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Application of the generalized potential energy function for solving the inverse spectroscopic problem for diatomic molecules
Volume
126, number
CHEMICAL
3,4
PHYSICS
LETTERS
9 May 1986
APPLICATION OF THE GENERALIZED POTENTIAL ENERGY FUNCTION FOR SOLVING THE INVERSE SPECTROSCOPIC PROBLEM FOR DIATOMIC MOLECULES
A.A. SURKUS &auliar K. Preikias
Teacher-Traming
Institute, &auliai 235419, USSR
R.J. RAKAUSKAS and A.B. BOLOTIN Vilnius V. Kapsukas Received
3 February
University, Faculty of Physics, Vilnius 232054, USSR 1986
The generalized potential energy function (GPEF) suggested by us in 1984 is used here to solve the inverse spectroscopic problem for diatomic molecules, and corresponding relationships are derived for the cases when GPEF defines (1) the total and (2) the electronic energy of diatomics. The permissible range of values for non-linear parameters of GPEF is discussed.
1. Introduction The potential energy function (PEF) yields basic information on the structure and properties of diatomic molecules. As PEF cannot be directly measured, a number of methods have been suggested for estimating it from experimentally obtained physical characteristics. Spectroscopic methods are of particular importance, as considerable progress has recently been made in laser and Fourier spectroscopy which has resulted in increased accuracy of measurements and the possibility of registering a greater number of spectral lines. Two general methods are usually employed to solve the inverse spectroscopic problem: (1) the RydbergKlein-Rees method [l] and (2) the method of empirical PEFs [2-41. In the latter case empirical expressions of two types are used for constructing the PEF: (a) with a fmed number of parameters [2,3], and (b) with a variable number of parameters [4-181. In the case of the latter type, the PEF is most frequently presented analytically as a power series:
tigiwi)
v=g()w2(l
(1)
3
where V is the PEF,gi are linear parameters of PEF (l), and w is a function of the internuclear separation R. There are several PEFs of type (1) differing from one another by the analytical expression of w [5,8,10-12,14, 16,17,19]. In ref. [19] we suggested the following expression for w: w = w(RIp, n) = s(p) (RP - RQ/(Rp
+ nR$) ,
(2)
where the parametersp and n are real numbers on condition that p # 0 and n # -1, and s(p) = 1 if p > 0, and s(p) = -1 ifp < 0. The PEF (1) has a simple analytical form, and it generalizes some of the PEFs suggested previously which can be obtained from (1) and (2) by substituting the corresponding values of p and n into (2). Thus, w(R I-1,0) yields Dunham’s PEF 151;w(R 11, 0) is Slmons-Parr-Finlan’s PEF [8] ; o(R jp, 0) is Thakkar’s 356
0 009-2614/86/$0350 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 126, number 3,4
CHEMICAL PHYSICS LETTERS
9 May 1986
PEF [l l] ; o(R1 1, 1) is Ogilvie-Tipping’s PEF [lo] ; w(R II, k/m), where k and m are integers, yields the family of Ogilvie’sPEFs 1171. Therefore all the properties of these functions can be viewed as the properties of the generalized PEF (GPEF) (1) and (2). This paper is concerned with applying GPEF to solve the inverse spectroscopic problem for diatomic molecules. Section 2 describes the method which was used to obtain the relationships required to solve it. In section 3 these relationships for GPEF (1) and (2) are given. In section 4 the variable w(R (p, n) is used to determine the function of electronic energy of diatomic molecules and the corresponding’expressions are listed. Section 5 deals with the permissible range of values for the non-linear parameters p and n.
2. Method The vibration-rotation
energy levels of a diatomic molecule Ed are given in ref. [5] as
(3) where Ykl are equilibrium spectroscopic constants and u and J are vibrational and rotational quantum numbers respectively. Ykl can be determined by adjusting (3) to the experimentally measured frequencies of the transitions. The spectroscopic constants Ykr are related by Dunham’s formulas [5] to coefficients aj whose values are defined by the derivatives of the PEF at the minimum: a0 = dRi(d2 V/dR2),=,,
,
ai ={Rc2/ [ao(i+ 2)!]} (di*2V/dRi+2)R,Re.
(4)
By successively estimating the derivatives with respect to R of the analytical expression of the PEF and substituting them into (4) we obtain the relationships between the parameters of this PEF and coefficients ai.
3. Relationships for GPEF (1) and (2)
Let us consider the case when p > 0. By estimating the derivatives of GPEF (1) and (2) and substituting them into (4), we obtain relationships between parametersgi and coefficients ai: go = aoa12, 81 =fZ~Wil
(54 - 020i2,
WI
82 = a2Cdi2 - $ C.4JibJi4- 5 W30i3
- ig1&J2Wi2 ,
g3 = a30i3 -i 02W3Wi5 - 6 W4Op - ag,UiU? g4 = a4wp 2 - &2(JJ2q
- & 0gOi6 -4-Z
- &W2W40i6
5g2W3Wi3
- &
- $g3W2Wi2
Us
(5 c) - tglU3Ui3
Wi5-
- hW5W1
,
3 4-g2W2W3Wi5-_g2W4W1 2gzw2w1
(54
tg~Ca_)~Cali6 - ~g1W203Ct_Ji5 - iglW4W14 (54
-5 _ ~W~WqW~‘-~W~W~W~7-~W~W~6-~8~W~w3W~71 85 = ap1 -5-L
- 2g2W2Wi2,
iig1W$Wi6-b?1W2W4Wi6
-4
5
-~g3W~Wi4-~g3W3Wi3-3g4W2Wi2,
(5f)
where
357
Volume 126, number 3,4
CHEMICAL PHYSICS LETTERS
ERe[dW(RlPtn)ldRlRsR, =P/(n+ 1)
~1
o2 = R; [d2w(RIp, n)/dR2]R,R,
= q(p
w3 = R; [d3w(RIp, n)/dR31RzRe= q(p
9 May 1986
(6a)
3
- 1) - 2~; ,
(6b)
- 1) (p - 2) - 60&p - 1) + 6~;)
(6~)
~4~R~[d4~(Rl~,~)/dR41~~R,=wl(~-l)(~-2)(p-3)-2w~(p-1)(7p-11)+36w~(p-1)-24~~,(6d) o5 = R: [d’w(RIp, n)/dR5]R,Re =q@-l)(p-2)...(p-4)-
lOw&l)@-2)(3p-5)
+ 3Ow& - 1) (5~ - 7) - 24Ow:@ - 1) t 120~; , a6 =R:[@%W,
+ 90&r
l)(p--
~)ldR61,=,e=wl(p-
(6e)
2)...(p-
5) - 2w;(p-1)&2)(31p2-132~
- 1)(6p2 - 19P+ 15) - 4Ow:(p - 1)(39R - 51) + 18OOw;(p - 1) - 720~: ,
o7 =R~[d7~(Rl~,~)/dR71R,Re=wl(P-l)0,-2)...(p-6)-
t 137) (60
14w;@-l)@-2)@-3)(9+39pt42)
~42w~o,-l)(p-2)(43~2-141p~116)-84O~~~-1)(1Op~-29p+21) + 42OOc& - 1)(4p - 5) - 1512Oo&p - 1) t 5040~; .
CW
Relationships for p < 6 can easily be obtained from (5) by substituting -gl, -g3, -g5 for gl , g3, g5 respectively. Thus, if the spectroscopic constants Ykl are known, we can calculate the coefficients uj with Dunham’s formulas [5], and by substituting ai into (5) and (6) we obtain the parametersgi of the GPEF (1) and (2).
4. Coulomb-subtracted GPEF Because of its flexibility, the variable o(Rlp, n) can be used to construct the electronic energy function of a diatomic molecule, vel = v-
z,z,/R,
(7)
where ZA and Z, are the charges of nuclei A and B. Let us present Vel as the power series of o(RIp, n). In this case the total energy function (PEF) will be as follows: M V = ,Fu r+J(R IpI n) + ZAZ,lR ,
(8)
where ri are parameters of PEF (8). The PEF (8) will be termed the Coulomb-subtracted GPEF (CS GPEF). Let us consider the case when p > 0. By estimating the derivatives of GS GPEF (8) and substituting them into (4) we obtain the relationships between parameters ri and coefficients ai: r,=-b, r1 = boil
(W ,
(9b)
r2 = (~0 - b)Wi2 - $r102wi2
,
r3 = (~a1 + b)Ui3
- ‘202Wl ’ 41 2-4
r4 = (uou2 - b)wF
358
- ir103wi3 - $r1w401
w -2
- zr2a2q
@d) -
ir2w3q
-3
3 - 5r3w2q
-2
,
(se)
r5 = (aOa3 + b)ui5 ‘6 = -
-
- tr2(d2w3(di5 2
- kr203wl
2 -4 3 ar3w2w3wl -5 _ I8 r3W401 - zwpl
2
-& r3w3wl -
&rlo5wi5
(aOa4 - b)wi6 - &rlw6wi6
r7 = (aoag + b)wi7
2
gr5w2w1
-6
-4
- &rlw7wi7
-$yd2w4w;6 -
9 May 1986
CHEMICAL PHYSICS LETTERS
Volume 126, number 3,4
5 ir5wq
-3
-6
- ~‘3&$
- ii r2w2w4wi6
2 - 5r4w3q
- $rZw3w4w;7 -$jr3W5Wi5
-
-4
- &r2u40r4
-
-3 -
- &r2w5wl
fr3f.~3&~~~->402~~~, (90
3 -6 -5 - i 8 r3°2w1
qr5w2w1 -2 ,
- &r2w2w5wi7 ir4wzwp
-
- &r2w6wi6
(W
- ir3w$w3wi7
- r4w2w3w1 -5 - tr4w4w1 -4
3r6w2wi2,
where b =2,2,/R,. Relationships for p < 0 can be obtained from (9) by substituting -rl, rl, r3, r5, r7 respectively.
-r3,
-r5,
-r7 for
5. Discussion Let us discuss qualitative considerations which should govern the choice of the values of parameters p and n in the GPEF and CS GPEF. The variable w(R(p, n) can be used with both positive and negative values of p. It is easy to show that w(R Ip, n) satisfies the equation w(RI-P,
l/n) = nw(RIp,
n) .
00)
On the other hand, the transformation formulas (5) and (9) possess the following property: if we obtain some coefikientsgi or ri with (5) or (9) for arbitrary values of p and n, then substituting the valuesp’= -p and n’=l/n into (5) or (9) yields coefficients gi or ri related to gi or ri by relationships gb =go1n2 , r;=rJni.
g,: = gilt+ 2
(114 (1 lb)
By substituting into (1) first w (R Ip, n) and then w(R I -p, I/n) and taking into account (10) and (1 la), we obtain the same expression for V in both cases. We also obtain the same PEFs in the case of substituting into (8) first w(R(p, n) and then w(Rl-p, I/n). In other words, to each set of parameters of the GPEF (p, n, R,, gi) or of CSGPEF (p, n, R,, ri) there corresponds a set of parameters (-p, l/n, R,,g,!) or (-p, l/n, R,, ri) respectively which yields the same PEFs (1) or (8). Thus, there is no need to use the GPEF (1) and (2) and CS GPEF (8) both with the positive and negative values of p: it is sufficient to use the values of p > 0. If we choose the value of n, its positive values should be preferred, because in the case of n < 0 the function w(R ip, n) has a non-physical vertical asymptote at the point R = InlllPR,, as the denominator of w(RIp, n) becomes zero. As the expressions of GPEF and CS GPEF depend on the values of p and n, it is obvious that in each particular case there are optimum values of p and n for which the path of the potential curve is reproduced with greatest accuracy. In fact, Simons-Parr-Fir&n’s PEF [8] and Ogilvie-Tipping’s PEF [lo] correspond to different procedures of GPEF parameterization in which p and n have fixed values. In Thakkar’s PEF [ 1 I] n = 0, and p may vary freely. According to Thakkar [l l] , the optimum value is p = -al - 1. For weakly bound molecules, it has been proposed in ref. [20] to use Thakkar’s PEF with such a minimum integer value of p for which the coefficientsgj are positive and form a uniformly decreasing sequence. Engelke constructed GPEF as a power series (1) of the variable w = X(R Ip, /3) [ 141 and suggested that the values of the parameters p and fl be determined (i) on condition that the coefficientsgl and g2 in (1) equal zero, or (ii) by taking p equal to the power of the leading term of the multipole expansion of the atom-atom interaction at large separation and /I = 1. Calculations for a number of diatomic molecules [4,8,11,14,17,20-221 reveal that it is difficult to distinguish the best method of parameterization 359
Volume 126, number 3,4
CHEMICAL PHYSICS LETTERS
9 May 1986
among all those suggested, because they field qualitatively different results for different molecules. Thus, we cannot consider the problem of choosing the optimum p and n in the GPEF and CS GPEF as solved. At present a careful investigation of this problem is being carried out. In ref. [ 191, an error was made in obtaining the third relationship in (8) for a2 which is corrected in this paper (relationship (SC)). Correspondingly, eq. (13) in ref. [ 191 should be changed as follows: p2-$+6a2-l=O, Both roots of this equation yield one and the same analytical expression of the potential Gl. In table 1 of ref. [19] the parameter values of the potentials Gl and G2 are changed; potential Gl: p = 1.1634, n = 0.3170,gu = 0.465369 au; potential G2: p = 1,n = 0.5,go = 0.817083 au,gl = -0.4050,g2 = -0.0096. Mean errors for both potentials Gl and G2 (table 2 of ref. 1191) obtained with these parameter values equal 5.3%. On p. 292 of ref. [19] line nine on the left should read n # -1 and p + 0.
Acknowledgement
The authors wish to thank M. StakvileviEius for useful discussions.
[l]
[2] [3] [4] [S] [6] [7] [8] [9]
[lo] [ll] [12] [13] [14] [15] [ 161 [17] [18] [ 191 [20] [Zl] [22]
360
R. Rydberg, Z.Physik 73 (1931) 376;80 (1933) 514; 0. Klein, Z. Physik 76 (1932) 226; A.L.G. Ret%, Proc. Phys. Sot. (London) A59 (1947) 998. Y.P. Varshni, Rev. Mod. Phys. 29 (1957) 664. D. Steele, E.R. Lippincott and J.T. Vanderslice, Rev. Mod. Phys. 34 (1962) 239. R. Engelke, J. Chem. Phys. 68 (1978) 3514. J.L. Dunharn,Phys. Rev. 41 (1932) 721. I. Sandeman, Proc. Roy. Sot. (Edinburgh) 60 (1940) 210. H.W. Woolley, J. Chem. Phys. 37 (1962) 1307; 56 (1972) 1792(E). G. Simons, R.G. Parr and J.M. FinIan, J.Chem. Phys. 59 (1973) 3229. K.D. Jordan, J.L. Kinsey and R. Silbey, J. Chem. Phys. 61(1974) 911. J.F. Ogilvie, Canadian Spectroscopy Symposium, Ottawa,Canada (October 1974); Paper ZS4, Symposium on Molcxuhu Spectroscopy, Columbus OH (18 June 1976). AJ. Thakkar, J. Chem. Phys. 62 (1975) 1693. J.N.Huffaker,J.Chem.Phys.64 (1976) 3175. R.F. Nalewajski and R.G. Parr, J.Chem. Phys. 67 (1977) 1324. R. Engelke, J. Chem. Phys. 70 (1979) 3745. V.S. Jorish and N.B. Shuherbak, Chem. Phys. Letters 67 (1979) 160. L. Mattera, C. Salvo, S. Terreni and F. Tommasini, J. Chem. Phys. 72 (1980) 6815. J.F. Ogilvie, Proc. Roy. Sot. A378 (1981) 287;A381 (1982)479(E). GA. Arteca, F.M. Fernandez and EA. Castro, J. Chem. Phys. 81(1984) 4540. AA. surkus, R.J. Rakauskas and A.B. Bolotin, Chem. Phys. Letters 105 (1984) 291. L. Vahala and M.D. Havey, J. Chem. Phys. 81(1984) 4867. J.H. Goble, DC. Hartman and J.S.Wii, J.Chem. Phys. 67 (1977) 4206. J.H. Goble and J.S. Winn, J. Chem. Phys. 70 (1979) 2051.
126, number
CHEMICAL
3,4
PHYSICS
LETTERS
9 May 1986
APPLICATION OF THE GENERALIZED POTENTIAL ENERGY FUNCTION FOR SOLVING THE INVERSE SPECTROSCOPIC PROBLEM FOR DIATOMIC MOLECULES
A.A. SURKUS &auliar K. Preikias
Teacher-Traming
Institute, &auliai 235419, USSR
R.J. RAKAUSKAS and A.B. BOLOTIN Vilnius V. Kapsukas Received
3 February
University, Faculty of Physics, Vilnius 232054, USSR 1986
The generalized potential energy function (GPEF) suggested by us in 1984 is used here to solve the inverse spectroscopic problem for diatomic molecules, and corresponding relationships are derived for the cases when GPEF defines (1) the total and (2) the electronic energy of diatomics. The permissible range of values for non-linear parameters of GPEF is discussed.
1. Introduction The potential energy function (PEF) yields basic information on the structure and properties of diatomic molecules. As PEF cannot be directly measured, a number of methods have been suggested for estimating it from experimentally obtained physical characteristics. Spectroscopic methods are of particular importance, as considerable progress has recently been made in laser and Fourier spectroscopy which has resulted in increased accuracy of measurements and the possibility of registering a greater number of spectral lines. Two general methods are usually employed to solve the inverse spectroscopic problem: (1) the RydbergKlein-Rees method [l] and (2) the method of empirical PEFs [2-41. In the latter case empirical expressions of two types are used for constructing the PEF: (a) with a fmed number of parameters [2,3], and (b) with a variable number of parameters [4-181. In the case of the latter type, the PEF is most frequently presented analytically as a power series:
tigiwi)
v=g()w2(l
(1)
3
where V is the PEF,gi are linear parameters of PEF (l), and w is a function of the internuclear separation R. There are several PEFs of type (1) differing from one another by the analytical expression of w [5,8,10-12,14, 16,17,19]. In ref. [19] we suggested the following expression for w: w = w(RIp, n) = s(p) (RP - RQ/(Rp
+ nR$) ,
(2)
where the parametersp and n are real numbers on condition that p # 0 and n # -1, and s(p) = 1 if p > 0, and s(p) = -1 ifp < 0. The PEF (1) has a simple analytical form, and it generalizes some of the PEFs suggested previously which can be obtained from (1) and (2) by substituting the corresponding values of p and n into (2). Thus, w(R I-1,0) yields Dunham’s PEF 151;w(R 11, 0) is Slmons-Parr-Finlan’s PEF [8] ; o(R jp, 0) is Thakkar’s 356
0 009-2614/86/$0350 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 126, number 3,4
CHEMICAL PHYSICS LETTERS
9 May 1986
PEF [l l] ; o(R1 1, 1) is Ogilvie-Tipping’s PEF [lo] ; w(R II, k/m), where k and m are integers, yields the family of Ogilvie’sPEFs 1171. Therefore all the properties of these functions can be viewed as the properties of the generalized PEF (GPEF) (1) and (2). This paper is concerned with applying GPEF to solve the inverse spectroscopic problem for diatomic molecules. Section 2 describes the method which was used to obtain the relationships required to solve it. In section 3 these relationships for GPEF (1) and (2) are given. In section 4 the variable w(R (p, n) is used to determine the function of electronic energy of diatomic molecules and the corresponding’expressions are listed. Section 5 deals with the permissible range of values for the non-linear parameters p and n.
2. Method The vibration-rotation
energy levels of a diatomic molecule Ed are given in ref. [5] as
(3) where Ykl are equilibrium spectroscopic constants and u and J are vibrational and rotational quantum numbers respectively. Ykl can be determined by adjusting (3) to the experimentally measured frequencies of the transitions. The spectroscopic constants Ykr are related by Dunham’s formulas [5] to coefficients aj whose values are defined by the derivatives of the PEF at the minimum: a0 = dRi(d2 V/dR2),=,,
,
ai ={Rc2/ [ao(i+ 2)!]} (di*2V/dRi+2)R,Re.
(4)
By successively estimating the derivatives with respect to R of the analytical expression of the PEF and substituting them into (4) we obtain the relationships between the parameters of this PEF and coefficients ai.
3. Relationships for GPEF (1) and (2)
Let us consider the case when p > 0. By estimating the derivatives of GPEF (1) and (2) and substituting them into (4), we obtain relationships between parametersgi and coefficients ai: go = aoa12, 81 =fZ~Wil
(54 - 020i2,
WI
82 = a2Cdi2 - $ C.4JibJi4- 5 W30i3
- ig1&J2Wi2 ,
g3 = a30i3 -i 02W3Wi5 - 6 W4Op - ag,UiU? g4 = a4wp 2 - &2(JJ2q
- & 0gOi6 -4-Z
- &W2W40i6
5g2W3Wi3
- &
- $g3W2Wi2
Us
(5 c) - tglU3Ui3
Wi5-
- hW5W1
,
3 4-g2W2W3Wi5-_g2W4W1 2gzw2w1
(54
tg~Ca_)~Cali6 - ~g1W203Ct_Ji5 - iglW4W14 (54
-5 _ ~W~WqW~‘-~W~W~W~7-~W~W~6-~8~W~w3W~71 85 = ap1 -5-L
- 2g2W2Wi2,
iig1W$Wi6-b?1W2W4Wi6
-4
5
-~g3W~Wi4-~g3W3Wi3-3g4W2Wi2,
(5f)
where
357
Volume 126, number 3,4
CHEMICAL PHYSICS LETTERS
ERe[dW(RlPtn)ldRlRsR, =P/(n+ 1)
~1
o2 = R; [d2w(RIp, n)/dR2]R,R,
= q(p
w3 = R; [d3w(RIp, n)/dR31RzRe= q(p
9 May 1986
(6a)
3
- 1) - 2~; ,
(6b)
- 1) (p - 2) - 60&p - 1) + 6~;)
(6~)
~4~R~[d4~(Rl~,~)/dR41~~R,=wl(~-l)(~-2)(p-3)-2w~(p-1)(7p-11)+36w~(p-1)-24~~,(6d) o5 = R: [d’w(RIp, n)/dR5]R,Re =q@-l)(p-2)...(p-4)-
lOw&l)@-2)(3p-5)
+ 3Ow& - 1) (5~ - 7) - 24Ow:@ - 1) t 120~; , a6 =R:[@%W,
+ 90&r
l)(p--
~)ldR61,=,e=wl(p-
(6e)
2)...(p-
5) - 2w;(p-1)&2)(31p2-132~
- 1)(6p2 - 19P+ 15) - 4Ow:(p - 1)(39R - 51) + 18OOw;(p - 1) - 720~: ,
o7 =R~[d7~(Rl~,~)/dR71R,Re=wl(P-l)0,-2)...(p-6)-
t 137) (60
14w;@-l)@-2)@-3)(9+39pt42)
~42w~o,-l)(p-2)(43~2-141p~116)-84O~~~-1)(1Op~-29p+21) + 42OOc& - 1)(4p - 5) - 1512Oo&p - 1) t 5040~; .
CW
Relationships for p < 6 can easily be obtained from (5) by substituting -gl, -g3, -g5 for gl , g3, g5 respectively. Thus, if the spectroscopic constants Ykl are known, we can calculate the coefficients uj with Dunham’s formulas [5], and by substituting ai into (5) and (6) we obtain the parametersgi of the GPEF (1) and (2).
4. Coulomb-subtracted GPEF Because of its flexibility, the variable o(Rlp, n) can be used to construct the electronic energy function of a diatomic molecule, vel = v-
z,z,/R,
(7)
where ZA and Z, are the charges of nuclei A and B. Let us present Vel as the power series of o(RIp, n). In this case the total energy function (PEF) will be as follows: M V = ,Fu r+J(R IpI n) + ZAZ,lR ,
(8)
where ri are parameters of PEF (8). The PEF (8) will be termed the Coulomb-subtracted GPEF (CS GPEF). Let us consider the case when p > 0. By estimating the derivatives of GS GPEF (8) and substituting them into (4) we obtain the relationships between parameters ri and coefficients ai: r,=-b, r1 = boil
(W ,
(9b)
r2 = (~0 - b)Wi2 - $r102wi2
,
r3 = (~a1 + b)Ui3
- ‘202Wl ’ 41 2-4
r4 = (uou2 - b)wF
358
- ir103wi3 - $r1w401
w -2
- zr2a2q
@d) -
ir2w3q
-3
3 - 5r3w2q
-2
,
(se)
r5 = (aOa3 + b)ui5 ‘6 = -
-
- tr2(d2w3(di5 2
- kr203wl
2 -4 3 ar3w2w3wl -5 _ I8 r3W401 - zwpl
2
-& r3w3wl -
&rlo5wi5
(aOa4 - b)wi6 - &rlw6wi6
r7 = (aoag + b)wi7
2
gr5w2w1
-6
-4
- &rlw7wi7
-$yd2w4w;6 -
9 May 1986
CHEMICAL PHYSICS LETTERS
Volume 126, number 3,4
5 ir5wq
-3
-6
- ~‘3&$
- ii r2w2w4wi6
2 - 5r4w3q
- $rZw3w4w;7 -$jr3W5Wi5
-
-4
- &r2u40r4
-
-3 -
- &r2w5wl
fr3f.~3&~~~->402~~~, (90
3 -6 -5 - i 8 r3°2w1
qr5w2w1 -2 ,
- &r2w2w5wi7 ir4wzwp
-
- &r2w6wi6
(W
- ir3w$w3wi7
- r4w2w3w1 -5 - tr4w4w1 -4
3r6w2wi2,
where b =2,2,/R,. Relationships for p < 0 can be obtained from (9) by substituting -rl, rl, r3, r5, r7 respectively.
-r3,
-r5,
-r7 for
5. Discussion Let us discuss qualitative considerations which should govern the choice of the values of parameters p and n in the GPEF and CS GPEF. The variable w(R(p, n) can be used with both positive and negative values of p. It is easy to show that w(R Ip, n) satisfies the equation w(RI-P,
l/n) = nw(RIp,
n) .
00)
On the other hand, the transformation formulas (5) and (9) possess the following property: if we obtain some coefikientsgi or ri with (5) or (9) for arbitrary values of p and n, then substituting the valuesp’= -p and n’=l/n into (5) or (9) yields coefficients gi or ri related to gi or ri by relationships gb =go1n2 , r;=rJni.
g,: = gilt+ 2
(114 (1 lb)
By substituting into (1) first w (R Ip, n) and then w(R I -p, I/n) and taking into account (10) and (1 la), we obtain the same expression for V in both cases. We also obtain the same PEFs in the case of substituting into (8) first w(R(p, n) and then w(Rl-p, I/n). In other words, to each set of parameters of the GPEF (p, n, R,, gi) or of CSGPEF (p, n, R,, ri) there corresponds a set of parameters (-p, l/n, R,,g,!) or (-p, l/n, R,, ri) respectively which yields the same PEFs (1) or (8). Thus, there is no need to use the GPEF (1) and (2) and CS GPEF (8) both with the positive and negative values of p: it is sufficient to use the values of p > 0. If we choose the value of n, its positive values should be preferred, because in the case of n < 0 the function w(R ip, n) has a non-physical vertical asymptote at the point R = InlllPR,, as the denominator of w(RIp, n) becomes zero. As the expressions of GPEF and CS GPEF depend on the values of p and n, it is obvious that in each particular case there are optimum values of p and n for which the path of the potential curve is reproduced with greatest accuracy. In fact, Simons-Parr-Fir&n’s PEF [8] and Ogilvie-Tipping’s PEF [lo] correspond to different procedures of GPEF parameterization in which p and n have fixed values. In Thakkar’s PEF [ 1 I] n = 0, and p may vary freely. According to Thakkar [l l] , the optimum value is p = -al - 1. For weakly bound molecules, it has been proposed in ref. [20] to use Thakkar’s PEF with such a minimum integer value of p for which the coefficientsgj are positive and form a uniformly decreasing sequence. Engelke constructed GPEF as a power series (1) of the variable w = X(R Ip, /3) [ 141 and suggested that the values of the parameters p and fl be determined (i) on condition that the coefficientsgl and g2 in (1) equal zero, or (ii) by taking p equal to the power of the leading term of the multipole expansion of the atom-atom interaction at large separation and /I = 1. Calculations for a number of diatomic molecules [4,8,11,14,17,20-221 reveal that it is difficult to distinguish the best method of parameterization 359
Volume 126, number 3,4
CHEMICAL PHYSICS LETTERS
9 May 1986
among all those suggested, because they field qualitatively different results for different molecules. Thus, we cannot consider the problem of choosing the optimum p and n in the GPEF and CS GPEF as solved. At present a careful investigation of this problem is being carried out. In ref. [ 191, an error was made in obtaining the third relationship in (8) for a2 which is corrected in this paper (relationship (SC)). Correspondingly, eq. (13) in ref. [ 191 should be changed as follows: p2-$+6a2-l=O, Both roots of this equation yield one and the same analytical expression of the potential Gl. In table 1 of ref. [19] the parameter values of the potentials Gl and G2 are changed; potential Gl: p = 1.1634, n = 0.3170,gu = 0.465369 au; potential G2: p = 1,n = 0.5,go = 0.817083 au,gl = -0.4050,g2 = -0.0096. Mean errors for both potentials Gl and G2 (table 2 of ref. 1191) obtained with these parameter values equal 5.3%. On p. 292 of ref. [19] line nine on the left should read n # -1 and p + 0.
Acknowledgement
The authors wish to thank M. StakvileviEius for useful discussions.
[l]
[2] [3] [4] [S] [6] [7] [8] [9]
[lo] [ll] [12] [13] [14] [15] [ 161 [17] [18] [ 191 [20] [Zl] [22]
360
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