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Floating spherical gaussian orbital (FSGO) studies with a model potential: some open-shell systems
FLOATING
SPHERICAL GAUSSIAN
SOME OPEN-SHELL S.P. MEHANDRU Department
15 February 1978
CHEMICAL PHYSICS LETTERS
Volume 54, number 1
ORBITAL (FSGO) STUDIES WITH A MODEL POTENTIAL:
SYSTEMS
and N.K. RAY
of Chemistry.
Univwsity of Delhi, Delhi-210007.
India
Received 14 September 1977
A gaussian based mcdcl potential is used within the FSGO formalism to study a series of open-shell systems (e.g. LiH+, Nail+, Lit, Naf and LiNs+). Results for calculated equrhbrium geometries and dissociation energies arc compared to the corresponding_.quantities from available all-electron ab initio studies and other more elaborate theoretical estimates. The _ overall agrcemcnt is quite satisfactory.
The electrons of a molecule can be broadly divided into two parts: the “core” electrons associated with the inner shells of atoms and ihe “valence” electrons in_ the valence orbit& of the molecule. Such a partitioning allows valence orbitsls to be treated by comparatively rigorous ab initio techniques with a minimal amount of labour. Methods dealing with only valence electrons are based on the pseudopotential theory [l-4]. The exact pseudopotentials can bc replaced with little loss of accuracy by simple operarors [S]. The valence electron calculations using these operators are referred to as model potential calculations. Valence electron studies with model potentials have drawn considerable attention in recent years [6-321. Earlier Ray and co-workers 122-271 have quite successfully used a gaussian based model potential within the floating spherical gaussian orbital (FSGO) formalism [33] to study a series of closed-shell systems. In the present work we have used the same model potential to study the equilibrium geometries and dissociation energies of a series of open-shel! systems. The core model potential used here has the form Pm = -ZJr+A,
exp(--yf
2)/r,
(0
where 2, is the nuclear charge minus the number of core electrons and A, is the model potential parameter for the s-state of the atom under consideration. Core model potential parameters needed for the present study have been taken from the work of Ray and 42
Switalski [23] and are listed in table 1. Only valence electrons are taken into consideration and the effect of the core is simulated through the use of the model potential given above. Since all the open-shell systems studied here (LiH+, NaH+, Liz, Na+2 and LiNa+) are one-valence-electron systems, only one single floating spherical gaussian orbital (p) cp= (2/7r~~)~/~
exp[-(r
- R)2/p2]
(2)
is used for each system to describe the molecular wavefunction. The valence energy (,5,_J can be expressed as a function of p, R and the nuclear coordinates. The best values for these variable parameters are obtained by minimizing the valence energy. The optimization is carried out by the subroutine STEPIT *. Table 2 summarizes the calculated valence energies, dissociation energies (De) and equilibrium geom* ObtGred from QCPE, Chemistry Department, Indiana University, Bloomington, Indiana 47401.
Table 1 Model potential parameters for the atoms a) Atom
As
Y
Li N3
1.1113 0.9915
0.3610 0.2721
a) Taken from ref. [ 231.
Table 2 Calculated equilibrium geometries (Re) and dissociation energies (De) System
Eval(au)
15 February 1978
CHEMICAL PHYSICS LETTERS
Volume 54. number 1
Re(au)
De(eV)
this work
earlier work a)
this work
earlier work a)
LiH+ Nati+
-0.426 1 -0.4255
5.10 5.84
4.5 5.4
0.05 0.03
0.09 0.03
Li+ N$
-0.2277 -0.2168
6.34 6.70
5.8 6.7 (==6.8)
0.86 0.79
1.22 0.97 (0.99)
LiNa+
-0.2224
6.52
6.5
0.72
0.92
a) Results for Litl+, NatI+ and Nag are taken from ref. [ 341 and those for Liz and LiNa+ arc taken from alMec:ron ab mitio studies [ 35,361. Numbers given in parentheses are obtained from the experimental work of Roach and Baybutt [37].
etries for all one-valence-electron diatomics studied here. Results of more elaborate studies using SCF LCAO MO methods are also shown in table 2 for the sake of comparison. The results of our present study are in good agreement with other theoretical estimates and available experimental data [34-371. The agreernent of our calculated geometry and dissociation energy is comparatively less satisfactory for Lit-i+. The error is partially due to the neglect of the p-portion of the model potential. The use of only s-type potential for these systems is not strictly justified and for some systems, such as LiW, it may be a poor approximation. Since the model potential in general has angular momentum dcpcndence [9,11,28,30,38,39], the valence orbital should be decomposed into its various angular components and proper treatment of angular momentum dependence should be taken into account. Our present approximation for the small atoms but becomes
is most severe
less important as we go to larger systems (e.g. LiNa+ and Nas) where a model potential method would be most useful. This work shows that the simple gaussian based model potential can be used quite successfully within the FSCO formalism to study a series of open-shell one-electron systems. Work is in progress in our laboratory to study other open-shell systems like trlatomic alkali metal clusters and other polyatomic radicals. Explicit introduction of angular momentum dependence in these calculations is also being studied. Re-
sults of these studies
will be reported
elsewhere.
One of the authors (SPM) thanks UGC, New Delhi for the grant of a Teacher-Fellowship. Thanks are also due to the staff of the Computer Centre, University of Delhi, for their valuable cooperation.
References
[ 11 P. Gombas, Z. rhysik 94 (1935) 473. [ 21 H. Hellmm, J. Chcm. Phyr 3 (1935) 61. [ 31 H. Preuss, Z. Naturforxh. 10e (1955) 365. [4] L. Szasz and G. McGinn, J. Chcm. Phys. 48 (1968) 2997, am! references therein. 151 J.D. Weeks, A. Iiazi and S.A. Rice, Advan. Chem. Phys. 16 (1969)
;83.
I61 G. Simons, Chem. Phyr Letters 18 (I 973) 3 15. 171 C. Simons, J. Cbern. Phys. 55 (197 1) 756. 181 J.C. Barthelat and Ph. Durand, Chcm. Phys. Letters I6 (1372) 63. 191 J.C. Barth&t and Ph. Dumnd, J. Chcm. Phys. 71 (1974) 505, 1105. 1101 hl. Kleincr and R. hIcWeeny, Chcm. Phys. Letters 19 (1973) 47?. IllI C.F. Melivs, B.D. OIafson and W.A. Goddard 111,Chem. Phys. Lettxs 28 (1974) 457. [I21 1I.A. Pohl and D.R. Fowler, Intern. J. Quantum Chcm. 8 (1974) 435. 1131 C.T. Mehiis and W.A. Goddard III, Phys. Rev. 10 (1974) 1528. (141 R. Gaspar and 1. Tomaqsy-Len?ei, Acta Phys. Acad. Sci. IIung. 33 (1973) 387. I151 J.D. Switalski, J.T.H. IIuang and hI.L. Schwartz, J. Chcm. Ph) s. 60 (1974) 2252. 1161 J.D. Switalski and ME. Schwartz, J. Cbcm. Phys. 62 (1975) 1521. 1171 h1.C. Zerner, hlol. Phrs. 23 (1972) 963. [W hf. Horn and J.N. hfurrell, J. Chem. Sot. Faraday II 70 (1974) 769. [W J.N. Murrell and LG. Vincent, J. Chem. SOL’.I araday II 71 (1975) 890. t2o1 A.M. Semkow, R.A. Suthers and J.W. I.innett, Chem. Phys. Letters 32 (1975) 116. I211 hI.E. Schwartz and J.D. Switalski, I. Chem. Phys. 57 (1972) 4125. 1221 N.K. Ray and J.D. Switalski, J. Chem. Phys 63 (1975) 5053. [231 N.K. Ray and J.D. Switalski. Theoret. Chim. Acta 41 (1976)
329.
[241 N.K. Ray, S.P. hlehandru and J.D. Switalski,
Chem.
Phys. Letters 47 (1977) 563. [251 S.P. Mehandru and N.K. Ray, Chem. Phys. Lcttcrs 50 (1977) 497. WI S.P. Mehandru and N.K. Ray, Indian J. Chem. 15A (1977) 145.
43
Volume
54. number I
CHEMICAL
PHYSICS
1271 S.P. Mehandru and N.K. Ray, Indiin J. Chem., to be published. [28] J.C. Barthelat and Ph. Durand. Chem. Phys. Letters 40 (1976) 407. [29] S. Topiol, A.A. Frost, M.A. Ratner and J.W. Moskowitz, J. Chem. Phys 65 (1976) 4467. 1301 S. Topiol, A.A. Frost, J.W. Mos’cowitz and M.A. Ratner, J. Am. Chem. Sot. 99 (1977) 4276. [31] V. Bonifacic and S. Huzinaga, J. Chem. Phys 60 (1974) 2779. 1321 P. Coffey, C.S. Xwig and J.R. van Wazer, J. Am. Chem. Sot. 97 (1975) 1656.
44
LE’ITERS
15 February
1978
[33] A.A. Frost, J. Chem. Phys 47 (1967) 3707. 1341 M.E. Schwartz and J.D. Switalski, J. Chem. Phys 57 (1972) 4132. [35] J.N. Barddey, Phys Rev. A3 (1971) 1317. [36] P.J. Bertoncini, G. Das and A.C. Wahl, J. Chcm. Phys 52 (1970) 5112. 1371 A.C. Roach and P. Baybutt, Chcm. Phys Letters 7 (1970) 7. [38] L.R. Kahn and W.A. Goddard III, J. Chem. Phys 56 (1972) 2685. 1391 L.R. Kahn, P. Baybutt and D.G. Truhlar, J. Chcm. Phys 65 (1976) 3826.
SPHERICAL GAUSSIAN
SOME OPEN-SHELL S.P. MEHANDRU Department
15 February 1978
CHEMICAL PHYSICS LETTERS
Volume 54, number 1
ORBITAL (FSGO) STUDIES WITH A MODEL POTENTIAL:
SYSTEMS
and N.K. RAY
of Chemistry.
Univwsity of Delhi, Delhi-210007.
India
Received 14 September 1977
A gaussian based mcdcl potential is used within the FSGO formalism to study a series of open-shell systems (e.g. LiH+, Nail+, Lit, Naf and LiNs+). Results for calculated equrhbrium geometries and dissociation energies arc compared to the corresponding_.quantities from available all-electron ab initio studies and other more elaborate theoretical estimates. The _ overall agrcemcnt is quite satisfactory.
The electrons of a molecule can be broadly divided into two parts: the “core” electrons associated with the inner shells of atoms and ihe “valence” electrons in_ the valence orbit& of the molecule. Such a partitioning allows valence orbitsls to be treated by comparatively rigorous ab initio techniques with a minimal amount of labour. Methods dealing with only valence electrons are based on the pseudopotential theory [l-4]. The exact pseudopotentials can bc replaced with little loss of accuracy by simple operarors [S]. The valence electron calculations using these operators are referred to as model potential calculations. Valence electron studies with model potentials have drawn considerable attention in recent years [6-321. Earlier Ray and co-workers 122-271 have quite successfully used a gaussian based model potential within the floating spherical gaussian orbital (FSGO) formalism [33] to study a series of closed-shell systems. In the present work we have used the same model potential to study the equilibrium geometries and dissociation energies of a series of open-shel! systems. The core model potential used here has the form Pm = -ZJr+A,
exp(--yf
2)/r,
(0
where 2, is the nuclear charge minus the number of core electrons and A, is the model potential parameter for the s-state of the atom under consideration. Core model potential parameters needed for the present study have been taken from the work of Ray and 42
Switalski [23] and are listed in table 1. Only valence electrons are taken into consideration and the effect of the core is simulated through the use of the model potential given above. Since all the open-shell systems studied here (LiH+, NaH+, Liz, Na+2 and LiNa+) are one-valence-electron systems, only one single floating spherical gaussian orbital (p) cp= (2/7r~~)~/~
exp[-(r
- R)2/p2]
(2)
is used for each system to describe the molecular wavefunction. The valence energy (,5,_J can be expressed as a function of p, R and the nuclear coordinates. The best values for these variable parameters are obtained by minimizing the valence energy. The optimization is carried out by the subroutine STEPIT *. Table 2 summarizes the calculated valence energies, dissociation energies (De) and equilibrium geom* ObtGred from QCPE, Chemistry Department, Indiana University, Bloomington, Indiana 47401.
Table 1 Model potential parameters for the atoms a) Atom
As
Y
Li N3
1.1113 0.9915
0.3610 0.2721
a) Taken from ref. [ 231.
Table 2 Calculated equilibrium geometries (Re) and dissociation energies (De) System
Eval(au)
15 February 1978
CHEMICAL PHYSICS LETTERS
Volume 54. number 1
Re(au)
De(eV)
this work
earlier work a)
this work
earlier work a)
LiH+ Nati+
-0.426 1 -0.4255
5.10 5.84
4.5 5.4
0.05 0.03
0.09 0.03
Li+ N$
-0.2277 -0.2168
6.34 6.70
5.8 6.7 (==6.8)
0.86 0.79
1.22 0.97 (0.99)
LiNa+
-0.2224
6.52
6.5
0.72
0.92
a) Results for Litl+, NatI+ and Nag are taken from ref. [ 341 and those for Liz and LiNa+ arc taken from alMec:ron ab mitio studies [ 35,361. Numbers given in parentheses are obtained from the experimental work of Roach and Baybutt [37].
etries for all one-valence-electron diatomics studied here. Results of more elaborate studies using SCF LCAO MO methods are also shown in table 2 for the sake of comparison. The results of our present study are in good agreement with other theoretical estimates and available experimental data [34-371. The agreernent of our calculated geometry and dissociation energy is comparatively less satisfactory for Lit-i+. The error is partially due to the neglect of the p-portion of the model potential. The use of only s-type potential for these systems is not strictly justified and for some systems, such as LiW, it may be a poor approximation. Since the model potential in general has angular momentum dcpcndence [9,11,28,30,38,39], the valence orbital should be decomposed into its various angular components and proper treatment of angular momentum dependence should be taken into account. Our present approximation for the small atoms but becomes
is most severe
less important as we go to larger systems (e.g. LiNa+ and Nas) where a model potential method would be most useful. This work shows that the simple gaussian based model potential can be used quite successfully within the FSCO formalism to study a series of open-shell one-electron systems. Work is in progress in our laboratory to study other open-shell systems like trlatomic alkali metal clusters and other polyatomic radicals. Explicit introduction of angular momentum dependence in these calculations is also being studied. Re-
sults of these studies
will be reported
elsewhere.
One of the authors (SPM) thanks UGC, New Delhi for the grant of a Teacher-Fellowship. Thanks are also due to the staff of the Computer Centre, University of Delhi, for their valuable cooperation.
References
[ 11 P. Gombas, Z. rhysik 94 (1935) 473. [ 21 H. Hellmm, J. Chcm. Phyr 3 (1935) 61. [ 31 H. Preuss, Z. Naturforxh. 10e (1955) 365. [4] L. Szasz and G. McGinn, J. Chcm. Phys. 48 (1968) 2997, am! references therein. 151 J.D. Weeks, A. Iiazi and S.A. Rice, Advan. Chem. Phys. 16 (1969)
;83.
I61 G. Simons, Chem. Phyr Letters 18 (I 973) 3 15. 171 C. Simons, J. Cbern. Phys. 55 (197 1) 756. 181 J.C. Barthelat and Ph. Durand, Chcm. Phys. Letters I6 (1372) 63. 191 J.C. Barth&t and Ph. Dumnd, J. Chcm. Phys. 71 (1974) 505, 1105. 1101 hl. Kleincr and R. hIcWeeny, Chcm. Phys. Letters 19 (1973) 47?. IllI C.F. Melivs, B.D. OIafson and W.A. Goddard 111,Chem. Phys. Lettxs 28 (1974) 457. [I21 1I.A. Pohl and D.R. Fowler, Intern. J. Quantum Chcm. 8 (1974) 435. 1131 C.T. Mehiis and W.A. Goddard III, Phys. Rev. 10 (1974) 1528. (141 R. Gaspar and 1. Tomaqsy-Len?ei, Acta Phys. Acad. Sci. IIung. 33 (1973) 387. I151 J.D. Switalski, J.T.H. IIuang and hI.L. Schwartz, J. Chcm. Ph) s. 60 (1974) 2252. 1161 J.D. Switalski and ME. Schwartz, J. Cbcm. Phys. 62 (1975) 1521. 1171 h1.C. Zerner, hlol. Phrs. 23 (1972) 963. [W hf. Horn and J.N. hfurrell, J. Chem. Sot. Faraday II 70 (1974) 769. [W J.N. Murrell and LG. Vincent, J. Chem. SOL’.I araday II 71 (1975) 890. t2o1 A.M. Semkow, R.A. Suthers and J.W. I.innett, Chem. Phys. Letters 32 (1975) 116. I211 hI.E. Schwartz and J.D. Switalski, I. Chem. Phys. 57 (1972) 4125. 1221 N.K. Ray and J.D. Switalski, J. Chem. Phys 63 (1975) 5053. [231 N.K. Ray and J.D. Switalski. Theoret. Chim. Acta 41 (1976)
329.
[241 N.K. Ray, S.P. hlehandru and J.D. Switalski,
Chem.
Phys. Letters 47 (1977) 563. [251 S.P. Mehandru and N.K. Ray, Chem. Phys. Lcttcrs 50 (1977) 497. WI S.P. Mehandru and N.K. Ray, Indian J. Chem. 15A (1977) 145.
43
Volume
54. number I
CHEMICAL
PHYSICS
1271 S.P. Mehandru and N.K. Ray, Indiin J. Chem., to be published. [28] J.C. Barthelat and Ph. Durand. Chem. Phys. Letters 40 (1976) 407. [29] S. Topiol, A.A. Frost, M.A. Ratner and J.W. Moskowitz, J. Chem. Phys 65 (1976) 4467. 1301 S. Topiol, A.A. Frost, J.W. Mos’cowitz and M.A. Ratner, J. Am. Chem. Sot. 99 (1977) 4276. [31] V. Bonifacic and S. Huzinaga, J. Chem. Phys 60 (1974) 2779. 1321 P. Coffey, C.S. Xwig and J.R. van Wazer, J. Am. Chem. Sot. 97 (1975) 1656.
44
LE’ITERS
15 February
1978
[33] A.A. Frost, J. Chem. Phys 47 (1967) 3707. 1341 M.E. Schwartz and J.D. Switalski, J. Chem. Phys 57 (1972) 4132. [35] J.N. Barddey, Phys Rev. A3 (1971) 1317. [36] P.J. Bertoncini, G. Das and A.C. Wahl, J. Chcm. Phys 52 (1970) 5112. 1371 A.C. Roach and P. Baybutt, Chcm. Phys Letters 7 (1970) 7. [38] L.R. Kahn and W.A. Goddard III, J. Chem. Phys 56 (1972) 2685. 1391 L.R. Kahn, P. Baybutt and D.G. Truhlar, J. Chcm. Phys 65 (1976) 3826.