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Hypervirial-perturbative approximation to the SCF eigenvalues of coupled oscillators
Volume 92. number
1
CHEMICAL PHYSICS LEI-I-ERS
HYPERMRIAL-PERTURBATIVE
8 October 1981
APPROXIMATION TO THE SCF EIGENVALUES
OF COUPLED OSCILLATORS
Gustavo A. ARTECA, Francisco hf. FERNhlDEZ and Eduardo A. CASTRO IIW~~. Szccidn Quimlcu Tedrrca, Sucursal4, Gsdh de correo 16. Ln PInta 1900. Agentma Recewed 7 June 1982
The hypenina relstions and perturbatton theory are used in order to obtain analytwl non-numertal e\prewons ior the SCF rlgenvalues oi coupled oscdlxors
1. Introduction Lately, the study of quantum properties of polyatomic molecules has given nse to a growing interest among investigators of non-separable systems. Owmg to thrs, at present there are several methods which allow one to obtam, with different degree of accuracy, the ergenfunctions and ergenvalues of coupled oscillators [l-36]. The self-consrstent field (SCF) theory [l-8] is one of the simplest approximations which has been used to study tius problem. Notwithstanding its sunplicity, the difficulties of the SCF method grow markedly with the increase of quantum numbers (because the functions become very oscillatory) and the degrees of freedom of the system. There have been proposed various alternative procedures in order to lessen the inconvemences associated to the calculatrons. One of them consists in introducing within the zero-order wavefunction certain parameters properly chosen, which in their turn are optimized in accordance with the variational principle [4,34-361. Thus, by way of extremely simple computation schemes [4] one can obtam results with a sumlar (or even larger) accuracy to the SCF ones [ 1,2]. Another simplificatron which has been recently developed [6,7] consrsts m dealing with the SCF equatrons in a semiclassical(X) manner. Thrs procedure allows us to take into consideranon the interaction of states [7], being essentially a semiclassical limit of the configuration-interaction method. The purpose of this communication is to present a new way for simplifying SCF calculations. Our method consists in treating the SCF equations by means of the perturbation theory with the purpose of obtaining analytical expressions for the energy associated to the different states. The perturbation corrections are deduced in a very simple form by application of the hypervirial-perturbative method (I-Phi). Thhnmethod was proposed by Swenson and Danforth [37] and Krhingbeck [38]. Owing to the fact that the HPM was developed for one-dimensional quantum systems [37] and central field systems [38], our application to coupled oscillators (via the SCF theory) represents a generahzation of it.
2. The method In general, our method is apphcable to any set of M coupled oscrllators whose potential has the following form: P g
M
6f
M
1c, 1
,T2
-mig,
2
vi,i2.
isxilxi2-
xis*
pa
?*
(0
s
43 0 009-2614/82/0000~000/$02.75
0 1982 North-Holland
Volume9,.
CHEMICAL PHYSlCS LETTERS
number 1
8 October
1982
However, in order to simphfy the wnting, we have chosen the H&ton--Heiles hamdtonian (HHh) [39] as ilhrstrative example: H = -$Gja.d
+ azja$)
t f +2
+ ; yiy2 t A,$
t r2x3
(2)
This simple two-dimensional system has been widely studied and Its eigenvalues are known up to a large degree of accuracy [2,10-12,15-17,19-21,25,30,33]. As a particular case, when n = 0 we obtain the Barbanis hamiltoman (Bh) [IO,1 1,14,17,20,27,40] which ~OSS~SSS exact SCF solutions [4,5,36]. The SCF method consists in calculating the extremum of the energy functional E(9) = WIHW,
(3)
*@,_I’) = @t(x) tiz(j’),
(4)
wtth respect to the whole set of vanations m the funcuons @t(x), Q$y) with the condition (4 14) = 1.
i= 1,2.
(5)
This procedure leads to a system of integro-differential equations, which have to be solved via an iterative method [l-3]: (64
where F, and F2 are the SCF operators and Xtkj = (4,
Y(k)
I.K”$$,
(7)
= cQzlykf$zl.
After the resolution of eqs. (6). the energy is calculated in a very sunple way wtth the formula ~=2~t~~_-(lj~(~)-~~(3).
(8)
Recently, Femdndez and Castro [36] have demonstrated that the SCF function \Ir satisfies all the dragonal hypervinal relanonzlups [al] that uwolve one-coorchnate hypervirial operators. In particular ([H,xNd/br])=
([F1,x#dldx])=(jH,xN]) = ([F,,xN])= 0,
([H, jlNdId~]>=([F?,yNd/d~])=
(94
([H,~~])=([F~,y~l)=o.
W9
Since
F, and F2 are both one-dimensional hamiltonians which are associated to the Scluikiinger equation (6). we can apply Swenson and Danforth’s method [37]. For this purpose, we get rid of all the derivative terms from eq. (9), and thus we obtain: $ N(N - l)(N - 2)JP-J) i,V(N_ 1)(&3)y’“-s’
t We,X(N-‘)
- wT(N t l)X(N+‘) - (2N + l)XX(NW
+7_/ve2yt~-tLw;(N+
t)y(N+t)_2(N+
- (2/V+ 3)17x(N+2) = 0, (lOa) l)XX(l)y~+I)-~~~y(N-t)X(3) = 0. (lob)
Making double series expansions in X and ;7 for the mean values X(nfj, YtN) and the eigenvalues el, ~1 [42,43]:
E,
=
c c EI”‘h”d,
i=Q
44
j=Q
E2 =
c c E(zi’WlJ,
r=Q I=‘3
(11’3
Volume 92. number 1
CHEMICAL PHYSICS LEITERS
8 October 1982
we deduce from eqs. (10) the following pair of equations r
5
$N(N - l)(N - 2)X;,t-3) + 2N c
c E~“kJ~,~;$ r=Oj-0 S-I
- (2N+3)XS!;)
. -
-(Iv+
SN(N- l)(N - 2)Y (N-3)+ s-l -
W+ 1),$
- CL@ + l&T’)
r
1) c cX$%$~i_,,r_i=O, r=Oj=o
3_N
5 h ‘2’(I ,)y(N-1) ss
I
X;:)Yjy.)I
‘_,
-
_ $(N+
])y'N+t'
s.r
r.r-1
I=0 j=O
,po
(13)
2N J=O 2
r-l
c Y;~-‘)x;~;
j=O
Furthermore, from the application of the Hellmann-Feynman
r_,_,
-
=
o.
(J2b)
theorem [+I] we have:
a~, /ax=caF,laxl=x(l)~(')+Ax(')aY(')/aX I
(13a)
a+hj=caF,ja17)=x(3)+ m(1Jay(2)/aq,
(13b)
a+3 =ta~~/ah)= x(‘)Y(‘)+ hyQ)axfl)lah+ qaxf3)lah,
(13c)
ae,/at,
=(a~?ial7)=~y(')ax(~)Jal7+x(~)+hax(~)/a~.
(13d)
Therefore we arrive at the following equations: s-l
r (141)
(14b)
(14c)
Eqs. (12) and (14) plus the normalization comhtion (15) a quick and simple manner, the whole set of perturbative corrections. Thus, we deduce for the total energy Emln2 the following approximate expression:
allow USto obtain, in
E n1nz = (II1 + $)w, + (n2 + ;,w, - 3(“, + i)(n2 + {)A&+,
- ($ •t 9’A2/2w:w; .
- $&J,
t 1_)’ + %]/3w; (16)
The form of this equation (i.e. a power series expansion in (n, t :), I = 1,2) is slrmlar (with differences in the coefficients) to that given by Swimm and Delos [33] applying the Birkhoff-Gustavson normal form. The difference between both series expansions rests on the fact that ours approximates the SCF energy levels, while Swimm
and Delos’ approximates the semiclassical levels. 45
Volums 92. number I
CHEMICAL PHYSICS LETTERS
8
October 1982
The calculation scheme is more simple for the Bh [q = 0 in eq. (2)] owing to the zero value for the parameter q. A power series expanston as stated previously, give us:
Unbke the previous case, now the solutions of the Bh can be obtained by using just two HR: (IH, d/dx I) = 0,
(18)
(IH,y d/dyl)= 0.
Thts inreresting feature takes place for any hamiltonian operator whose associated SCF operators contain only linear and quadratrc terms, because in these cases the SCF solutions are derived exactly and they are determined by a finite number of HR [36].
3, Results The stmcture ot‘eqs. (16) and (17) shows us clearly that the accuracy of our results must decrease for large quantum numbers and large values of the coupling parameters. Especially, eq. (16) takes into account only first and second order corrections, so that it will be valid for the lowest energy levels within the range of weak couplings. In table 1 we present the SCF eigenvalues corresponding to the tint three levels of the HHb for different X and q values. We can apprecrate that our results (En”,+ ) make up a good approximation to the numerical ones (I$, na) ]2] whenever h and ~7values remain small. We have included the independent osctllators eigenvahres (Ef,:,) in order to know the extent of the interaction among the chosen oscillators. Table 2 shows the error percent (a) of our perturbative SCF results @:,,a) with respect to the numerical values (f$‘,,r,) for the first sixteen HHh energy levels. Although a few perturbative terms were considered (second order in IJ and fourth order III A) the error is less than 1.7%. Bringing tius section to a close, we compare in table 3 the fust six SCF energy levels of the Bh calculated from eq. (17) (Ei,,2) with regard to the esact SCF etgenvalues (E,” n, ) and the independent osctlfators model ones (E,““,,).Wesee that the accuracy of eq. (17) is large enough for ihe X, wt and wz values considered in the present calckion.
Table 1 SCF
~pproumn~mn t-orthe l-rustthree levelsof HHh -?.
n,
0 06 0.08 0 10 0 12 0 14
=I+
=Q
n = -A’, w I
n, = b
q
0 7. wz = 1 3 ‘1
nt = O.n2 = 1
l.??, = 0 b
Eoo
Gil
El0
4
EOI
0 99902 0.99794 0.99623 0 99372 0.99020
0 99902 0 99793 0 99619 0.99359 0.98984
1.69789 169493 1.68974 168143 166897
1.69788 169489 1.68954 168070 1.66671
2.29392
2.29390
2.28841 2.28061 2.27014 2.25660
2.18832 2.28028 2.26915 2.25103
E,“,” = 1.7 3) Ea,,,nz
46
irom
eq. (16)
b
Gl
(sdudm$
tie
WRI
m ?.’
of eq. (17).
E,b ,“=.
from
ref. [21.
Table 2 SCF approuimauon for the fust stxteen levelr of HHh. A = -0.1116. r) = -0.00939, w; = 0 19375. w; = 2.12581 a)
0 I 2 0 3 1 2 0 3 1 2 0 3 1 2 3
8 October
CHEMICAL PHYSICS LETTERS
Volume 92. number I
0
0.9927
0.992410
0 0 1 0 1 1 2 1 2 2 3 2 3 3 3
I 5’03 2.0401 2 4236 2.5524 2.9376 3.4439 3 8332 3.9426 43337 4 8265 5.2209 5 3116 5.7079 6.1871 66581
1519022 2.036567 2.421187 2.544272 2.93 1229 3 430472 3.825223 3.917662 4316161 4.793862 5.201049 5.256299 5.669154 6 120199 6549758
0 03 0.08 0 17 0.10 0.32 0.22 0.39 0.2 1 0.64 041 068 0.38 1.05 0.68 1.09 1.66
1 .ooooo 1 1541989 2.083916 2 458017 2.625963 3.0000Ll~ 3.541991 3 916033 4.083978 4.458020 5 000007 5.374049 5.541994 5.916036 6.458023 7.000010
1982
Table 3 SCF approulmation for the fust SLYlevels of Bh. A = -0.08, w, = 1.6. w2 = 0.g3)
0
0
0 1 0 1 2
: 2 I 0
1.1062403 2 05OU48 2 3711513 2.9901084 3.3153559 3 6360624
a) Ea n,n2 irom eq. (17).Ei,,,2
1.1062403 2.0501UB 2.3711513 2.9901082 3 3153559 3 6360624
1.106797 2.055480 2.371708 3.00-I 164 3.32G391 3.636619
exact SCF, rrf. [Sl.
a)Eanrn, from eq. (16) (mcludmg the term m A4 of eq (17)). Eb- - from ref. 131. b),??100(E,3,,2
- E;,,,)/E;,,,.
KIllI
_
c) Zerolh
order e~gcrt-
4. Conclusions
We deem that the method presented in this paper is interesting and possesses a marked usefulness because its application is simple enough and it enables us to obtain analytic expressions for the elgenvalues and the matrix elements of the coordinate moments. In those cases where it is necessary to calculate a larger number of terms, this Iabor can be made numerically, because the recursion formulas (12) and (14) are programmed easily. For the HHh, It is possible to set aside the use of the double perturbation theory when the coupbng potential V,(x,y) is written in itsusual form [l-4,10-13,15-17,19-21,25,29-31,331 V&y)
= A($+
QX3) )
(1%
and the expansions (1 I) are replaced by power series in a sole parameter A [42]. For those systems with many freedom degrees, whose potentials have the general form (I), both alternatives are feanble: multiple perturbation theory [43], and a power series expansion UIa sole parameter [42].
References [ 11 J.M. Bowman, J. Chem. Phys. 68 (1978) 608. 12) M. Cohen, S Creita and R.P. McEachran. Cbem Phys Letters 60 (1979) 445. [3] T.L.Tobm and J.M. Bowman, Chem. Phys. 47 (1980) 151. [4] S. Maluendes, G. Arteca, F.M. retindez and E.A. Qsuo. Mol. Phys 45 (1982) 511. (51 FM. Femindez, G.A. Artem, S.A. Maluendes, H.O. Vdlar and E A Castro, Rev. Erasd. l-is, to bcpubhshed [6] R.B. Gerber and MA. Ratner. Chem. Phys. Letters 68 (1979) 195. [71 MA Ratncr. V. Buch and R.B. Gerber, Chem. Phys. 53 (1980) 345.
Volume 92. number
1
CHEMICAL PHYSICS LETTERS
[S] G.D. Camry, L L. Sprandel and C.W. Kern, Advan
[9] 1101 i 1 l] 112) [ 131 (I41 [ 151 [ 161 1171 f 181 [ 19) [ZOj 1211 122j [ 231 1241 1251 1261 1271
[ISI 129) 1301 [31] 1321 1331 f 341 f35] f361 1371 [3B] 1391 1401 j41 j I-121 [a31 [44l
48
Chem. Phys. 37 (1978) 305. I C. PcrsivaJ. J. Phys. B 6 (1973) L229. KS J Nordholm and S A. RICC.J. Chem. Phys. 61 (1974) 203. K.SJ. NordhoIm and S.A. Rice. J. Chem. Phys. 61 (1974) 768. W. Eastes and R.A Marcus, J. Chem Phys 61(1974) 4301. N. Pomphrey, J. Phys. B 7 (1974) 1909. R S.J. Nordhohn and S A I&e. J. Chem. Phys 62 (1975) 157. D.W. Nold and R A. Marcus, J. Chem. Phys. 62 (1975) 2119. S. Chapman. B G. Garrett and W H. MdJcr. J. Chem. Phys. 64 (1976) SO?. K S.Sorbm, hlol. Phys. 32 (1976) 1577. 1 C. PercrvaJ and N. Pompbrey, Mol. Phys 31 (1976) 97. D.W. Noid and R A. hfarcus, J. Chem Phys 67 (1977) 559. KS. Sorbic and NC. Handy, kfol. Phys. 33 (1977) 1319. D.W. Notd. h¶ L. Koszykowski and R.A. Marcus, J. Chem Phyr 67 (1977) 404 hl.G Bucknell, N C. Handy and S F. Boys, bfol Phys. 28 (1974) 7.59. hi G. Bucknell and N C Handy, hlol Phys 28 (1974) 777. K D Hansel, Chem Phys. Letters 57 (1978) 619 hl J. Davrsand E-l. HclJer, J.Chem. Phys. 71(1979) 3383. SW. McDonaJd and A N. Kaufman, Phys Rev. Letters42 (1979) 1189. R.A¶ Suatt, NC Handy and W H MJIer, J. Chem. Phys. 71 (1979) 3311. R A. Harris. J. Chem. Phys 72 (1980) 1776. M. Duff, H Rabrtz. A Askar, A Wtmak and M Ablowbz, I. Chem. Phys 72 (1980) 1543 J.B Delos and R.T. Swimm, Chem Phys. Letters 47 (1977) 76. R A. Pullen and AR. Edmonds. J. Phys A 13 (1981) L319. Y F. Charm, hi Tabor. J. Welssand G.Corhss, Phys. Letters85A (1981) 211. R.T. Swunmand J.B. Delos. J.Chem. Phys. 71 (1979J1706. F.hl Fernindez and E A. Castro. Rev. Bnsd. Fis. 11 (1981) 145. E.A Castro and F-hi. Fernindez, Lertere Nuovo Cunento 28 (1980) 484. FM. Femrindu and EA. Castro, Cbem. Phys 58 (1981) 65. RJ Swenson and S H. Danfonh. J. Chem Phys. 57 (1972) 1734 J. Ktingbeck. Phys Letrers65A (1978) 87. hl. H&on and C. Hedes, Astronom. J. 69 (1964) 73. B. Barbann, Asuonom. J 71 (1966) 415. J.O. Huschfelder. J. Cbem. Phys. 33 (1960) 1462. G A Arreca. FM. Fernindez and E.A. Castro, Phys Letters 86A (1981) 133. A. Requem. R. Peiia and A. Serna. An&s de Qufm 77 (1981) 231. S.T. EpStf~% neorrt. Calm. Acta 55 (1980) 51, and references therem
8 October
1982
1
CHEMICAL PHYSICS LEI-I-ERS
HYPERMRIAL-PERTURBATIVE
8 October 1981
APPROXIMATION TO THE SCF EIGENVALUES
OF COUPLED OSCILLATORS
Gustavo A. ARTECA, Francisco hf. FERNhlDEZ and Eduardo A. CASTRO IIW~~. Szccidn Quimlcu Tedrrca, Sucursal4, Gsdh de correo 16. Ln PInta 1900. Agentma Recewed 7 June 1982
The hypenina relstions and perturbatton theory are used in order to obtain analytwl non-numertal e\prewons ior the SCF rlgenvalues oi coupled oscdlxors
1. Introduction Lately, the study of quantum properties of polyatomic molecules has given nse to a growing interest among investigators of non-separable systems. Owmg to thrs, at present there are several methods which allow one to obtam, with different degree of accuracy, the ergenfunctions and ergenvalues of coupled oscillators [l-36]. The self-consrstent field (SCF) theory [l-8] is one of the simplest approximations which has been used to study tius problem. Notwithstanding its sunplicity, the difficulties of the SCF method grow markedly with the increase of quantum numbers (because the functions become very oscillatory) and the degrees of freedom of the system. There have been proposed various alternative procedures in order to lessen the inconvemences associated to the calculatrons. One of them consists in introducing within the zero-order wavefunction certain parameters properly chosen, which in their turn are optimized in accordance with the variational principle [4,34-361. Thus, by way of extremely simple computation schemes [4] one can obtam results with a sumlar (or even larger) accuracy to the SCF ones [ 1,2]. Another simplificatron which has been recently developed [6,7] consrsts m dealing with the SCF equatrons in a semiclassical(X) manner. Thrs procedure allows us to take into consideranon the interaction of states [7], being essentially a semiclassical limit of the configuration-interaction method. The purpose of this communication is to present a new way for simplifying SCF calculations. Our method consists in treating the SCF equations by means of the perturbation theory with the purpose of obtaining analytical expressions for the energy associated to the different states. The perturbation corrections are deduced in a very simple form by application of the hypervirial-perturbative method (I-Phi). Thhnmethod was proposed by Swenson and Danforth [37] and Krhingbeck [38]. Owing to the fact that the HPM was developed for one-dimensional quantum systems [37] and central field systems [38], our application to coupled oscillators (via the SCF theory) represents a generahzation of it.
2. The method In general, our method is apphcable to any set of M coupled oscrllators whose potential has the following form: P g
M
6f
M
1c, 1
,T2
-mig,
2
vi,i2.
isxilxi2-
xis*
pa
?*
(0
s
43 0 009-2614/82/0000~000/$02.75
0 1982 North-Holland
Volume9,.
CHEMICAL PHYSlCS LETTERS
number 1
8 October
1982
However, in order to simphfy the wnting, we have chosen the H&ton--Heiles hamdtonian (HHh) [39] as ilhrstrative example: H = -$Gja.d
+ azja$)
t f +2
+ ; yiy2 t A,$
t r2x3
(2)
This simple two-dimensional system has been widely studied and Its eigenvalues are known up to a large degree of accuracy [2,10-12,15-17,19-21,25,30,33]. As a particular case, when n = 0 we obtain the Barbanis hamiltoman (Bh) [IO,1 1,14,17,20,27,40] which ~OSS~SSS exact SCF solutions [4,5,36]. The SCF method consists in calculating the extremum of the energy functional E(9) = WIHW,
(3)
*@,_I’) = @t(x) tiz(j’),
(4)
wtth respect to the whole set of vanations m the funcuons @t(x), Q$y) with the condition (4 14) = 1.
i= 1,2.
(5)
This procedure leads to a system of integro-differential equations, which have to be solved via an iterative method [l-3]: (64
where F, and F2 are the SCF operators and Xtkj = (4,
Y(k)
I.K”$$,
(7)
= cQzlykf$zl.
After the resolution of eqs. (6). the energy is calculated in a very sunple way wtth the formula ~=2~t~~_-(lj~(~)-~~(3).
(8)
Recently, Femdndez and Castro [36] have demonstrated that the SCF function \Ir satisfies all the dragonal hypervinal relanonzlups [al] that uwolve one-coorchnate hypervirial operators. In particular ([H,xNd/br])=
([F1,x#dldx])=(jH,xN]) = ([F,,xN])= 0,
([H, jlNdId~]>=([F?,yNd/d~])=
(94
([H,~~])=([F~,y~l)=o.
W9
Since
F, and F2 are both one-dimensional hamiltonians which are associated to the Scluikiinger equation (6). we can apply Swenson and Danforth’s method [37]. For this purpose, we get rid of all the derivative terms from eq. (9), and thus we obtain: $ N(N - l)(N - 2)JP-J) i,V(N_ 1)(&3)y’“-s’
t We,X(N-‘)
- wT(N t l)X(N+‘) - (2N + l)XX(NW
+7_/ve2yt~-tLw;(N+
t)y(N+t)_2(N+
- (2/V+ 3)17x(N+2) = 0, (lOa) l)XX(l)y~+I)-~~~y(N-t)X(3) = 0. (lob)
Making double series expansions in X and ;7 for the mean values X(nfj, YtN) and the eigenvalues el, ~1 [42,43]:
E,
=
c c EI”‘h”d,
i=Q
44
j=Q
E2 =
c c E(zi’WlJ,
r=Q I=‘3
(11’3
Volume 92. number 1
CHEMICAL PHYSICS LEITERS
8 October 1982
we deduce from eqs. (10) the following pair of equations r
5
$N(N - l)(N - 2)X;,t-3) + 2N c
c E~“kJ~,~;$ r=Oj-0 S-I
- (2N+3)XS!;)
. -
-(Iv+
SN(N- l)(N - 2)Y (N-3)+ s-l -
W+ 1),$
- CL@ + l&T’)
r
1) c cX$%$~i_,,r_i=O, r=Oj=o
3_N
5 h ‘2’(I ,)y(N-1) ss
I
X;:)Yjy.)I
‘_,
-
_ $(N+
])y'N+t'
s.r
r.r-1
I=0 j=O
,po
(13)
2N J=O 2
r-l
c Y;~-‘)x;~;
j=O
Furthermore, from the application of the Hellmann-Feynman
r_,_,
-
=
o.
(J2b)
theorem [+I] we have:
a~, /ax=caF,laxl=x(l)~(')+Ax(')aY(')/aX I
(13a)
a+hj=caF,ja17)=x(3)+ m(1Jay(2)/aq,
(13b)
a+3 =ta~~/ah)= x(‘)Y(‘)+ hyQ)axfl)lah+ qaxf3)lah,
(13c)
ae,/at,
=(a~?ial7)=~y(')ax(~)Jal7+x(~)+hax(~)/a~.
(13d)
Therefore we arrive at the following equations: s-l
r (141)
(14b)
(14c)
Eqs. (12) and (14) plus the normalization comhtion (15) a quick and simple manner, the whole set of perturbative corrections. Thus, we deduce for the total energy Emln2 the following approximate expression:
allow USto obtain, in
E n1nz = (II1 + $)w, + (n2 + ;,w, - 3(“, + i)(n2 + {)A&+,
- ($ •t 9’A2/2w:w; .
- $&J,
t 1_)’ + %]/3w; (16)
The form of this equation (i.e. a power series expansion in (n, t :), I = 1,2) is slrmlar (with differences in the coefficients) to that given by Swimm and Delos [33] applying the Birkhoff-Gustavson normal form. The difference between both series expansions rests on the fact that ours approximates the SCF energy levels, while Swimm
and Delos’ approximates the semiclassical levels. 45
Volums 92. number I
CHEMICAL PHYSICS LETTERS
8
October 1982
The calculation scheme is more simple for the Bh [q = 0 in eq. (2)] owing to the zero value for the parameter q. A power series expanston as stated previously, give us:
Unbke the previous case, now the solutions of the Bh can be obtained by using just two HR: (IH, d/dx I) = 0,
(18)
(IH,y d/dyl)= 0.
Thts inreresting feature takes place for any hamiltonian operator whose associated SCF operators contain only linear and quadratrc terms, because in these cases the SCF solutions are derived exactly and they are determined by a finite number of HR [36].
3, Results The stmcture ot‘eqs. (16) and (17) shows us clearly that the accuracy of our results must decrease for large quantum numbers and large values of the coupling parameters. Especially, eq. (16) takes into account only first and second order corrections, so that it will be valid for the lowest energy levels within the range of weak couplings. In table 1 we present the SCF eigenvalues corresponding to the tint three levels of the HHb for different X and q values. We can apprecrate that our results (En”,+ ) make up a good approximation to the numerical ones (I$, na) ]2] whenever h and ~7values remain small. We have included the independent osctllators eigenvahres (Ef,:,) in order to know the extent of the interaction among the chosen oscillators. Table 2 shows the error percent (a) of our perturbative SCF results @:,,a) with respect to the numerical values (f$‘,,r,) for the first sixteen HHh energy levels. Although a few perturbative terms were considered (second order in IJ and fourth order III A) the error is less than 1.7%. Bringing tius section to a close, we compare in table 3 the fust six SCF energy levels of the Bh calculated from eq. (17) (Ei,,2) with regard to the esact SCF etgenvalues (E,” n, ) and the independent osctlfators model ones (E,““,,).Wesee that the accuracy of eq. (17) is large enough for ihe X, wt and wz values considered in the present calckion.
Table 1 SCF
~pproumn~mn t-orthe l-rustthree levelsof HHh -?.
n,
0 06 0.08 0 10 0 12 0 14
=I+
=Q
n = -A’, w I
n, = b
q
0 7. wz = 1 3 ‘1
nt = O.n2 = 1
l.??, = 0 b
Eoo
Gil
El0
4
EOI
0 99902 0.99794 0.99623 0 99372 0.99020
0 99902 0 99793 0 99619 0.99359 0.98984
1.69789 169493 1.68974 168143 166897
1.69788 169489 1.68954 168070 1.66671
2.29392
2.29390
2.28841 2.28061 2.27014 2.25660
2.18832 2.28028 2.26915 2.25103
E,“,” = 1.7 3) Ea,,,nz
46
irom
eq. (16)
b
Gl
(sdudm$
tie
WRI
m ?.’
of eq. (17).
E,b ,“=.
from
ref. [21.
Table 2 SCF approuimauon for the fust stxteen levelr of HHh. A = -0.1116. r) = -0.00939, w; = 0 19375. w; = 2.12581 a)
0 I 2 0 3 1 2 0 3 1 2 0 3 1 2 3
8 October
CHEMICAL PHYSICS LETTERS
Volume 92. number I
0
0.9927
0.992410
0 0 1 0 1 1 2 1 2 2 3 2 3 3 3
I 5’03 2.0401 2 4236 2.5524 2.9376 3.4439 3 8332 3.9426 43337 4 8265 5.2209 5 3116 5.7079 6.1871 66581
1519022 2.036567 2.421187 2.544272 2.93 1229 3 430472 3.825223 3.917662 4316161 4.793862 5.201049 5.256299 5.669154 6 120199 6549758
0 03 0.08 0 17 0.10 0.32 0.22 0.39 0.2 1 0.64 041 068 0.38 1.05 0.68 1.09 1.66
1 .ooooo 1 1541989 2.083916 2 458017 2.625963 3.0000Ll~ 3.541991 3 916033 4.083978 4.458020 5 000007 5.374049 5.541994 5.916036 6.458023 7.000010
1982
Table 3 SCF approulmation for the fust SLYlevels of Bh. A = -0.08, w, = 1.6. w2 = 0.g3)
0
0
0 1 0 1 2
: 2 I 0
1.1062403 2 05OU48 2 3711513 2.9901084 3.3153559 3 6360624
a) Ea n,n2 irom eq. (17).Ei,,,2
1.1062403 2.0501UB 2.3711513 2.9901082 3 3153559 3 6360624
1.106797 2.055480 2.371708 3.00-I 164 3.32G391 3.636619
exact SCF, rrf. [Sl.
a)Eanrn, from eq. (16) (mcludmg the term m A4 of eq (17)). Eb- - from ref. 131. b),??100(E,3,,2
- E;,,,)/E;,,,.
KIllI
_
c) Zerolh
order e~gcrt-
4. Conclusions
We deem that the method presented in this paper is interesting and possesses a marked usefulness because its application is simple enough and it enables us to obtain analytic expressions for the elgenvalues and the matrix elements of the coordinate moments. In those cases where it is necessary to calculate a larger number of terms, this Iabor can be made numerically, because the recursion formulas (12) and (14) are programmed easily. For the HHh, It is possible to set aside the use of the double perturbation theory when the coupbng potential V,(x,y) is written in itsusual form [l-4,10-13,15-17,19-21,25,29-31,331 V&y)
= A($+
QX3) )
(1%
and the expansions (1 I) are replaced by power series in a sole parameter A [42]. For those systems with many freedom degrees, whose potentials have the general form (I), both alternatives are feanble: multiple perturbation theory [43], and a power series expansion UIa sole parameter [42].
References [ 11 J.M. Bowman, J. Chem. Phys. 68 (1978) 608. 12) M. Cohen, S Creita and R.P. McEachran. Cbem Phys Letters 60 (1979) 445. [3] T.L.Tobm and J.M. Bowman, Chem. Phys. 47 (1980) 151. [4] S. Maluendes, G. Arteca, F.M. retindez and E.A. Qsuo. Mol. Phys 45 (1982) 511. (51 FM. Femindez, G.A. Artem, S.A. Maluendes, H.O. Vdlar and E A Castro, Rev. Erasd. l-is, to bcpubhshed [6] R.B. Gerber and MA. Ratner. Chem. Phys. Letters 68 (1979) 195. [71 MA Ratncr. V. Buch and R.B. Gerber, Chem. Phys. 53 (1980) 345.
Volume 92. number
1
CHEMICAL PHYSICS LETTERS
[S] G.D. Camry, L L. Sprandel and C.W. Kern, Advan
[9] 1101 i 1 l] 112) [ 131 (I41 [ 151 [ 161 1171 f 181 [ 19) [ZOj 1211 122j [ 231 1241 1251 1261 1271
[ISI 129) 1301 [31] 1321 1331 f 341 f35] f361 1371 [3B] 1391 1401 j41 j I-121 [a31 [44l
48
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1982