- Email: [email protected]
Application of the generalized potential energy function for solving the inverse spectroscopic problem: the ground-state potential of SiF+
Volume 180, number 3
CHEMICAL PHYSICS LETTERS
17May 1991
Application of the generalized potential energy function for solving the inverse spectroscopic problem: the ground-state potential of SiF+ A. A. surkus @I Firikos kntedra, P. Vifinskiogt.. 25, 235419siauliai, Lilhuania, USSR Received
18December 1990
The previously suggested generalized potential energy function (GPEF) is used for calculation of the X ‘E+ state potential curve of SiF+ from the molecular constants. The procedure of the calculations is based on the optimization of nonlinear parameters p and n of GPEF. The obtained GPEF for SiF+ yields the experimental value of dissociation energy, has no nonphysical maximum, has a qualitatively correct R-’ behavior, and is more. accurate than the Simons-Parr-Finlan, Thakkar, Ogilvie-Tipping and Huffaker (PMO) potentials calculated from the same set of molecular constants.
1. Introduction
1.I. The generalized potential energyfimction Dunham in his classical work [ 1 ] solved the vibrational-rotational Schrtiinger equation with the vibrational potential,
where a, = w:/4B,, R is an internuclear separation, R, is an equilibrium internuclear separation; further, he derived the relationships between the parameters of potential ( 1) and the coefficients Y,, describing the spacing of the vibrational and rotational energy levels of a diatomic molecule. Dunham’s theory is widely used for solving the inverse spectroscopic problem for diatomic molecules. At the same time, the Dunham (D) potential energy function (PEF) has the following essential shortcoming: the potential ( 1) becomes infinite when R-t w, so that D-PEF ( 1) yields the correct results only for the region of the minimum of the potential curve. Several investigations [ 2-41 have been performed in an effort to avoid this shortcoming: Simons, Parr and Finlan (SPF) [ 21 changed the variable 230
(R-R,)/R, in (1) into the variable(R-R,)/R; Thakkar (T) [3] used the variable s(p)[ l(RJR)“]; Ogilvie and Tipping (OT) [4] used the variable (R -R, ) / (R + R,) . The parameters of these new PEFs may be obtained from the parameters of D-PEF with the help of simple algebraic relationships. SPF-, T- and OT-PEFs yield good results in a wider interval of internuclear separation than D-PEF, but these PEFs usually reproduce incorrectly the path of a potential curve in the region of large intemuclear separations [ 5,6]. It is difficult to indicate the best PEF among the three proposed in refs. [ 2-41, because they yield qualitatively different results for different molecules. In refs. [ 7-91, it has been demonstrated that D, SPF, T and OT potentials are different cases of generalized potentials. We have suggested [ 9; 13 p. 3601 that one define the generalized PEF (GPEF) in the form
(2) (3) wherep#O, n# - 1, s(p)= 1 ifp>O, ands(p)= - 1 ifp
0009-26 14/9 l/S 03.50 8 199 1 - Elsevier Science Publishers B.V. (North-Holland
)
Volume180,number 3
17May1991
CHEMICALPHYSICSLETTERS
yields SPF-PEF [ 21, u( R 1p, 0) yields T-PEF [ 31, and w(R] 1, 1) yields OT-PEF [4]. The generalization of several PEFs is interesting from the theoretical point of view, but the possibilities, which appear in the application of GPEF for solving the inverse spectroscopic problem, have not yet been studied. The aim of this work is the investigation of these possibilities. GPEF (2) and (3) will be used for calculation of the ground-state potential of SiF+, and it will be demonstrated that GPEF, thanks to its flexibility, yields more accurate results than SPF, T, OT and perturbed Morse oscillator (PMO) [ lo] potentials. 1.2. The spectrum and the molecular constants of SiF+ The SiF + molecule ion may be produced in plasma when silicon substrate is etched by fluorine-containing gas. The first spectroscopic observation of SiF+ has been carried out by Petrmichl, Peterson and Woods [ 111: the frequencies of 69 rotational transitions (microwave spectrum) spanning a range of rotational quantum-number J values from 1 to 14 and all vibrational states from v= 0 to 15 have been precisely measured. The parameters of D-PEF have been fitted to the measured spectrum in ref. [ 111, and the values of B,, o, and a, (i= 1, .... 5 ) have been determined for the ground state of z8Si19Ff. The equilibrium internuclear separation R, has also been calculated from the value of B, in ref. [ 111. D-PEF in ref. [ 111 describes qualitatively correctly the region of minimum of the potential (approximately 400/bof the potential), but for R~2.1 A, this PEF yields incorrect results. The infrared spectrum of SiF+ (34 lines in the v= 1t0 and v=2t 1 bands) has been observed by Akiyama, Tanaka and Tanaka [ I2 1. The set of derived molecular constants in ref. [ 121 is smaller than the set in ref. [ 111; therefore, our calculations of PEF for SiF+ will be based on the molecular constants B,, R,,w,anda,(i=l,..., 5)fromref. [II].
2. Calculations of GPEF for SiF+ In ref. [ 131, we have considered the application of GPEF (2) and (3) for solving the inverse spec-
troscopic problem and the relationships have been derived for calculation of the parameters gi of GPEF from the parameters ai of D-PEF ( 1). These relationships will be used in this work. It has also been demonstrated in ref. [ 13 ] that only the values p> 0 and n > 0 are useful in practical calculations. It is obvious that searching for optimal values of p and n in (3) is the main difficulty in the application of GPEF for solving the inverse spectroscopic problem. Let us analyse this problem for the ground state of SiF+ in detail. Diatomic PEFs of bound electronic states have many similar features; therefore, Varshni has formulated the criteria that a good PEF must satisfy [ 14 1. These criteria have been used in our calculations. (a) PEF must satisfy the condition lim V(R) =D, , R-m
(4)
where D, is a dissociation energy. The value of D, for the ground state of SiFe is known [ 51: D, = (5.23*0.14)~10~cm-‘.Itshouldbenotedthatthis value coincides with the values of D, reported in refs. [ 16,171 within the experimental errors. By substituting V, (2) and (3), into (4), we obtain the following condition for the parameters g, of GPEF: go 1+ (
:g,
r=l
=R>
(5)
With the help of the relationships (5), and (6) from ref. [ 131, it is possible to calculate from ai (i=O, ..., 5) the equal number of parameters g,; therefore, in our calculations N= 5 in (2) and (5). In the first stage of the calculations, we have found the values of p and n for which GPEF satisfies condition ( 5 ). The multitude of these values forms the curve AD in the (p, n) plane, fig. 1. (b) PEF of a diatomic molecule must not have maxima in the region R < R,, and must yield the large positive values of Vat R+O. Thus, it is necessary to separate from the p and n values, obtained in the previous stage of the calculations, the values for which GPEF satisfies this condition. We have carried out the calculations, and found that this criterion is satisfied in the case the values of p and n are placed in the part BD of the curve AD, fig. 1. Point B in fig. 1 corresponds to p=5.19 and n=0.48. 231
Volume
180,number3
17May1991
CHEMICAL PHYSICSLETTERS
Table 1 Parameters of the GPEF, (2) and (3), for the X ‘Z+ state of 2sSi’~F+
6
5 ‘A
p:
0
0.5
1
1.5
n
Fig. 1.The values of p and n (the curve AD) for the X ‘Z+ state of28Si’9Fc at which GPEF, (2) and (3), with N=5 satisfies the condition (5). p and n values disposed above the curve AD correspond to the condition lim,,, V(R) -CD., and for the p and n values disposed below the curve AD, limn,, V(R) > 0,.
(c) PEF of SF’ must not have maxima in the region R>R,. This requirement is derived from the
results of ab initio calculations for the ground electronic state of SiF+ [ 181. The analysis of the behavior of GPEF for SiF+ has shown that this requirement is satisfied in the case the values ofp and n are placed in the part AC of the curve AD, fig. 1. Point C in fig. 1 corresponds to p=3.84 and n= 0.8682. While summarising the obtained results, we have found that the requirements (a), (h) and (c) are satisfied in the case p and n values are placed in the segment BC of the curve AD, fig. 1. Now, we must find the optimal values of p and n among these values We propose to make this search from the analysis of the long-range properties of GPEF. The long-range part of PEF of a diatomic molecule can be represented with a sufficient accuracy as [19,201 V=D,-C,/Rk,
P
4 0.8678 1.5264950 94127.8 -0.81342 0.89538 -0.69619 0.64688 -0.47702
with k=4 the following boundary conditions for the PEF of SiF+: d”V
2-t
----=O! d( l/R)”
m=l,2,3.
(7)
Thus, GPEF for SiF + must satisfy these conditions. The study of the long-range properties of GPEF, ( 2 ) and (3), has shown [22] that GPEF has proper Rm4 behavior at the large internuclear separations, and satisfies boundary conditions (7) in the casep=4 in (3). The corresponding value of n is n=0.8678, fig. 1. The values p=4 and n=0.8678 are among the previously obtained possible values of p and n (segment BC in fig. 1). The parameters of GPEF, (2) and (3), for the ground state of SiF+ with these fmal values of p and n are presented in table 1.
3. Discussion and conclusions Let us compare GPEF from table I with SPF, T, OT and PM0 potentials for SiF+. The form of PM0 is [lo]
(6)
where the integer number k is the power of the leading term of the multipole expansion of the interatomic potential, and C, is the corresponding dispersion coefficient. The dissociation products for the ground state of SiF+ are Si+ (2P) and F (‘P) [ 211; therefore, k=4 in (6) because of the ion-induced dipole interaction [ 19 1. It is easy to derive from (6) 232
Value
:. (A) g0 (cm-‘) g1 & g3 g, g5
2 ?
Parameter
l= 1 -exp[ -a(R-RJ]
.
(8)
The parameters of this potential may be calculated from the parameters of D-PEF ( 1) by using the relationships ( 15) from ref. [lo]. For the T-PEF, we have used the fixed value p= -a, - 1~2.0251, which had been proposed as optimal by Thakkar [ 3 1. (It should be noted that T-PEF with ~~4.5732 yields
17May1991
CHEMICALPHYSICSLETTERS
Volume180, number3
the experimental value of D,, but at the same time this potential has a nonphysical maximum in the region R < R, and does not yield R -4 long-range behavior.) From the set of known D parameters (R,, a,, .... a,), it is possible to calculate an equal number of SPF, T, OT and PM0 parameters; therefore, N= 5 in (2) and K= 7 in (8). We have performed these calculations and the results obtained for SiF+ are presented in the upper part of table 2 and in the left part of table 3. All four calculated potentials do not yield the experimental value of D,, because this value has not been used in the calculations. SPF, T with fixed p, OT and PM0 potentials may satisfy condition (4) only by including in them the additional parameters calculated from the value of De. In our case, one additional parameter for the SPF-, T- and OT-PEFs may be calculated from the De value by the following
formula, which is derived from eq. (5 ):
It is easy to derive a similar formula for one additional parameter of PMO: h g=---;
-1e
i
ni.
(10)
i=4
The values of these additional parameters are presented in the lower part of table 2 and the characteristics of SPF, T, OT and PM0 potentials with one additional parameter are presented in the right part of table 3. Table 3 shows that the calculated potentials with one additional parameter incorrectly reproduce the ground-state potential curve of SF +: SPF-, T- and OT-PEFs have two nonphysical maxima and PM0
Table 2 Parameters of the SPF, T, OT and PM0 potentials for the X ‘Z+ state of %“F+ SPF
T 1 0
1.5264950 431695 -1.0251 0.1187 0.4854 -0.7010 0.3235
OT
PM0
2.0251 0 1.5264950 105265 0 -0.03721 0.02106 -0.00056 - 0.00747
I .5264950 1726780 -4.0502 9.6254 - 14.3172 -0.3820 89.6470
-0.47898
-81.4927
I I 1.5264950
Re(A)
47173.5 1.98173 0.09352 0.07075 0.04707 0.03913
V, (cm-‘) * (A-‘) h, hS h, h7
Additional parameters -0.0803
&
Table 3 Characteristics of the PEF
hs
-0.14179
SPF,T, OT and PM0 potentials for SiF+ from table 2 a) Calculated from R. and a, b, tim,,, V(R) (cm-‘)
maximum in region R
SPF T OT
86987 102720 1.41x10’
+ +
PM0
58989
+
Calculated from R., a, ‘) and 0. maximum
maximum
maximum
in region
in region
in region
R>R.
R
R>R,
t -
t + +
t + +
+
-
‘) The symbol t denotes the presence of a maximum and the symbol - denotes the absence of a maximum. b, i=o, ___,5.
233
Volume 180, number 3
CHEMICAL
PHYSICSLETTERS
has one nonphysical maximum. Conversely, GPEF for SiF+ from table 1 does not contain the additional nonspectroscopic parameters, yields the experimental value of the dissociation energy [ 151, has no nonphysical maximum, and has a qualitatively correct long-range behavior. Thus, the advantages of GPEF from table I are evident. The most reliable potentials for bound states of diatomic molecules may be obtained by using RKR [ 23 ] or IPA [ 24,25 ] methods, but for the ground state of SiF+, it is impossible to use these methods because of the absence of sufficiently complete experimental data concerning the spacing of vibrational levels. The set of molecular constants of SiF+ from ref. [ 111 is the largest at the present time; therefore, we believe that the calculated GPEF from this set in table 1 is now the best PEF for the ground state of SiF+. GPEF, (2) and (3), can represent the potential curves of diatomic molecules more accurately than SPF, OT and PM0 potentials, because the quality of GPEF may be improved by varying the parameters p and n in w(R Ip, n), while the analytic forms of w in SPF- and OT-PEFs (2 ) and { in PM0 (8) are fixed. GPEF is more flexible than T-PEF. Thus, the use of GPEF for solving the inverse spectroscopic problem for diatomic molecules is preferable to the use of SPF, T, OT and PM0 potentials.
References
[ 11J.L.Dunham, Phys. Rev. 41 (1932) 721.
234
17May 1991
[ 21 G. Simons, R.G. Parr and J.M. Finlan, J. Chem. Phys. 59 (1973) 3229. [3] A.J. Thakkar, J. Chem. Phys. 62 (1975) 1693. [4] J.F. Ogilvie, Canadian Spectroscopy Symposium, 1974 (unpublished); Molecular Spectroscopy Symposium, Ohio State University, 1976 (unpublished); Proc. Roy. Sot. A 378 (1981)287;A381 (1982)479(E). [ 5) C.L.Beck& J. Chem. Phys. 65 (1976) 4319. (61 R. Engelke,J. Chem. Phys. 68 (1978) 3514. [ 71 R. Engelke, J. Cbem. Phys. 70 ( 1979) 3745. [S] L. Mattera, C. Salvo, S. Terreni and F. Tommasini, J. Chem. Phys. 72 (1980) 6815. [91 A.A. surkus, R.J. Rakauskas and A.B. Bolotin, Chem. Phys. Letters 105 (1984) 291. [ 101 J.N. Huffaker, J. Chem. Phys. 64 ( 1976) 3 175. [ 1I ] R.H. Petrmichl, K.A. Peterson and R.C. Woods, I. Chem. Phys. 89 (1988) 5454. [ 121 Y. Akiyama,.K, Tanaka and T. Tanaka, Chem.Phys. Letters 155 (1989) 15. [ 131 A.A. surkus, R.J. Rakauskas and A.B. Bolotin, Chem. Phys. Letters 126 (I 986) 356. [ 141 Y.P. Varshni, Rev. Mod. Phys. 29 (1957) 664. [IS] KS. Krasnov et al., Molecular constants of inorganic combinations, Handbook (Chimia, Leningrad, 1979) [in Russian]. [ 161 M.E. WeberandP.B.Armentrout, J.Chem. Phys. 88 (1988) 6898. [ 171 K.A. Peterson and R.C. Woods, J. Chem. Phys. 89 (1988) 4929. [ 181 B.J. Garrison and W.A. Goddard III, J. Chem. Phys. 87 (1987) 1307. [ 191 R.J. LeRoy and R.B. Bernstein, J. Chem. Phys. 52 (1970) 3869. [ZO] W.C. S&valley, Contemp. Phys. 19 ( 1978) 65. [21] S.P. Kamaand F.Grein, J. Mol. Spectry. 122 (1987) 28. [22 1A.A. Surkus, to be published. [23] R. Rydberg, 2. Physik73 (1931) 376; 80 (1933) 514; 0. Klein, Z. Physik 76 (1932) 226; A.L.G. Rees, Proc. Phys. Sot. (London) A 59 (1947) 998. [24] W.M. Kosman and J. Hinze, J. Mol. Spectry. 56 (1975) 93. [25] C.R. Vidal andH. Scheingraber, J. Mol. Spectry. 65 (1977) 46.
CHEMICAL PHYSICS LETTERS
17May 1991
Application of the generalized potential energy function for solving the inverse spectroscopic problem: the ground-state potential of SiF+ A. A. surkus @I Firikos kntedra, P. Vifinskiogt.. 25, 235419siauliai, Lilhuania, USSR Received
18December 1990
The previously suggested generalized potential energy function (GPEF) is used for calculation of the X ‘E+ state potential curve of SiF+ from the molecular constants. The procedure of the calculations is based on the optimization of nonlinear parameters p and n of GPEF. The obtained GPEF for SiF+ yields the experimental value of dissociation energy, has no nonphysical maximum, has a qualitatively correct R-’ behavior, and is more. accurate than the Simons-Parr-Finlan, Thakkar, Ogilvie-Tipping and Huffaker (PMO) potentials calculated from the same set of molecular constants.
1. Introduction
1.I. The generalized potential energyfimction Dunham in his classical work [ 1 ] solved the vibrational-rotational Schrtiinger equation with the vibrational potential,
where a, = w:/4B,, R is an internuclear separation, R, is an equilibrium internuclear separation; further, he derived the relationships between the parameters of potential ( 1) and the coefficients Y,, describing the spacing of the vibrational and rotational energy levels of a diatomic molecule. Dunham’s theory is widely used for solving the inverse spectroscopic problem for diatomic molecules. At the same time, the Dunham (D) potential energy function (PEF) has the following essential shortcoming: the potential ( 1) becomes infinite when R-t w, so that D-PEF ( 1) yields the correct results only for the region of the minimum of the potential curve. Several investigations [ 2-41 have been performed in an effort to avoid this shortcoming: Simons, Parr and Finlan (SPF) [ 21 changed the variable 230
(R-R,)/R, in (1) into the variable(R-R,)/R; Thakkar (T) [3] used the variable s(p)[ l(RJR)“]; Ogilvie and Tipping (OT) [4] used the variable (R -R, ) / (R + R,) . The parameters of these new PEFs may be obtained from the parameters of D-PEF with the help of simple algebraic relationships. SPF-, T- and OT-PEFs yield good results in a wider interval of internuclear separation than D-PEF, but these PEFs usually reproduce incorrectly the path of a potential curve in the region of large intemuclear separations [ 5,6]. It is difficult to indicate the best PEF among the three proposed in refs. [ 2-41, because they yield qualitatively different results for different molecules. In refs. [ 7-91, it has been demonstrated that D, SPF, T and OT potentials are different cases of generalized potentials. We have suggested [ 9; 13 p. 3601 that one define the generalized PEF (GPEF) in the form
(2) (3) wherep#O, n# - 1, s(p)= 1 ifp>O, ands(p)= - 1 ifp
0009-26 14/9 l/S 03.50 8 199 1 - Elsevier Science Publishers B.V. (North-Holland
)
Volume180,number 3
17May1991
CHEMICALPHYSICSLETTERS
yields SPF-PEF [ 21, u( R 1p, 0) yields T-PEF [ 31, and w(R] 1, 1) yields OT-PEF [4]. The generalization of several PEFs is interesting from the theoretical point of view, but the possibilities, which appear in the application of GPEF for solving the inverse spectroscopic problem, have not yet been studied. The aim of this work is the investigation of these possibilities. GPEF (2) and (3) will be used for calculation of the ground-state potential of SiF+, and it will be demonstrated that GPEF, thanks to its flexibility, yields more accurate results than SPF, T, OT and perturbed Morse oscillator (PMO) [ lo] potentials. 1.2. The spectrum and the molecular constants of SiF+ The SiF + molecule ion may be produced in plasma when silicon substrate is etched by fluorine-containing gas. The first spectroscopic observation of SiF+ has been carried out by Petrmichl, Peterson and Woods [ 111: the frequencies of 69 rotational transitions (microwave spectrum) spanning a range of rotational quantum-number J values from 1 to 14 and all vibrational states from v= 0 to 15 have been precisely measured. The parameters of D-PEF have been fitted to the measured spectrum in ref. [ 111, and the values of B,, o, and a, (i= 1, .... 5 ) have been determined for the ground state of z8Si19Ff. The equilibrium internuclear separation R, has also been calculated from the value of B, in ref. [ 111. D-PEF in ref. [ 111 describes qualitatively correctly the region of minimum of the potential (approximately 400/bof the potential), but for R~2.1 A, this PEF yields incorrect results. The infrared spectrum of SiF+ (34 lines in the v= 1t0 and v=2t 1 bands) has been observed by Akiyama, Tanaka and Tanaka [ I2 1. The set of derived molecular constants in ref. [ 121 is smaller than the set in ref. [ 111; therefore, our calculations of PEF for SiF+ will be based on the molecular constants B,, R,,w,anda,(i=l,..., 5)fromref. [II].
2. Calculations of GPEF for SiF+ In ref. [ 131, we have considered the application of GPEF (2) and (3) for solving the inverse spec-
troscopic problem and the relationships have been derived for calculation of the parameters gi of GPEF from the parameters ai of D-PEF ( 1). These relationships will be used in this work. It has also been demonstrated in ref. [ 13 ] that only the values p> 0 and n > 0 are useful in practical calculations. It is obvious that searching for optimal values of p and n in (3) is the main difficulty in the application of GPEF for solving the inverse spectroscopic problem. Let us analyse this problem for the ground state of SiF+ in detail. Diatomic PEFs of bound electronic states have many similar features; therefore, Varshni has formulated the criteria that a good PEF must satisfy [ 14 1. These criteria have been used in our calculations. (a) PEF must satisfy the condition lim V(R) =D, , R-m
(4)
where D, is a dissociation energy. The value of D, for the ground state of SiFe is known [ 51: D, = (5.23*0.14)~10~cm-‘.Itshouldbenotedthatthis value coincides with the values of D, reported in refs. [ 16,171 within the experimental errors. By substituting V, (2) and (3), into (4), we obtain the following condition for the parameters g, of GPEF: go 1+ (
:g,
r=l
=R>
(5)
With the help of the relationships (5), and (6) from ref. [ 131, it is possible to calculate from ai (i=O, ..., 5) the equal number of parameters g,; therefore, in our calculations N= 5 in (2) and (5). In the first stage of the calculations, we have found the values of p and n for which GPEF satisfies condition ( 5 ). The multitude of these values forms the curve AD in the (p, n) plane, fig. 1. (b) PEF of a diatomic molecule must not have maxima in the region R < R,, and must yield the large positive values of Vat R+O. Thus, it is necessary to separate from the p and n values, obtained in the previous stage of the calculations, the values for which GPEF satisfies this condition. We have carried out the calculations, and found that this criterion is satisfied in the case the values of p and n are placed in the part BD of the curve AD, fig. 1. Point B in fig. 1 corresponds to p=5.19 and n=0.48. 231
Volume
180,number3
17May1991
CHEMICAL PHYSICSLETTERS
Table 1 Parameters of the GPEF, (2) and (3), for the X ‘Z+ state of 2sSi’~F+
6
5 ‘A
p:
0
0.5
1
1.5
n
Fig. 1.The values of p and n (the curve AD) for the X ‘Z+ state of28Si’9Fc at which GPEF, (2) and (3), with N=5 satisfies the condition (5). p and n values disposed above the curve AD correspond to the condition lim,,, V(R) -CD., and for the p and n values disposed below the curve AD, limn,, V(R) > 0,.
(c) PEF of SF’ must not have maxima in the region R>R,. This requirement is derived from the
results of ab initio calculations for the ground electronic state of SiF+ [ 181. The analysis of the behavior of GPEF for SiF+ has shown that this requirement is satisfied in the case the values ofp and n are placed in the part AC of the curve AD, fig. 1. Point C in fig. 1 corresponds to p=3.84 and n= 0.8682. While summarising the obtained results, we have found that the requirements (a), (h) and (c) are satisfied in the case p and n values are placed in the segment BC of the curve AD, fig. 1. Now, we must find the optimal values of p and n among these values We propose to make this search from the analysis of the long-range properties of GPEF. The long-range part of PEF of a diatomic molecule can be represented with a sufficient accuracy as [19,201 V=D,-C,/Rk,
P
4 0.8678 1.5264950 94127.8 -0.81342 0.89538 -0.69619 0.64688 -0.47702
with k=4 the following boundary conditions for the PEF of SiF+: d”V
2-t
----=O! d( l/R)”
m=l,2,3.
(7)
Thus, GPEF for SiF + must satisfy these conditions. The study of the long-range properties of GPEF, ( 2 ) and (3), has shown [22] that GPEF has proper Rm4 behavior at the large internuclear separations, and satisfies boundary conditions (7) in the casep=4 in (3). The corresponding value of n is n=0.8678, fig. 1. The values p=4 and n=0.8678 are among the previously obtained possible values of p and n (segment BC in fig. 1). The parameters of GPEF, (2) and (3), for the ground state of SiF+ with these fmal values of p and n are presented in table 1.
3. Discussion and conclusions Let us compare GPEF from table I with SPF, T, OT and PM0 potentials for SiF+. The form of PM0 is [lo]
(6)
where the integer number k is the power of the leading term of the multipole expansion of the interatomic potential, and C, is the corresponding dispersion coefficient. The dissociation products for the ground state of SiF+ are Si+ (2P) and F (‘P) [ 211; therefore, k=4 in (6) because of the ion-induced dipole interaction [ 19 1. It is easy to derive from (6) 232
Value
:. (A) g0 (cm-‘) g1 & g3 g, g5
2 ?
Parameter
l= 1 -exp[ -a(R-RJ]
.
(8)
The parameters of this potential may be calculated from the parameters of D-PEF ( 1) by using the relationships ( 15) from ref. [lo]. For the T-PEF, we have used the fixed value p= -a, - 1~2.0251, which had been proposed as optimal by Thakkar [ 3 1. (It should be noted that T-PEF with ~~4.5732 yields
17May1991
CHEMICALPHYSICSLETTERS
Volume180, number3
the experimental value of D,, but at the same time this potential has a nonphysical maximum in the region R < R, and does not yield R -4 long-range behavior.) From the set of known D parameters (R,, a,, .... a,), it is possible to calculate an equal number of SPF, T, OT and PM0 parameters; therefore, N= 5 in (2) and K= 7 in (8). We have performed these calculations and the results obtained for SiF+ are presented in the upper part of table 2 and in the left part of table 3. All four calculated potentials do not yield the experimental value of D,, because this value has not been used in the calculations. SPF, T with fixed p, OT and PM0 potentials may satisfy condition (4) only by including in them the additional parameters calculated from the value of De. In our case, one additional parameter for the SPF-, T- and OT-PEFs may be calculated from the De value by the following
formula, which is derived from eq. (5 ):
It is easy to derive a similar formula for one additional parameter of PMO: h g=---;
-1e
i
ni.
(10)
i=4
The values of these additional parameters are presented in the lower part of table 2 and the characteristics of SPF, T, OT and PM0 potentials with one additional parameter are presented in the right part of table 3. Table 3 shows that the calculated potentials with one additional parameter incorrectly reproduce the ground-state potential curve of SF +: SPF-, T- and OT-PEFs have two nonphysical maxima and PM0
Table 2 Parameters of the SPF, T, OT and PM0 potentials for the X ‘Z+ state of %“F+ SPF
T 1 0
1.5264950 431695 -1.0251 0.1187 0.4854 -0.7010 0.3235
OT
PM0
2.0251 0 1.5264950 105265 0 -0.03721 0.02106 -0.00056 - 0.00747
I .5264950 1726780 -4.0502 9.6254 - 14.3172 -0.3820 89.6470
-0.47898
-81.4927
I I 1.5264950
Re(A)
47173.5 1.98173 0.09352 0.07075 0.04707 0.03913
V, (cm-‘) * (A-‘) h, hS h, h7
Additional parameters -0.0803
&
Table 3 Characteristics of the PEF
hs
-0.14179
SPF,T, OT and PM0 potentials for SiF+ from table 2 a) Calculated from R. and a, b, tim,,, V(R) (cm-‘)
maximum in region R
SPF T OT
86987 102720 1.41x10’
+ +
PM0
58989
+
Calculated from R., a, ‘) and 0. maximum
maximum
maximum
in region
in region
in region
R>R.
R
R>R,
t -
t + +
t + +
+
-
‘) The symbol t denotes the presence of a maximum and the symbol - denotes the absence of a maximum. b, i=o, ___,5.
233
Volume 180, number 3
CHEMICAL
PHYSICSLETTERS
has one nonphysical maximum. Conversely, GPEF for SiF+ from table 1 does not contain the additional nonspectroscopic parameters, yields the experimental value of the dissociation energy [ 151, has no nonphysical maximum, and has a qualitatively correct long-range behavior. Thus, the advantages of GPEF from table I are evident. The most reliable potentials for bound states of diatomic molecules may be obtained by using RKR [ 23 ] or IPA [ 24,25 ] methods, but for the ground state of SiF+, it is impossible to use these methods because of the absence of sufficiently complete experimental data concerning the spacing of vibrational levels. The set of molecular constants of SiF+ from ref. [ 111 is the largest at the present time; therefore, we believe that the calculated GPEF from this set in table 1 is now the best PEF for the ground state of SiF+. GPEF, (2) and (3), can represent the potential curves of diatomic molecules more accurately than SPF, OT and PM0 potentials, because the quality of GPEF may be improved by varying the parameters p and n in w(R Ip, n), while the analytic forms of w in SPF- and OT-PEFs (2 ) and { in PM0 (8) are fixed. GPEF is more flexible than T-PEF. Thus, the use of GPEF for solving the inverse spectroscopic problem for diatomic molecules is preferable to the use of SPF, T, OT and PM0 potentials.
References
[ 11J.L.Dunham, Phys. Rev. 41 (1932) 721.
234
17May 1991
[ 21 G. Simons, R.G. Parr and J.M. Finlan, J. Chem. Phys. 59 (1973) 3229. [3] A.J. Thakkar, J. Chem. Phys. 62 (1975) 1693. [4] J.F. Ogilvie, Canadian Spectroscopy Symposium, 1974 (unpublished); Molecular Spectroscopy Symposium, Ohio State University, 1976 (unpublished); Proc. Roy. Sot. A 378 (1981)287;A381 (1982)479(E). [ 5) C.L.Beck& J. Chem. Phys. 65 (1976) 4319. (61 R. Engelke,J. Chem. Phys. 68 (1978) 3514. [ 71 R. Engelke, J. Cbem. Phys. 70 ( 1979) 3745. [S] L. Mattera, C. Salvo, S. Terreni and F. Tommasini, J. Chem. Phys. 72 (1980) 6815. [91 A.A. surkus, R.J. Rakauskas and A.B. Bolotin, Chem. Phys. Letters 105 (1984) 291. [ 101 J.N. Huffaker, J. Chem. Phys. 64 ( 1976) 3 175. [ 1I ] R.H. Petrmichl, K.A. Peterson and R.C. Woods, I. Chem. Phys. 89 (1988) 5454. [ 121 Y. Akiyama,.K, Tanaka and T. Tanaka, Chem.Phys. Letters 155 (1989) 15. [ 131 A.A. surkus, R.J. Rakauskas and A.B. Bolotin, Chem. Phys. Letters 126 (I 986) 356. [ 141 Y.P. Varshni, Rev. Mod. Phys. 29 (1957) 664. [IS] KS. Krasnov et al., Molecular constants of inorganic combinations, Handbook (Chimia, Leningrad, 1979) [in Russian]. [ 161 M.E. WeberandP.B.Armentrout, J.Chem. Phys. 88 (1988) 6898. [ 171 K.A. Peterson and R.C. Woods, J. Chem. Phys. 89 (1988) 4929. [ 181 B.J. Garrison and W.A. Goddard III, J. Chem. Phys. 87 (1987) 1307. [ 191 R.J. LeRoy and R.B. Bernstein, J. Chem. Phys. 52 (1970) 3869. [ZO] W.C. S&valley, Contemp. Phys. 19 ( 1978) 65. [21] S.P. Kamaand F.Grein, J. Mol. Spectry. 122 (1987) 28. [22 1A.A. Surkus, to be published. [23] R. Rydberg, 2. Physik73 (1931) 376; 80 (1933) 514; 0. Klein, Z. Physik 76 (1932) 226; A.L.G. Rees, Proc. Phys. Sot. (London) A 59 (1947) 998. [24] W.M. Kosman and J. Hinze, J. Mol. Spectry. 56 (1975) 93. [25] C.R. Vidal andH. Scheingraber, J. Mol. Spectry. 65 (1977) 46.