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Ab initio calculation of the dipole moment and frequency-dependent polarizability for silicon monoxide and its anion
Journal ofMolecular Structure, 243 (1991) 147-162 Elsevier Science Publishers B.V., Amsterdam
147
AB INITIO CALCULATION OF THE DIPOLE MOMENT AND FREQUENCY-DEPENDENT POLARIZABILITY FOR SILICON MONOXIDE AND ITS ANION
TOSHIHIRO INOUE and SUEHIRO IWATA Department of Chemistry, Faculty of Science and Technology, Keio University, 3-14-l Hiyoshi, Kohoky Yokohama 223 (Japan) (Received 12 April 1990)
ABSTRACT The potential energy curve and the bond-length dependence of the electric properties for SiO and its anion are calculated using ab initio MO methods. The static and frequency-dependent polarizabilities are evaluated directly from the large CI Hamiltonian matrix without the explicit knowledge of the eigenvectors of the matrix. Two states of the anion SiO- (‘%+ and 2H) lie closely by the ground state X’Z+ of SiO. The equilibrium internuclear distance R, of the 2Z+ state is nearly equal to that of the X’Z+ SiO state, while the R, of the 211state is substantially longer than the latter. The electron affinity is estimated to be negative or nearly zero. The bond-length dependence of the dipole moment, the quadrupole moment and the polarizability tensor is expected to be remarkably different among three states X’I;+, *H and *Z’.
INTRODUCTION
Silicon monoxide, SiO, is one of a few well-known interstellar molecules with strong maser phenomena [l-4]. One of the possible excitation mechanisms of the maser action is the electron-molecule collision. To study the electronmolecule interaction, knowledge of the electric properties of ground state SiO and their bond length dependence is required. Besides, to examine the possibility of the resonance interaction, one has to know the relative energy of SiOanion to the neutral SiO as well as the properties of the anion. Although several ab initio studies of SiO molecule have been reported [5-g], studies for the anion are not found. In the present work, we study the potential energy, electric dipole moment, electric quadrupole moment and dipole polarizability curves for the X12+ state of SiO and for the lowest 211and ‘xc+ states of SiO-. The electron affinity of SiO is evaluated. The potential energies and dipole moments are calculated by using the restricted SCF and CI methods. Dipole polarizabilities are calculated with the frequency-dependent moment method [ 10,111, which enables us to calculate
148
the polarizability tensor directly from the large CI hamiltonian matrix without the explicit knowledge of the eigenvectors of the matrix. We can evaluate not only the static polarizability but also the frequency-dependent polarizability with this method. The frequency-dependent polarizability is related to the optical properties such as the refractive index, Raman and Rayleigh scattering intensities. The static polarizabilities are calculated also with the finite-field method. At the SCF level, the electric quadrupole moments are also estimated. METHOD
Our own ab initio program package MOLYX was used in the calculations. Most of the calculations have been carried out at the Computer Center of Institute for Molecular Science, and some on our local engineering work station, SONY NEWS 831. SCF calculations The MIDI4* basis set of Tatewaki and Huzinaga [ 121 augmented with two diffuse s and two diffusep on each atom (0 (x,’ = 0.060, cy,” = 0.020, aP’ = 0.060, or,“=O.O20; Si a,‘=0.027, a!,” =0.009, c~!,‘=O.O27, a!,,” =0.009) is used. The total number of CGTO is 50. After the closed shell SCF calculation, the molecular orbitals of the ground state X’C+ of SiO are used as the initial guess orbitals for the restricted open-shell SCF calculation of SiO-. The dipole and quadrupolemoments of SiO and SiO- are calculated by using these SCF wavefunctions. Molecular coordinates are defined as follows. The origin is placed on the center of mass of the SiO- ion and the molecular axis from 0 to Si is selected as the z axis. For example, if the bond length R is 1.5 A,0 is at (X=0.0,Y=O.O,Z= -0.956)andsiisat (x:=O.O,y=O.O,z=O.544). The finite field SCF method [ 13-151 is also used for calculation of dipole moment from the perturbed electronic energy. The electric field is applied in the direction of the molecular axis (z axis). Configuration interaction calculations The molecular orbitals are divided into four types to classify the configuration state functions (SCFs); (1)“frozen” core orbitals, which are always doubly occupied, ( 2 ) “core orbitals” {c}, (3 ) “active orbitals” (a}, and (4) “excited orbitals” {e}. In the case of the ground state of SiO molecule, CSFs are based on the closed shell SCF MOs. However, CSFs for 217and 2Z+ states of SiOare based on each open-shell SCF MO. The orbitals are classified as {c} = {lo, 20,3a, In, 4a} and {a}={5cr, 2x, 60,70,3n, 80). Consequently, CSFs are classified into 9 types; (0) all of the possible CSFs generated by filling active orbitals (CAS), (1) a2-e2, (2) a-e, (3) ca-e*, (4)
149
c-e, (5) c2-e2, (6) c-a, (7) c2-ae, (8) c2-a2. In the present study, the configuration space consists of the types (0) and (2). As a result, the total CSFs are 4535 for X1x+, 2734 for 211and 2856 for 2Z+. The thresholds for eigenvalue and for vectors are 1.0~ 10m6hartree and 3.0~ 10v4. The CI calculations are carried out with C2” symmetry. Then the dipole moment function is calculated for each state. Polurizability calculation The polarizability tensor is calculated from the CI hamiltonian matrix. The frequency-dependent polarizability CYBA (hu) is expressed as
where B and A are the dipole length operators, (2, Y, or X) and Ek and jk) are an eigenvalue and an eigenvector of the hamiltonian matrix. To sum up the contributions from all of the eigenstates of the hamiltonian matrix, it appears that knowledge of ik) and Ek for all states is required. However, the sum in eqn. (1) can be evaluated by solving a simultaneous linear equation, instead of diagonalizing the hamiltonian matrix. This method, named the frequencydependent moment method, was reported by Iwata and applied to H,O [lo] andtoN,andO, [ll]. RESULTS AND DISCUSSION
Potential energy curves for the ground state X’C+ of SiO and the two lowest doublet states of SiOThe potential energy curves near the equilibrium bond distance obtained by the ab initio Cl calculation are shown in Fig. 1. The curves of the ground state X1x+ and the lowest ‘II state of SiO, and the lowest 2H and 2Z+ states of SiOare all shown in the figure. The potential curve of SiO ground state has a minimum at the bond distance R= 1.55 6, while the experimental value of the equilibrium bond distance is R e = 1.51 A [ 161. Thus the calculated R, is slightly longer than the experimental one. At the closed-shell SCF level, R,= 1.51 A. Previously Werner et al. [ 61 reported the calculated R, value was 1.48 A in the restricted SCF and 1.52 A in the MCSCF CI. The lowest ‘II state has the (70+3x) character and its energy minimum is at R, = 1.70 A in the present work. The vertical and adiabatic excitation energies from the ground state are 5.63 eV and 5.33 eV, respectively. Experimentally [ 17,181, R, = 1.62 A and the adiabatic excitation energy T, is 5.29 eV. Previously, Heil and Schaefer [8] studied several excited states for SiO molecule. Their calculated values for the lowest ‘H state are R, = 1.75 A and T, = 4.79
&f:+ 1
1.6
1.4 INTERNUCLEAR
DISTANCE
1.6 _ R/W
Fig. 1. Potential energy curves for the ground state X’Z+ lowest Tl and ‘Z + states of SiO- .
and the lowest ‘II state of SiO, and the
eV. Robbe et al. [9] studied the SiO valence states by CI calculations, and reported that the T, for the first ‘II state was 5.07 eV. Our results for the lowest optically allowed state reproduces the experimental results reasonably well. Two states of the anion SiO- (‘Z+ and 211) lie closely by the ground state SiO. In the CI calculations two curves cross at R= 1.70 A; at shorter bond lengths, the ‘C+ state is the lowest state. This is also true in the SCF calculation. The equilibrium bond distance R, of ‘C.+ happens to be nearly equal to that of the X’Z+ SiO state; R, = 1.55 A in the CI calculation and R,= 1.51 A in the SCF calculation. As will be discussed in the following section, an extra electron in the ‘Z+ state is very diffuse and non-bonding. No change in the bond distance is consistent with this nature of the state. In contrast to the 2Z+ state, the R, of the 211state is substantially longer than that of the X’Z+ state, and the diffuseness of electron distribution is not much different from that of the X1x+ state. Near R,, an extra electron in the 211state is trapped in the valence antibonding K* orbital. The calculated adiabatic energy difference between the ‘C+ and X1x+ states is 1.11 eV in the CI calculation and 0.25 eV in the ASCF calculation. The corresponding values for the 211and X’IZ+ states are 1.60 and 0.24 eV. Thus, the calculated electron affinity EA is negative. It is well known that the absolute value of EA is very difficult to evaluate. The ASCF method generally tends to overestimate the electron affinity; that is, the SCF energy of anion is too low, but the basis set deficiency might be more serious. Using the similar basis set, the calculated electron affinity of oxygen was 0.51 eV in the ASCF calculation, while the experimental is 1.465 eV [ 191.
151
Dipole momentfunction The electric dipole moment is calculated in three methods; (1) with the SCF MO for each state, (2) with CI wavefunctions, (3) with the finite-field SCF method. Since the SCF wavefunction obeys the Hellmann-Feynman theorem with respect to field strength [ 201, the results of the SCF and the finite-field SCF should be equal to each other in principle. In Fig. 2, the dipole moment functions on the internuclear distance are shown. The origin of moment for anions is the center of mass. The difference of charge distribution among three states X ’ 2 +, TI and ‘Z+ is remarkable. In the ground
‘I+ I
14
1.6 INTERNUCLEAR DISTANCE
1.8 R/ii
Fig. 2. Dipole moment functions, (a) calculated using the SCF MO method, (b) calculated with CI wavefunctions.
152
state XIZ+ of SiO, 0 is charged with 6- and Si is charged with S+ at any bond distance. The absolute value of moment function becomes larger with the stretching of the bond distance. This tendency is similar to that of CO near the equilibrium bond distance. In Table 1, the calculated dipole moments obtained by the SCF method and the finite-field SCF method are shown together. For the X1x+ state, they are equal to each other up to three digits. The values of the finite-field method for the other two states are in good agreement with those of the SCF. The good estimation of dipole moments for SiO and SiOwas attained with the applied finite field strength. For the neutral molecule, the feature of the result of the CI method is different from that of the SCF method. Though the moment function in the SCF result increases monotonously, the CI result shows a flat change and has a peak at R= 1.55 A. Thus, the correlation effect is substantial in the charge distribution of XlZ+. In Table 2, the bond length dependence of dipole moments for the three states is shown. The first derivative of the dipole moment p on the bond distance at R, shows a remarkable change in the X’C+ state. The experimental dipole moTABLE 1 Calculated dipole moments g (debye )
1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.80
SCF
Finite-field SCF
SCF
Finite-field SCF
SCF
Finite-field SCF
3.153 3.428 3.712 4.002 4.192 4.489 4.787 5.377
3.153 3.428 3.712 4.009 4.191 4.489 4.787 5.377
-2.585 - 2.212 - 1.631 - 0.630 0.075 0.739 1.305 2.281
- 2.603 -2.309 - 1.674 - 0.657 0.090 0.751 1.315 2.288
-
-
12.81 13.16 13.40 13.56 13.73 13.74 13.68 13.41
12.52 13.04 13.43 13.67 13.90 13.91 13.95 13.51
TABLE 2 Bond length dependence of dipole moment Parameter
Method
X’Z+
R, 6)
SCF CI SCF CI SCF CI
1.51 1.55 3.766 2.519 5.103 - 0.348
N,) (debye ) (a~laRo)R. (debye A-‘)
*rI 1.55 1.67 - 0.630 1.780 15.36 13.87
?z+ 1.515 1.55 - 13.47 - 13.43 - 3.958 - 3.431
153
ment ,Dis 3.098 debye [ 211, which is between the calculated values of the SCF (3.77 debye) and the CI (2.52 debye) methods. It was previously reported [ 221 that a dipole moment greater than 1.625 debye must have a stable anion within the Born-Oppenheimer approximation. Thus, the dipole moment of SiO is large enough for the neutral molecule to capture an electron in the electrostatic field. Werner et al. [6 ] studied the dipole moment function for the X’C+ state theoretically. In their SCF calculation, the dipole moment is 3.581 debye at I$= 1.48 A, and for the MCSCF CI calculation, that is 3.010 debye at R,= 1.52 A. The dipole moment p at R, tends to be smaller in the CI calculation than in the SCF calculation. The first derivative of the moment at R, is 3.344 debye A-‘. The dipole moment of the 211state changes its sign from minus to plus as R increases. This means that the center of charge moves from silicon to oxygen. In the SC! result, the center of mass coincides with the center of charge at about 1.6 A, but at 1.55 A in the CI result. The first derivative of the dipole moment ,u on the bond distance at R, has a small difference between the two methods. The sign of the moment of the ‘C+ state is always minus, and the absolute value is very large ( z 13 debye) when the bond length is around the equilibrium distance, both in the SCF and CI procedures. The difference between the positions of the center of charge and the center of mass is about 2.8 A in the region shown in Fig. 2. The center of charge is placed on the Si side and beyond the position of the Si atom. This phenomenon can be reduced to the large contribution of diffuse s-type functions for Si to 80 MO. In this state, an electron is trapped in the electrostatic field of SiO. The first derivative of the dipole momentp on the bond distance at R, has small difference between two methods as that of the ‘II state. Electric quadrupole moment The electric quadrupole moment function e,,(R) evaluated with the SCF wavefunction is shown in Fig. 3. The origin of the moment for the anion is the center of mass. The sign of the moment is always minus for each state at any bond length near R,. The absolute value of the moment increases with the bond length for each state. It is clear that the moment of ‘Z+ state is much larger than that of the other two states. At the shorter bond distance, the 211state has a quite small value. As R becomes longer, the difference between X’Z+ and ‘II gradually decreases. From these results, it can be concluded that the electron density distribution for the ‘C.+ state is very diffuse and the other two states have a more closely packed charge distribution. The diffuseness and the non-bonding character of the highest energy electron in the ‘C+ state seem to bring about this effect in the quadrupole moment. In Table 3, the bond length dependence of the quadrupole moment function is shown. The experimental
154 r
I
I
INTERNUCLEAR
DISTANCE
Rlli
Fig. 3. Quadrupole moment functions calculated using the SCF MO method. TABLE 3 Bond length dependence of quadrupole moment X’C+
Parameter
R, 6) 8,M.)
o (debyeA)
(W,IaR)R.
(debye)
1.51 -325.9 -390.8
ZrI 1.55 -287.5 -1231
%+ 1.515 -611.1 -790.1
values of the electric quadrupole moment for SiO molecule and SiO- anion are not reported. Frequency-dependent
polarizability
The frequency-dependent polarizabilities for SiO molecule and SiO- anion are evaluated by the frequency-dependent moment method [ 10,111. At o= 0, eqn. (1) gives the static polarizability. The bond length dependence of the static polarizabilities for three states is shown in Fig. 4 and in Table 4. The polarizability of each state has its characteristic feature. For the X1x+ state, the molecular axis component of polarizability cr,,(O) increases with the bond distance and the perpendicular component (Y,..(0 ) keeps almost constant near R,. Both components are very small compared with those for the other two states. For the ‘II state, ‘y_ (0) has a minimum at 1.55 A, and CY!,(0) shows a sudden decrease. At 1.4 A, (Y, (0) is about eight times as large as (Y,,(0). At about 1.64 A, the two curves cross each other and at longer bond distances, (Y,,(0) becomes larger than (Y, (0). For the ‘C+ state, the static polarizability
155
80! / QZZ
60
40
axx 20-
01
I
I
1.4
I
1+6 1.8 INTERNUCLEAR DISTANCE R/i
I
I
I
,
1.4 INTERNUCLEZ I
DISTANCE “??dt I
I I
8m: \ : I- \ aXX
2001
, 1.4
I 1.6 INTERNUCLEAR
I 1.8 DISTANCE R/A
Fig. 4. Static polarizability functions, (a) X1x+ state, (b) 3-I state, (c) ‘C+ state.
156 TABLE 4 Bond length dependence of static polarizahility Parameter
R, (A, %(R,) (au) o (aa,,/aR)R, (auA_‘) a&C) (au) o (aa,/aR)R, (auA_‘)
1.55 59.61 94.87 23.63 - 0.4886
1.67 182.3 418.6 117.9 -565.3
1.55 408.8 - 759.3 758.7 - 830.0
feature is very simple. Both cy,, (0) and cx,,(O) decrease gradually with the bond length. a!,, (0) is much larger than a,,(O), which is contrary to X’Z+. The order of magnitude of these two components for ‘Z+ is larger by one order than that for X1x+. It seems that the diffuseness and non-bonding character of the highest energy electron of the %+ state cause very large polarization. The static polarizabilities are also calculated with the finite-field SCF method in two ways, such as the energy derivative and the dipole moment derivative. Similarly to the dipole moment, correspondence between the results of the two methods is checked. These results are shown in Table 5 with those of the moment method. a, (0) values of the anions are not evaluated, because distinction is not possible as to whether the initial MO of open shell SCF is from the 21Jstate or from the 2Z+ state if the electric field is applied in the x direction. For the X’Cf state, the two types of finite-field SCF calculation result in good agreement with each other. Though the results for a,,(O) are about half of those of the moment method, the increasing behavior with the bond distance also appears in the finite-field results. The results of cr,(O) are equal to the moment method result up to two digits. For the %l state, the difference between two finite-field results of a,,(O) is very small. They decrease near the equilibrium point as the bond length becomes longer, while the moment method result has a minimum at R= 1.55 A. For the 2Z+ state, the correspondence between AE and Ap is not as good as the other two states, and those values are about ten times larger than the moment method value. As the polarizability is derived from the second-order perturbation about the field strength, numerical error tends to be more marked than in the dipole moment calculation. Thus the finite-field SCF calculation gives unreliable static polarizability values for the open shell SCF case particularly in the 2Z+ state. The frequency-dependent polarizability for SiO X1x+ at R,= 1.55 A is shown in Fig. 5. The calculated vertical excitation energy of 1% is 5.63 eV. Both components increase gradually with the photon energy until fiicc, = 0.16 au (4.35 eV). The oscillator strength is 0.023289 for the ‘II transition. Since the absorption band is weak in this transition, the resonance in the frequency-dependent polarizability is not significant. In Fig. 6, the frequency-dependence
157
TABLE 5 Calculated static polarizability (au)
R
(A)
1.40 1.45 1.50 1.55 1.60 1.65 1.70
X1X+
%
CI
Finite-field AE
46.40 50.39 54.80 59.61 64.70 70.17 75.85
Finite-field
CI
Finite-field AE
Finite-field
23.25 23.24 23.25 23.27 23.32 23.38 23.46
23.30 23.28 23.29 23.32 23.36 23.42 23.50
Finite-field AE
Finite-field
5894 5402 4876 4332 3790 3270 2772
5748 4812 3936 3160 2530 2042 1684
23.66 23.66 23.65 23.63 23.60 23.56 23.50
CI 193.0 177.3 159.2 151.3 158.7 175.3 198.7
Finite-field AE
Finite-field
316.9 280.3 220.6 167.9 123.3 93.77 79.19
274.3 234.5 196.7 161.8 109.8 85.40 80.78
CI
AP 554.0 498.4 450.0 408.8 374.4 346.0 321.4
60
0
AP
2c+
w
1.40 1.45 1.50 1.55 1.60 1.65 1.70
%-
AP 26.68 27.95 29.33 30.81 32.41 34.05 35.94
26.73 28.00 29.38 30.98 32.47 34.18 36.00
X'Z+
I
005 PHOTON
I
01 ENERGY
1
0.15 la.u.
Fig. 5. Frequency-dependent polarizability for SiO X’Z+ state at R,= 1.55 A.
AP
158
80 -
2 > I5
70-
z c! %
60-
$
50-
R=ld
40
I 0.05 PHOTON ENERGY
0
I 010 hvlau. I
I
axx
,/
-----R=l.f,~ ----z t. 2
Rz1.7A
26-
$ c! 5
0
I 005 PHOTON ENERGY
Fig. 6. Frequency-dependence
I 0.10 hvl a.u.
of polarizability tensors for X’C+ state, (a) (Y,, (b) (Y,.
of the polarizability tensor for the X’C+ state is shown at 1.40 A, 1.55 A and 1.70 A. It can be seen that the global feature of the polarizability tensor in the small photon energy region does not change with bond distance. In Table 6, the components of the polarizability-derivative tensor are shown. All the derivatives exhibit an increasing trend with the photon energy. The Raman intensity can be evaluated from these derivatives. For the 211and ‘Z+ states, the frequency-dependence of the polarizability tensor is shown in Figs. 7 and 8, and the components of the polarizabilityderivative tensor are shown in Tables 7 and 8. Because the energy-gap between ‘II and ‘ECf is very small, convergence of a,, is not good for each state, so reliable values of cy,, are obtained only for very low frequencies. (Y,, of the ‘II
159 TABLE 6 Polarizability derivatives for X’C+ hu (au)
(aa,,IaR)R,
0.00 0.06
(a%zIamR,
94.87 98.62
0.12
(au A-l)
- 0.4886 -0.4221
112.6
0.6493
I
I
I
240 -
R =1.6A 160 -
I 0
1
I
I
002
004
0.06
PHOTON
ENERGY
I
hvlau.
R=1.6A 0
0
I 0.01 PHOTON
I 0.02 ENERGY
I c-03 hvla.u.
I 0.04
Fig. 7. Frequency-dependence of polarizability tensors for ‘II state, (a) apz, (b) CY,.
160 1
1
!
I
az2
1500 -
500 -
0
2000
:
-
I 002 PHOTON
I 0.04 ENERGY
I
I
I 0.06 hvl au.
t
L 0.08
I
aXX
1500 -
ti iii !?I ti 1000 -
700 L 0
I 0.01 PHOTON
I 0.02 ENERGY
I 0.03 hv /a.u.
I 0.04
Fig. 8. Frequency-dependence of polarizability tensors for ‘C+ state, (a) CY,,,(b) (Y,.
TABLE 7 Polarizability derivatives for 211 (au A-‘) hv (au)
(~%Im~,
(aG/wR,
0.00 0.02 0.04
459.1 463.3 476.8
- 565.3 - 586.5 -657.8
161 TABLE 8 Polar&ability derivatives for %+ (au A-‘) hv (au)
(aff,IaR)R,
(aff,IaR)R,
0.00 0.02
- 759.3 -805.1
- 830.0 - 754.1
state shows a sharp increase at R= 1.4 A, and as the bond length becomes longer, the increase is more moderate. The change of the frequency-dependence of (Y, of 211with the bond distance is more drastic, if R is shorter than the crossing point. At R = 1.4 A, the difference between ay,. (0) and ay,, (0.04) is about 2100 au, while at R= 1.6 A, it is 31.3 au. (Y,, of ‘C+ shows behavior similar to that of 211.cy,, of 2Z+ shows a different behavior from that of 211.At R = 1.5 A, the increase is sharper than at R= 1.4 A, because of the small energy gap between two states. The experimental values of polarizabilities for SiO and SiO- have not been reported as far as we know. CONCLUSION
We have studied the dipole and quadrupole moments and the frequencydependent polarizabilities for SiO ground state and 211and ‘C+ states of SiO-. The present calculations of electric properties tell us that the charge distributions in X’C+, 211and 2E+ states are remarkably different from each other. Additionally, potential energy curves for X’Z+ and ‘II states of SiO reproduce the experimental results well. ACKNOWLEDGEMENT
Most of the calculations were carried out at the Computer Center of the Institute for Molecular Science.
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L.E. Snyder and D. Buhl, Astrophys. J., 197 (1975) 329. N. Kaifu, D. Buhl and L.E. Snyder, Astrophys. J., 195 (1975) 359. S. Deguchi and T. Iguchi, Pub]. Astron. Sot. Jpn., 28 (1976) 307. G. Winnewisser, in A.C.P. Alves et al. (Ed.), Frontiers af Laser Spectroscopy of Gases, Kluwer, Dordrecht, 1988, p. 491. C. Wood and I.H. Hillier, Mol. Phys., 37 (1979) 1329. H.-J. Werner, P. Rosmus and M. Grimm, Chem. Phys., 73 (1982) 169. S.R. Langhoff and J.O. Arnold, J. Chem. Phys., 70 (1979) 852. T.G. Heil and H.F. Schaefer III, J. Chem. Phys., 56 (1972) 958.
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J.M. Robbe, J. Schamps, H. Lefebvre-Brion and G. Raseev, J. Mol. Spectrosc., 74 (1979) 375. S. Iwata, Chem. Phys. Lett., 102 (1983) 544. T. Inoue, S. Matsushima and S. Iwata, Mol. Phys., 56 (1985) 1097. H. Tatewaki and S. Huzinaga, J. Comput. Chem., 1 (1980) 205. H.D. Cohen and C.C.J. Roothaan, J. Chem. Phys., 43 (1965) 534. A. Shweig, Chem. Phys. Lett., 1 (1967) 163. N.S. Hush and M.L. Williams, Chem. Phys. Lett., 5 (1970) 507. R.H. Tipping and C. Chackerian, Jr., J. Mol. Spectrosc., 88 (1981) 352. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, New York, 1950. A. Lagerqvist and U. Uhler, Ark. Fys., 6 (1953) 95. J. Irisawa and S. Iwata, Chem. Phys., to be published. R.E. Stanton, J. Chem. Phys., 36 (1962) 1298. J.W. Raymonda, J.S. Muenter and W.A. Klemperer, J. Chem. Phys., 52 (1970) 3458. K.D. Jordan, Act. Chem. Res., 12 (1979) 36.
147
AB INITIO CALCULATION OF THE DIPOLE MOMENT AND FREQUENCY-DEPENDENT POLARIZABILITY FOR SILICON MONOXIDE AND ITS ANION
TOSHIHIRO INOUE and SUEHIRO IWATA Department of Chemistry, Faculty of Science and Technology, Keio University, 3-14-l Hiyoshi, Kohoky Yokohama 223 (Japan) (Received 12 April 1990)
ABSTRACT The potential energy curve and the bond-length dependence of the electric properties for SiO and its anion are calculated using ab initio MO methods. The static and frequency-dependent polarizabilities are evaluated directly from the large CI Hamiltonian matrix without the explicit knowledge of the eigenvectors of the matrix. Two states of the anion SiO- (‘%+ and 2H) lie closely by the ground state X’Z+ of SiO. The equilibrium internuclear distance R, of the 2Z+ state is nearly equal to that of the X’Z+ SiO state, while the R, of the 211state is substantially longer than the latter. The electron affinity is estimated to be negative or nearly zero. The bond-length dependence of the dipole moment, the quadrupole moment and the polarizability tensor is expected to be remarkably different among three states X’I;+, *H and *Z’.
INTRODUCTION
Silicon monoxide, SiO, is one of a few well-known interstellar molecules with strong maser phenomena [l-4]. One of the possible excitation mechanisms of the maser action is the electron-molecule collision. To study the electronmolecule interaction, knowledge of the electric properties of ground state SiO and their bond length dependence is required. Besides, to examine the possibility of the resonance interaction, one has to know the relative energy of SiOanion to the neutral SiO as well as the properties of the anion. Although several ab initio studies of SiO molecule have been reported [5-g], studies for the anion are not found. In the present work, we study the potential energy, electric dipole moment, electric quadrupole moment and dipole polarizability curves for the X12+ state of SiO and for the lowest 211and ‘xc+ states of SiO-. The electron affinity of SiO is evaluated. The potential energies and dipole moments are calculated by using the restricted SCF and CI methods. Dipole polarizabilities are calculated with the frequency-dependent moment method [ 10,111, which enables us to calculate
148
the polarizability tensor directly from the large CI hamiltonian matrix without the explicit knowledge of the eigenvectors of the matrix. We can evaluate not only the static polarizability but also the frequency-dependent polarizability with this method. The frequency-dependent polarizability is related to the optical properties such as the refractive index, Raman and Rayleigh scattering intensities. The static polarizabilities are calculated also with the finite-field method. At the SCF level, the electric quadrupole moments are also estimated. METHOD
Our own ab initio program package MOLYX was used in the calculations. Most of the calculations have been carried out at the Computer Center of Institute for Molecular Science, and some on our local engineering work station, SONY NEWS 831. SCF calculations The MIDI4* basis set of Tatewaki and Huzinaga [ 121 augmented with two diffuse s and two diffusep on each atom (0 (x,’ = 0.060, cy,” = 0.020, aP’ = 0.060, or,“=O.O20; Si a,‘=0.027, a!,” =0.009, c~!,‘=O.O27, a!,,” =0.009) is used. The total number of CGTO is 50. After the closed shell SCF calculation, the molecular orbitals of the ground state X’C+ of SiO are used as the initial guess orbitals for the restricted open-shell SCF calculation of SiO-. The dipole and quadrupolemoments of SiO and SiO- are calculated by using these SCF wavefunctions. Molecular coordinates are defined as follows. The origin is placed on the center of mass of the SiO- ion and the molecular axis from 0 to Si is selected as the z axis. For example, if the bond length R is 1.5 A,0 is at (X=0.0,Y=O.O,Z= -0.956)andsiisat (x:=O.O,y=O.O,z=O.544). The finite field SCF method [ 13-151 is also used for calculation of dipole moment from the perturbed electronic energy. The electric field is applied in the direction of the molecular axis (z axis). Configuration interaction calculations The molecular orbitals are divided into four types to classify the configuration state functions (SCFs); (1)“frozen” core orbitals, which are always doubly occupied, ( 2 ) “core orbitals” {c}, (3 ) “active orbitals” (a}, and (4) “excited orbitals” {e}. In the case of the ground state of SiO molecule, CSFs are based on the closed shell SCF MOs. However, CSFs for 217and 2Z+ states of SiOare based on each open-shell SCF MO. The orbitals are classified as {c} = {lo, 20,3a, In, 4a} and {a}={5cr, 2x, 60,70,3n, 80). Consequently, CSFs are classified into 9 types; (0) all of the possible CSFs generated by filling active orbitals (CAS), (1) a2-e2, (2) a-e, (3) ca-e*, (4)
149
c-e, (5) c2-e2, (6) c-a, (7) c2-ae, (8) c2-a2. In the present study, the configuration space consists of the types (0) and (2). As a result, the total CSFs are 4535 for X1x+, 2734 for 211and 2856 for 2Z+. The thresholds for eigenvalue and for vectors are 1.0~ 10m6hartree and 3.0~ 10v4. The CI calculations are carried out with C2” symmetry. Then the dipole moment function is calculated for each state. Polurizability calculation The polarizability tensor is calculated from the CI hamiltonian matrix. The frequency-dependent polarizability CYBA (hu) is expressed as
where B and A are the dipole length operators, (2, Y, or X) and Ek and jk) are an eigenvalue and an eigenvector of the hamiltonian matrix. To sum up the contributions from all of the eigenstates of the hamiltonian matrix, it appears that knowledge of ik) and Ek for all states is required. However, the sum in eqn. (1) can be evaluated by solving a simultaneous linear equation, instead of diagonalizing the hamiltonian matrix. This method, named the frequencydependent moment method, was reported by Iwata and applied to H,O [lo] andtoN,andO, [ll]. RESULTS AND DISCUSSION
Potential energy curves for the ground state X’C+ of SiO and the two lowest doublet states of SiOThe potential energy curves near the equilibrium bond distance obtained by the ab initio Cl calculation are shown in Fig. 1. The curves of the ground state X1x+ and the lowest ‘II state of SiO, and the lowest 2H and 2Z+ states of SiOare all shown in the figure. The potential curve of SiO ground state has a minimum at the bond distance R= 1.55 6, while the experimental value of the equilibrium bond distance is R e = 1.51 A [ 161. Thus the calculated R, is slightly longer than the experimental one. At the closed-shell SCF level, R,= 1.51 A. Previously Werner et al. [ 61 reported the calculated R, value was 1.48 A in the restricted SCF and 1.52 A in the MCSCF CI. The lowest ‘II state has the (70+3x) character and its energy minimum is at R, = 1.70 A in the present work. The vertical and adiabatic excitation energies from the ground state are 5.63 eV and 5.33 eV, respectively. Experimentally [ 17,181, R, = 1.62 A and the adiabatic excitation energy T, is 5.29 eV. Previously, Heil and Schaefer [8] studied several excited states for SiO molecule. Their calculated values for the lowest ‘H state are R, = 1.75 A and T, = 4.79
&f:+ 1
1.6
1.4 INTERNUCLEAR
DISTANCE
1.6 _ R/W
Fig. 1. Potential energy curves for the ground state X’Z+ lowest Tl and ‘Z + states of SiO- .
and the lowest ‘II state of SiO, and the
eV. Robbe et al. [9] studied the SiO valence states by CI calculations, and reported that the T, for the first ‘II state was 5.07 eV. Our results for the lowest optically allowed state reproduces the experimental results reasonably well. Two states of the anion SiO- (‘Z+ and 211) lie closely by the ground state SiO. In the CI calculations two curves cross at R= 1.70 A; at shorter bond lengths, the ‘C+ state is the lowest state. This is also true in the SCF calculation. The equilibrium bond distance R, of ‘C.+ happens to be nearly equal to that of the X’Z+ SiO state; R, = 1.55 A in the CI calculation and R,= 1.51 A in the SCF calculation. As will be discussed in the following section, an extra electron in the ‘Z+ state is very diffuse and non-bonding. No change in the bond distance is consistent with this nature of the state. In contrast to the 2Z+ state, the R, of the 211state is substantially longer than that of the X’Z+ state, and the diffuseness of electron distribution is not much different from that of the X1x+ state. Near R,, an extra electron in the 211state is trapped in the valence antibonding K* orbital. The calculated adiabatic energy difference between the ‘C+ and X1x+ states is 1.11 eV in the CI calculation and 0.25 eV in the ASCF calculation. The corresponding values for the 211and X’IZ+ states are 1.60 and 0.24 eV. Thus, the calculated electron affinity EA is negative. It is well known that the absolute value of EA is very difficult to evaluate. The ASCF method generally tends to overestimate the electron affinity; that is, the SCF energy of anion is too low, but the basis set deficiency might be more serious. Using the similar basis set, the calculated electron affinity of oxygen was 0.51 eV in the ASCF calculation, while the experimental is 1.465 eV [ 191.
151
Dipole momentfunction The electric dipole moment is calculated in three methods; (1) with the SCF MO for each state, (2) with CI wavefunctions, (3) with the finite-field SCF method. Since the SCF wavefunction obeys the Hellmann-Feynman theorem with respect to field strength [ 201, the results of the SCF and the finite-field SCF should be equal to each other in principle. In Fig. 2, the dipole moment functions on the internuclear distance are shown. The origin of moment for anions is the center of mass. The difference of charge distribution among three states X ’ 2 +, TI and ‘Z+ is remarkable. In the ground
‘I+ I
14
1.6 INTERNUCLEAR DISTANCE
1.8 R/ii
Fig. 2. Dipole moment functions, (a) calculated using the SCF MO method, (b) calculated with CI wavefunctions.
152
state XIZ+ of SiO, 0 is charged with 6- and Si is charged with S+ at any bond distance. The absolute value of moment function becomes larger with the stretching of the bond distance. This tendency is similar to that of CO near the equilibrium bond distance. In Table 1, the calculated dipole moments obtained by the SCF method and the finite-field SCF method are shown together. For the X1x+ state, they are equal to each other up to three digits. The values of the finite-field method for the other two states are in good agreement with those of the SCF. The good estimation of dipole moments for SiO and SiOwas attained with the applied finite field strength. For the neutral molecule, the feature of the result of the CI method is different from that of the SCF method. Though the moment function in the SCF result increases monotonously, the CI result shows a flat change and has a peak at R= 1.55 A. Thus, the correlation effect is substantial in the charge distribution of XlZ+. In Table 2, the bond length dependence of dipole moments for the three states is shown. The first derivative of the dipole moment p on the bond distance at R, shows a remarkable change in the X’C+ state. The experimental dipole moTABLE 1 Calculated dipole moments g (debye )
1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.80
SCF
Finite-field SCF
SCF
Finite-field SCF
SCF
Finite-field SCF
3.153 3.428 3.712 4.002 4.192 4.489 4.787 5.377
3.153 3.428 3.712 4.009 4.191 4.489 4.787 5.377
-2.585 - 2.212 - 1.631 - 0.630 0.075 0.739 1.305 2.281
- 2.603 -2.309 - 1.674 - 0.657 0.090 0.751 1.315 2.288
-
-
12.81 13.16 13.40 13.56 13.73 13.74 13.68 13.41
12.52 13.04 13.43 13.67 13.90 13.91 13.95 13.51
TABLE 2 Bond length dependence of dipole moment Parameter
Method
X’Z+
R, 6)
SCF CI SCF CI SCF CI
1.51 1.55 3.766 2.519 5.103 - 0.348
N,) (debye ) (a~laRo)R. (debye A-‘)
*rI 1.55 1.67 - 0.630 1.780 15.36 13.87
?z+ 1.515 1.55 - 13.47 - 13.43 - 3.958 - 3.431
153
ment ,Dis 3.098 debye [ 211, which is between the calculated values of the SCF (3.77 debye) and the CI (2.52 debye) methods. It was previously reported [ 221 that a dipole moment greater than 1.625 debye must have a stable anion within the Born-Oppenheimer approximation. Thus, the dipole moment of SiO is large enough for the neutral molecule to capture an electron in the electrostatic field. Werner et al. [6 ] studied the dipole moment function for the X’C+ state theoretically. In their SCF calculation, the dipole moment is 3.581 debye at I$= 1.48 A, and for the MCSCF CI calculation, that is 3.010 debye at R,= 1.52 A. The dipole moment p at R, tends to be smaller in the CI calculation than in the SCF calculation. The first derivative of the moment at R, is 3.344 debye A-‘. The dipole moment of the 211state changes its sign from minus to plus as R increases. This means that the center of charge moves from silicon to oxygen. In the SC! result, the center of mass coincides with the center of charge at about 1.6 A, but at 1.55 A in the CI result. The first derivative of the dipole moment ,u on the bond distance at R, has a small difference between the two methods. The sign of the moment of the ‘C+ state is always minus, and the absolute value is very large ( z 13 debye) when the bond length is around the equilibrium distance, both in the SCF and CI procedures. The difference between the positions of the center of charge and the center of mass is about 2.8 A in the region shown in Fig. 2. The center of charge is placed on the Si side and beyond the position of the Si atom. This phenomenon can be reduced to the large contribution of diffuse s-type functions for Si to 80 MO. In this state, an electron is trapped in the electrostatic field of SiO. The first derivative of the dipole momentp on the bond distance at R, has small difference between two methods as that of the ‘II state. Electric quadrupole moment The electric quadrupole moment function e,,(R) evaluated with the SCF wavefunction is shown in Fig. 3. The origin of the moment for the anion is the center of mass. The sign of the moment is always minus for each state at any bond length near R,. The absolute value of the moment increases with the bond length for each state. It is clear that the moment of ‘Z+ state is much larger than that of the other two states. At the shorter bond distance, the 211state has a quite small value. As R becomes longer, the difference between X’Z+ and ‘II gradually decreases. From these results, it can be concluded that the electron density distribution for the ‘C.+ state is very diffuse and the other two states have a more closely packed charge distribution. The diffuseness and the non-bonding character of the highest energy electron in the ‘C+ state seem to bring about this effect in the quadrupole moment. In Table 3, the bond length dependence of the quadrupole moment function is shown. The experimental
154 r
I
I
INTERNUCLEAR
DISTANCE
Rlli
Fig. 3. Quadrupole moment functions calculated using the SCF MO method. TABLE 3 Bond length dependence of quadrupole moment X’C+
Parameter
R, 6) 8,M.)
o (debyeA)
(W,IaR)R.
(debye)
1.51 -325.9 -390.8
ZrI 1.55 -287.5 -1231
%+ 1.515 -611.1 -790.1
values of the electric quadrupole moment for SiO molecule and SiO- anion are not reported. Frequency-dependent
polarizability
The frequency-dependent polarizabilities for SiO molecule and SiO- anion are evaluated by the frequency-dependent moment method [ 10,111. At o= 0, eqn. (1) gives the static polarizability. The bond length dependence of the static polarizabilities for three states is shown in Fig. 4 and in Table 4. The polarizability of each state has its characteristic feature. For the X1x+ state, the molecular axis component of polarizability cr,,(O) increases with the bond distance and the perpendicular component (Y,..(0 ) keeps almost constant near R,. Both components are very small compared with those for the other two states. For the ‘II state, ‘y_ (0) has a minimum at 1.55 A, and CY!,(0) shows a sudden decrease. At 1.4 A, (Y, (0) is about eight times as large as (Y,,(0). At about 1.64 A, the two curves cross each other and at longer bond distances, (Y,,(0) becomes larger than (Y, (0). For the ‘C+ state, the static polarizability
155
80! / QZZ
60
40
axx 20-
01
I
I
1.4
I
1+6 1.8 INTERNUCLEAR DISTANCE R/i
I
I
I
,
1.4 INTERNUCLEZ I
DISTANCE “??dt I
I I
8m: \ : I- \ aXX
2001
, 1.4
I 1.6 INTERNUCLEAR
I 1.8 DISTANCE R/A
Fig. 4. Static polarizability functions, (a) X1x+ state, (b) 3-I state, (c) ‘C+ state.
156 TABLE 4 Bond length dependence of static polarizahility Parameter
R, (A, %(R,) (au) o (aa,,/aR)R, (auA_‘) a&C) (au) o (aa,/aR)R, (auA_‘)
1.55 59.61 94.87 23.63 - 0.4886
1.67 182.3 418.6 117.9 -565.3
1.55 408.8 - 759.3 758.7 - 830.0
feature is very simple. Both cy,, (0) and cx,,(O) decrease gradually with the bond length. a!,, (0) is much larger than a,,(O), which is contrary to X’Z+. The order of magnitude of these two components for ‘Z+ is larger by one order than that for X1x+. It seems that the diffuseness and non-bonding character of the highest energy electron of the %+ state cause very large polarization. The static polarizabilities are also calculated with the finite-field SCF method in two ways, such as the energy derivative and the dipole moment derivative. Similarly to the dipole moment, correspondence between the results of the two methods is checked. These results are shown in Table 5 with those of the moment method. a, (0) values of the anions are not evaluated, because distinction is not possible as to whether the initial MO of open shell SCF is from the 21Jstate or from the 2Z+ state if the electric field is applied in the x direction. For the X’Cf state, the two types of finite-field SCF calculation result in good agreement with each other. Though the results for a,,(O) are about half of those of the moment method, the increasing behavior with the bond distance also appears in the finite-field results. The results of cr,(O) are equal to the moment method result up to two digits. For the %l state, the difference between two finite-field results of a,,(O) is very small. They decrease near the equilibrium point as the bond length becomes longer, while the moment method result has a minimum at R= 1.55 A. For the 2Z+ state, the correspondence between AE and Ap is not as good as the other two states, and those values are about ten times larger than the moment method value. As the polarizability is derived from the second-order perturbation about the field strength, numerical error tends to be more marked than in the dipole moment calculation. Thus the finite-field SCF calculation gives unreliable static polarizability values for the open shell SCF case particularly in the 2Z+ state. The frequency-dependent polarizability for SiO X1x+ at R,= 1.55 A is shown in Fig. 5. The calculated vertical excitation energy of 1% is 5.63 eV. Both components increase gradually with the photon energy until fiicc, = 0.16 au (4.35 eV). The oscillator strength is 0.023289 for the ‘II transition. Since the absorption band is weak in this transition, the resonance in the frequency-dependent polarizability is not significant. In Fig. 6, the frequency-dependence
157
TABLE 5 Calculated static polarizability (au)
R
(A)
1.40 1.45 1.50 1.55 1.60 1.65 1.70
X1X+
%
CI
Finite-field AE
46.40 50.39 54.80 59.61 64.70 70.17 75.85
Finite-field
CI
Finite-field AE
Finite-field
23.25 23.24 23.25 23.27 23.32 23.38 23.46
23.30 23.28 23.29 23.32 23.36 23.42 23.50
Finite-field AE
Finite-field
5894 5402 4876 4332 3790 3270 2772
5748 4812 3936 3160 2530 2042 1684
23.66 23.66 23.65 23.63 23.60 23.56 23.50
CI 193.0 177.3 159.2 151.3 158.7 175.3 198.7
Finite-field AE
Finite-field
316.9 280.3 220.6 167.9 123.3 93.77 79.19
274.3 234.5 196.7 161.8 109.8 85.40 80.78
CI
AP 554.0 498.4 450.0 408.8 374.4 346.0 321.4
60
0
AP
2c+
w
1.40 1.45 1.50 1.55 1.60 1.65 1.70
%-
AP 26.68 27.95 29.33 30.81 32.41 34.05 35.94
26.73 28.00 29.38 30.98 32.47 34.18 36.00
X'Z+
I
005 PHOTON
I
01 ENERGY
1
0.15 la.u.
Fig. 5. Frequency-dependent polarizability for SiO X’Z+ state at R,= 1.55 A.
AP
158
80 -
2 > I5
70-
z c! %
60-
$
50-
R=ld
40
I 0.05 PHOTON ENERGY
0
I 010 hvlau. I
I
axx
,/
-----R=l.f,~ ----z t. 2
Rz1.7A
26-
$ c! 5
0
I 005 PHOTON ENERGY
Fig. 6. Frequency-dependence
I 0.10 hvl a.u.
of polarizability tensors for X’C+ state, (a) (Y,, (b) (Y,.
of the polarizability tensor for the X’C+ state is shown at 1.40 A, 1.55 A and 1.70 A. It can be seen that the global feature of the polarizability tensor in the small photon energy region does not change with bond distance. In Table 6, the components of the polarizability-derivative tensor are shown. All the derivatives exhibit an increasing trend with the photon energy. The Raman intensity can be evaluated from these derivatives. For the 211and ‘Z+ states, the frequency-dependence of the polarizability tensor is shown in Figs. 7 and 8, and the components of the polarizabilityderivative tensor are shown in Tables 7 and 8. Because the energy-gap between ‘II and ‘ECf is very small, convergence of a,, is not good for each state, so reliable values of cy,, are obtained only for very low frequencies. (Y,, of the ‘II
159 TABLE 6 Polarizability derivatives for X’C+ hu (au)
(aa,,IaR)R,
0.00 0.06
(a%zIamR,
94.87 98.62
0.12
(au A-l)
- 0.4886 -0.4221
112.6
0.6493
I
I
I
240 -
R =1.6A 160 -
I 0
1
I
I
002
004
0.06
PHOTON
ENERGY
I
hvlau.
R=1.6A 0
0
I 0.01 PHOTON
I 0.02 ENERGY
I c-03 hvla.u.
I 0.04
Fig. 7. Frequency-dependence of polarizability tensors for ‘II state, (a) apz, (b) CY,.
160 1
1
!
I
az2
1500 -
500 -
0
2000
:
-
I 002 PHOTON
I 0.04 ENERGY
I
I
I 0.06 hvl au.
t
L 0.08
I
aXX
1500 -
ti iii !?I ti 1000 -
700 L 0
I 0.01 PHOTON
I 0.02 ENERGY
I 0.03 hv /a.u.
I 0.04
Fig. 8. Frequency-dependence of polarizability tensors for ‘C+ state, (a) CY,,,(b) (Y,.
TABLE 7 Polarizability derivatives for 211 (au A-‘) hv (au)
(~%Im~,
(aG/wR,
0.00 0.02 0.04
459.1 463.3 476.8
- 565.3 - 586.5 -657.8
161 TABLE 8 Polar&ability derivatives for %+ (au A-‘) hv (au)
(aff,IaR)R,
(aff,IaR)R,
0.00 0.02
- 759.3 -805.1
- 830.0 - 754.1
state shows a sharp increase at R= 1.4 A, and as the bond length becomes longer, the increase is more moderate. The change of the frequency-dependence of (Y, of 211with the bond distance is more drastic, if R is shorter than the crossing point. At R = 1.4 A, the difference between ay,. (0) and ay,, (0.04) is about 2100 au, while at R= 1.6 A, it is 31.3 au. (Y,, of ‘C+ shows behavior similar to that of 211.cy,, of 2Z+ shows a different behavior from that of 211.At R = 1.5 A, the increase is sharper than at R= 1.4 A, because of the small energy gap between two states. The experimental values of polarizabilities for SiO and SiO- have not been reported as far as we know. CONCLUSION
We have studied the dipole and quadrupole moments and the frequencydependent polarizabilities for SiO ground state and 211and ‘C+ states of SiO-. The present calculations of electric properties tell us that the charge distributions in X’C+, 211and 2E+ states are remarkably different from each other. Additionally, potential energy curves for X’Z+ and ‘II states of SiO reproduce the experimental results well. ACKNOWLEDGEMENT
Most of the calculations were carried out at the Computer Center of the Institute for Molecular Science.
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