam was 0.92380. The present muffin-tin-restricted version of the MS Xcumethod has been shown to yield inadequate potential energy curves, dissociation energies, and bond lengths not only in diatomics, such as Ne2 [ 171 and N,, 02, F, [ 181, but also in more globular molecules, such as XeF, [ 191. This is to be expected, however, because the neglected non-muffin-tin part of the molecular potential both inside the muffin-tin spheres, where all nonspherical overlap is left out, and in the intersphere region, where the overlap is approximated by a constant, could rnake nonnegligible contributions to the total energy that are strong 1 functions of the internuclear separation. In our attempts to establish an equilibrium hlS Xa geometry for GeH4 we encountered difficulties similar to those discussed in the previous paragraph. Our stndy of the total statistical energy as a function of R and of the ratio of the muffin-tin radii R, and RGe is shown in table 5. In the first set of calculations, the value for RH was fned at the same value as that used in the Cl-I4 and SiH, calculations and RG, was varied until an optimum ratio, R,/RG,= 0.2326, was obtained. With this ratio for the muffin-tin radii, the total energy was calculated as a function of R and

the results are given in the second set of entries in table 5. From these results, an equilibrium R was obtained which is only 6% larger than the experimental value; however, the molecule is not bound at this internuclear separation (separated-atom energy is -4154.7180 Ry). In an attempt to improve on these results, the bond distance was fL.ed at the experimental value of 2.834 au [2] in the third set of calculations, and RH and Rc~ were varied (keeping the atomic spheres tangent to one another). An equilibrium ratio, RH/RGe = 0.1759, was established, and with this ratio several points were calculated on

Volume 20, number 5

CHEMICAL PHYSICS LETTERS

Table 5 Total statistical energy (in Ry) of GeH, as a function of R for various choices of the muffin-tin radii R

RGe

2.788 2.988 3.188 3.588 3.3 17 = R,

E

2.162 2.362 2.562 2.962 2.691

RH 0.626 0.626 0.626 0.626 0.626

-4154.3196 -4154.4726 -4154.5255 -4154.4740 -4154.5404

0.2326

2.788 2.988 3.188 3.290 3.008 = R,

2.262 2.424 2.586 2.669 2.440

0.526 0.564 0.602 0.621 0.568

-4154.5669 -4154.6179 -4154.5861 -4 154.5403 -4154.6183

0.2326 0.2326 0.2326 0.2326 0.2326

2.834 2.834 2.834 2.834 2.834 = R,

2.508 2.408 2.308 2.208 2.410

0.326 0.426 3.526 0.626 0.424

-4154.6109 -4154.7001 -4154.6076 -4154.3685 -4154.7002

0.1759

2.788 2.834 2.988 3.188 3.290 3.588 3.195 = R,

2.371 2.408 2.541 2.711 2.798 3.051 2.71-i

0.417 0.426

-4154.6438 -4154.7001 -4154.7732 -4154.7989 -4154.7884 -4154.6865 -4154.7989

0.1759 0.1759 0.1759 0:1759 0.1759 0.1759 0.1759

0.447

0.477 0.492 0.537 0.478

1 July 1973

Table 6 Variation cf selected germane ionization potenti& with internuclear experiment

distmce

(md RH), and comparison

(in ev) with

RH&k

the potential curve. These E(R) points are given in the fourth set of entries in table 5. From these, we caiculated a positive dissociation energy of 1.l eV, which

is in wide disagreement from the experimental value of 12 eV [20], and an equilibrium separation which is off by 13% from the experimental value. Even though these disagreements with experimental values are rather large, they are nevertheless smaller than the discrepancies reported in refs. [ 17,181 for diatomic molecules. This may be explained by the fact that the muffin-tin approximation in the inter-

sphere region is less drastic for tetrahedral molecules than it is for diatomics. Seiected ionization potentials were calculated for germane at several internuclear separations and radii parameters, since our limited study of potential curves did not yield an optimum equilibrium geometry for this molecule. In table 6 we give these resul!s for two particular geometries and compare the calculated values to experimentai data. The calculated ionization potentials for all molecular geometries are in better relative agreement with experimental values

R(m)

RH(FU)

4%

42,

3%

le

2b

2.788

0.626 OS26

12.4 11.7

18.9 18.5

37.9

37.9

122

0.626

11.6

17.9

38.7

38.7

123

0.564

11.1

17.7

2.988

Expcrimcntal 12.46”) 11.98b)

18.4al c)

124.7 :29.7d)

a) Photoelectron spectroscopy, ref. [ 11. b) Photoelectron spectroscopy, ref. [21_ c) This transition was not observed in tkis experiment, ei’cn tbougb it is below the 21.22 eV photon energy used in this work. d) Absorption, ref. [ 31.

than are bond lengths and binding energies. Also, these ionization potentials appear to be more sensitive to the internuclearseparation than to the choice of R, and R,,. The 42, and 4t, IP’s cdculated at the internuclear separation R = 2.788 au, which is closest to the experimental value, are in better agreement with the experimental data than are the IP’s calculated at R = 2.988, which is an internuclear separation close to an MS XCY minimum (see table 5). The 3t2 and le ionization potentials are inc!uded in table 6 since they are sufficiently low in energy that they may be observed in photoelectron spectra with He II (40.8 eV) photons. The IP for the deep-lying 2t2 state shows a discrepancy of 2-3 eV compared to the absorption data [3] ) which in addition show a large (5 e’v’)spinorbit splitting. The corresponding discrepancy of the 1t2 state of SiH, (see table 3) is only 0.5 eV. In this case the experimental spin-orbit spiitting is only 0.65 eV. From this comparison, it appears that it is necessary to generalize the MS Xczmethod to include relativistic effects - with Darwin and mass-veIocity corrections included in addition to spin-orbit effects - at least for molecules with high Z atoms. (The multiple-scattering method for band-structure calculations in solids, known as the KKR method 1201, has been generalized to include such correcticns [21] .) Transition-state calculations for the 4a, and 4t, 477

Volume 20, number 5

CHEMICALPHYSICSLEZ-I’ERS

orbitals to the excited Sa,, 5t,, and -2e orbitals were carried out at R = 2.988and R, = 0.626. Transition enemies obtained were 7.07,7.35 and 8.45 eV for the St*, 5aI and 2e excitations of the 4t, orbital, and 13.2, 13.6 and 14.7 eV for analogous transitions of the 4aI orbital. An analysis of the charge distributions in the molecular regions and the partial-wave components for the excited-state orbitals showed that all are mainly concentrated in the region outside R,,,, and therefore are Rydberg in nature (although the 9, orbital does show some valence character). The partial-wave components (referred to the center of the molecule) of the excited-state orbitals indicate that the 5aI orbital is mainly s-like, the 2e d-like, and the St2 mostly d-like (with some mixture of s character around the H centers). The present study of the group IV hydrides points out the need for extending the present version of the MS Xo method - however successful it may be for the calculation of ionization potentials - to include non-muffin-tin corrections, which seem to be important in potential-curve calculations, and relativistic effects for molecules wkh high 2 atoms.

Acknowledgement The authors wish to thank Professor K.H. Johnson and Dr. F.C. Smith Jr. for making their SCF MS Xcr programs available.

References [ 11 A.W. Potts and WC. Price, Proc. Roy. Sec. A326 (1972) 165. [2] B.P. Pullen, T.A. Carlson, W.E. Moddemnn, G.K. Scheitzer, W.E. Bull and F.A. Grimm, J. Chcm. Phys. 53 (1970) 768.

1 July 1973

131 W. Hayes and F.C. Brown, Phys. Rev. A6 (1972) 21; W. Hayes, F.C. Brown and A.B. Kunz, Phys. Rev. Letters 27 (1971) 774; W. Hayes, Contemp.Phys. 13 (1972) 441. 141 E. Clementi and H. Popkie, J. Am. Chem. Sot. 94 (1972) 4057. 151 S. Rothenberg, R.H. Young and H.F. Schaefer HI, J. Am. Chem. Sot. 92 (1970) 3243. [6] J.B. Danese, Intern. J. Quantum Chem. 6 (1972) 209. ]71 K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Heddn, K. Hamrin, U. Gelius, T. Bergmark, L.O. Werme, R. Manne and Y. Baer, ESCA applied to free molecules (North-Holland, Amsterdam, 1969). 181 R. Janoschek, G. Diercksen and H. Preuss, Intern. J. Quantum Chem. 1 (1967) 373. 191 K.H. Johnson and F.C. Smith Jr., Chem. Phys. Letters 7 (1970) 541. 1101 K.H. Johnson and F.C. Smith Jr., Chem. Phys. Letters 10 (1971) 219. III1 K.H. Johnson, in: Advances in quantum chemistry, Vol. 7, cd. P.-O. Lawdin (Academic Press, New York, 1973). 1121 J.C. Slater, in: Advances in quantum chemistry, Vol. 6, ed. P.-O. Liiwdin (Academic Press, New York, 1972). 1131 K. Schwarz, Phys. Rev. B5 (1972) 2466. [I41 J.C. Slater, Quantum theory of molecules and solids, Vol. 2 (McGraw-Hill, New York, 1965) ch. 4. [I51 D.R.J. Boyd, J.Chem. Phys. 23 (1955) 922. __. [ 16 J S.R. Gunn and L.G. Green, J. Phys. Chem. 65 (1961) 779. [ 171 D.D. Konowalow. P. Weinbereer. J.L. Calais and J.W.D. Connolly, Chem. Phys. Letter; 16 (1972) 81. 1181P. Weinberger and D.D. Konowalow, to be published. 1191E.N. Phillips, J.W.D. Connolly and S.B. Trickey, Chem. Phys. Letters 17 (1972) 203. [201 J. Xorringa, Physica 13 (1947) 392; W. Kohn and N. Rostoker, Phys. Rev. 124 (1961) 1786. [211 Y. Onodera and M. Okazaki, J. Phys. Sot. Japan 21 (1966) 1273; S. Takada, Progr. Theoret. Phys. 36 (1966) 224. 1221 H. Neuert and H. Giasen, Z. Naturforsch. A7 (1952) 410. [231 T. Moccis, J. Chem. Phys. 40 (1964) 2164. [24] D.B. Cook and P. Pahnieri, Chem. Phys. Letters 3 (1959) 219.