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Perturbational and variational treatments of the Morse oscillator
67
Chemical Physics 112 (1987) 67-75 North-Holland, Amsterdam
PERTURBATIONAL
AND VARIATIONAL
TREATMENTS
OF THJI MORSE OSCILLATOR
I.L. COOPER School of Chemistry,
The University, Newcastle-upon-Tyne
Received
1986
9 October
NEI 7RU, UK
The Morse oscillator hamiltonian is expressed as an infinite expansion in powers of a natural perturbation parameter, the square root of the anharmonicity constant, relative to the simple harmonic oscillator as zeroth-order hamiltonian. A transformation of variables leads to a hamiltonian which involves terms no higher than second order in this natural perturbation parameter. In both cases, the exact bound state eigenvalues of the Morse oscillator are given by second-order perturbation theory. The S&r&linger equation corresponding to the transformed Morse hamiltonian is solved variationally, via a complete set expansion in simple harmonic oscillator eigenstates. Accurate approximations to the exact eigenvalues and eigenfunctions of bound states of the Morse oscillator can be obtained for all but the very highest levels.
1. Introduction There is considerable current interest in the application of the Morse oscillator model to the investigation of local vibrational modes, with particular emphasis on those highly excited vibrational levels which are accessible by laser spectroscopy [l]. In a recent study of the relation between local and normal modes, Mills and Robiette [2] noted that a perturbational analysis of the Schrbdinger equation corresponding to the Morse potential, when based on an expansion in powers of the relative vibrational coordinate, reproduced the exact bound state energy eigenvalues when the potential was truncated at the quartic term and when the quartic and cubic terms were treated by first- and second-order perturbation theory respectively. The present paper is designed to investigate the perturbational approach to the Morse oscillator based upon the simple harmonic oscillator as zeroth-order hamiltonian. It will be shown that the square root of the anharmonicity constant acts as a natural perturbational parameter and that the exact bound state eigenvalues appear exactly at second order in this parameter. This result, which becomes obvious with hindsight, does not appear to be well known. A simple transformation of variable will be shown to lead to yet another form of the Morse hamiltonian, which depends on both position and momentum coordinates in the new variable. This form of the Morse hamiltonian, which does not appear to have been previously identified, is amenable to a similar perturbational expansion in the square root of the anharmonicity and again reproduces the exact bound state eigenvalues at second order. The transformed hamiltonian can also be rewritten in terms of the familiar raising and lowering operators appropriate to the simple harmonic oscillator in the transformed variable. A variational approach to the Morse oscillator problem using a basis of simple harmonic oscillator eigenstates becomes feasible since all necessary matrix elements can be calculated in a straightforward manner using the well-known algebra of the simple harmonic oscillator. This provides a computational approach to the generation of accurate approximations to the exact Morse eigenstates which may prove to be more convenient than the exact solutions for certain applications. 0301-0104/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
I.L. Cooper / Perturbational
68
approach to the Morse oscillator
2. Preliminaries
We shall find it convenient to express the Morse hamiltonian depends solely on the harmonic vibrational frequency and hamiltonian has the form [3]
Hh,f= -
in dimensionless the anharmonicity
coordinates, constant.
such that it The Morse
$ -j$ +D,(l - epux)*,
(1)
where x = R - R, is the displacement from equilibrium, D, is the dissociation energy from equilibrium, (Y is related to the harmonic force constant at equilibrium and /A is the appropriate reduced mass for the vibrational motion of the diatomic molecule represented by the Morse potential. Using the standard definitions, we = (2De,‘#*a
(2)
x, = Ao,/4D,,
(3)
and
we can express
ITM in the form + $1
2P%
Converting
-eeax)*
dx*
to the dimensionless
e coordinate
q,
1.
defined
(4) by
q = (P%/W2X,
(5)
the Morse hamiltonian
becomes -I
H,
= +ho,
- -$
+ &-- (1 - exp[ - (2x,)“*q] e
,‘J,
(6)
where we have used eqs. (2) and (3) to write ax = (,u/2D,)“‘w,( Since x, is generally natural perturbation
&-{le
A/,uc+)“*q
= (h,/2D,)“*q
= (2xe)l’*q,
(7)
small, the exponential in eq. (6) can be expanded in powers of xi/*, which acts as a parameter, to generate an infinite perturbational expansion in the form
exp[ -(2x,)“*q])*
= &[(2x,)“*q-
x,q*
= q* - (2Xey2q3
+ $,(2xe)3’2q3
+ ;x,q4
+ . ..I’
+ ....
(8)
Hence H,-hu.[+(
-$+q*)
-($xe)1’2q3+$xeq4+
. ..I.
where the leading term corresponds to the simple harmonic oscillator Aw,(u + t), where u = 0, 1, 2,. . . . Since the eigenvalues of H, are [3]
(9) hamiltonian,
with
E*=Ao,[(v+f)-x,(u+f)*], we note that only terms quadratic
eigenvalues
(10) in the natural
perturbation
parameter
xi/* can contribute
to the energy.
I. L. Cooper / Perturbational
69
approach to the Morse oscillator
It should be noted at this point that we in eq. (10) denotes the angular harmonic frequency. Customary spectroscopic notation uses w, to denote the harmonic vibrational wavenumber (i.e. Aw, is replaced by hew, in eq. (lo)), while Mills and Robiette [2] use a notation in which tzw, is replaced by hew, and --x,w, by hex,. The remainder of the present paper is devoted to a perturbational and variational analysis of the Schrodinger equation for the Morse oscillator. Section 2 will consider the perturbational analysis based on eq. (9) and section 3 will identify a simple transformation of variables which yields an alternative representation of the Morse hamiltonian, also amenable to a perturbational analysis. Section 4 will treat this transformed hamiltonian variationally using the simple harmonic oscillator eigenstates as basis. The paper will conclude with a discussion of the results, and identification of some possible further applications.
3. Perturbational treatment of the Morse oscillator The Morse hamiltonian, eq. (9) is represented as an infinite perturbational expansion in powers of the hamiltonatural perturbation parameter x,l/2 , based upon the simple harmonic oscillator as zeroth-order nian. The familiar Rayleigh-Schrodinger perturbation theory [4] requires only slight modification to accomodate such an expansion of the hamiltonian, and the approach may be summarised as follows: We express the Schriidinger equations for the Morse oscillator and the simple harmonic oscillator respectively as &l
I 4,) = E” I $“)Y
(11)
H,,Iu)=E,'~'Iu), using an obvious
notation.
(12) The energy
shift A E, for each vibrational
state u is expressible
in the form
AE,~E,-E,c')=(uIH'(\CI,), where H' = HM - Ho and we have assumed
(13) that (u I 4,) = 1. Rewriting
[E,~H,,-H/+(E,-E,Jo))]IJ/,)=o
eq. (11) in the form
04)
yields the identity
I$,)=(E,'"'-Ho)-l(H'-AE,)I$,). Introducing
the projection
operator
(15)
FU by the relation
i” I$“> = I u> and writing
06)
&, = 1 - kU, then
I~“)=(~“+d”hu =tlIu)+
[~u/(E,“‘-Ho)](H’-AE,)I~,)
= I u> + %W’where R,, the reduced R,=
AK)
resolvent
&‘(E,‘“‘-Ho).
I +,>, for state u, is defined
(17) by
(18)
70
I.L Cooper / Perturbational approach to the Morse oscillator
On iteration,
we find
09) and AE,=
f (u(H’[R”(H’-AE,)]“(u). n=O case, with xt/* as the perturbation
In the present H’
cx1/*7/(1)
+
e
v
V(2) +
x
parameter,
we have the following
expansions:
. . .
(21)
e
=X’/*E(‘)+~
AE
(20)
e
e
u
EC*)+
. . .
(22)
”
where ~/cl)/h~~
=
_
2-112~3,
and so on. Hence, substituting parameter, we find
v(*)/~O,
=
hi44
(23)
eqs. (21) and (22) into
eq. (20), and equating
like powers
of the perturbation
E,(')=(vlV('+)=O,
(24)
E,'*' = (uIV(*) Iu)+ (u(V'(')R,V(') Iu) =hw,[~(u*+u+~)-_( u*+u+ g>] = -Ao,(u+ g', where we have utilised order in xi/*, we have
standard
integrals
over simple
harmonic
oscillator
(25) eigenstates.
Hence,
E,=E,'"'+~','2E,"'+xeE;2'=~w,(u+~)[1-x,(u+~)].
to second
(26)
This is the exact energy eigenvalue expression for bound states of the Morse oscillator, and so all higher order terms provide identically vanishing contributions to the energy. (All energy corrections of odd order vanish by symmetry; those of even order higher than second become increasingly complex, although ultimately non-contributing. This can be checked out at fourth order, where it is observed that the disconnected and connected terms identically cancel. This must also be the case for higher-order even terms.) We have thus identified the origin of the result of Mills and Robiette [2], who treated the term in q3 by second-order perturbation theory, and the term in q4 by first-order perturbation theory, while dropping all higher terms in the expansion of the potential. In view of the assumption of the so-called harmonically coupled anharmonic-oscillator model [5] that the coupling between bond modes may be approximated, at least for small values of vibrational quantum number, by interactions between the corresponding harmonic oscillator states, it is of interest to examine the first-order corrections to the wavefunction. From eq. (19) we note that
(27) where the prime on the summation
symbol
denotes
that the term n = u is to be excluded.
I. L. Cooper / Perturbational
Since v(l) a q3, only terms involving
71
approach to the Morse oscillator
n = u * 1 and n = u _+ 3 are non-vanishing,
and we find that
~,@)=~[(~+1)(~+2)(~1+3)]“~~~+3)+~(u+l)~’~~~+l) -~~“‘~u-1)-~[~(~-1)(u-2)]“~~~-33).
(28)
If we choose x, to be 0.023, which is a typical value for XH diatomic approximated to first order in the perturbation parameter as I$,)
molecules,
then the lowest states are
= IO) + 0.114 11) + 0.03 13),
I I//,) = -0.114 I$,)=
IO) + 11) + 0.322 12) + 0.062 14),
-0.32211)+
~2)+d.591~3)+0.098~5),
14,) = -0.031
IO) - 0.59112)
I+,)
11) - 0.910 13) + 14) + 1.272 IS) + 0.18317).
= -0.062
+ 13) + 0.910 14) + 0.138 16), (29)
Note that the first-order correction overrides the zeroth-order term when u = 4, indicating that the harmonic coupling of anharmonic bond modes will only model the behaviour of the very lowest eigenstates, and fails to provide an adequate description of the more highly anharmonic states. This matter will be discussed further elsewhere.
4. Transformation
of variables
The Morse hamiltonian, represented by eq. (6), has been analysed using the expanded form displayed in eq. (9). This infinite expansion is amenable to a perturbational treatment based on the simple harmonic oscillator as zeroth-order hamiltonian, and the exact energy eigenvalues are obtained when the energy is evaluated to second order in the square root of the anharmonicity parameter. The fact that this hamiltonian expansion is an infinite one is unsatisfactory, and it is of interest to seek a transformation of variables such that the resultant transformed hamiltonian contains only a finite number of terms. This objective may be achieved in the following manner. We define a new variable, Q, by the relation Q = (2~,)-“~(
1 - exp[ - (2x,)“2q]
where we note that Q + q in the harmonic
), limit,
(30) x, + 0. The conjugate
momentum,
d p= is related 2
by
(31)
-% to the original
P, defined
momentum,
= exp[ - (2x,)“2q]
p = -id/dq,
as follows:
Since (32)
= 1 - (~x~)~‘~Q,
we have
=f[P(1-t2~,)“~Q)+(1-(2x,)“~Q)P]
=P-($Xe)1’2[Q,
p]+,
(33)
where
IQ,
PI+= QP+PQ
(34)
I.L. Cooper / Perturbational approach io the Morse oscillator
12
represents momentum
the anticommutator of P and Q, and the symmetrisation operator retains its hermitian character. Since
f’,[Q, PI+ ] + + b,[Q,
P* = P* - (b,)“‘[
ensures
that
PI:
the
transformed
(35)
we have H&w,
= 4 ( P*
+Q~)=~(~~+Q~)-t(t~,)~‘~[~~[Q,
p]+]++ix,[Q,
PI:.
(36)
Eq. (36) represents the desired expansion, in which the transformed Morse hamiltonian now contains terms no higher than quadratic in the natural perturbation parameter, xi/*, and the correction terms to the zeroth-order hamiltonian now depend on both position and momentum. It should be noted at this point that we have been treating s-state solutions of the full three-dimensional Morse oscillator. The coordinate q thus ranges from 0 to infinity, and this implies, from eq. (30) that Q if aR, = In 2, ranges from - (2xe)-l/*(eaRe - 1) to +(2x )-l/*. This range will only be symmetrical whereas in general we must have aR, < In 5 so that the potential function becomes repulsive at small internuclear separations. Since we are only concerned here with bound state solutions we may consider Q to lie in the symmetrical range -(2x,)-‘/* G Q G (2x,)-‘/*. with respect to the harmonic potential given by V(Q)/Ao,
= +Q’.
(37)
As pointed out by ter Haar [6], the analytical solutions obtained by Morse [3] are only approximate since they fail to exhibit the correct boundary condition at q = 0 (viz. the wavefunctions must vanish there). The error ought to be negligible since it corresponds physically to penetration of the (finite) potential barrier at small internuclear separations, and the resulting wavefunctions should decay rapidly to zero. We can investigate the validity of the perturbational approach by noting that the number of harmonic oscillator states with energy less than the dissociation limit is given by u0 + 1 where u0 + $5 1/4x,, i.e. the number of bound harmonic levels is Int(l/4x, + i). No bound states exist if 1/2x, 2 1, corresponding physically to the relation D,/( iho,)
5 I
(38)
and is equivalent to one of the conditions quoted solutions become appreciable. Finally, we note that Us + 1, where the vibrational quantum number Us is Int (1/2x, + 3) and no bound state exists if xi’ D,/( $Aw,) 5 I.
by ter Haar [6] at which deviations from the rigorous the number of bound Morse state solutions is given by satisfies uh? + f ,< 1/2x,. The number of bound states 5 1, i.e. (39)
We note that the number of bound states in zeroth order is approximately half the number of bound states supported by the Morse potential and this confirms that the perturbational approach must rapidly break down with increasing value of the vibrational quantum number. Although the perturbational treatment of the transformed Morse hamiltonian, eq. (36), may be carried out in terms of the variables P and Q, we shall find it convenient to transform to the familiar raising and lowering operators of the simple harmonic oscillator, i.e. A = 2_l/*(Q A+=
2_l/*(Q
+ iP),
(40)
- iP),
(41)
such that [A, A+] =/t/I+-A+A
= 1.
(42)
I. L. Cooper / Perturbational
(Note that the ground exp( - Q ‘/2).)
state
of this harmonic
73
approach io the Morse oscillator
oscillator
satisfies
the relation
A IO) = 0 so that
10) a
Hence
H,/liw,
= Ho + x;“~H~ + x, H2,
(43)
where Ho= +(P’+ Hi=
Q’) =A+A+
-2-3/2[P,[Q,
+,
(44)
P]+]+=f(A3+(~+)3-A+~A+-~~+~),
H,=+[Q, P]:=$[-i(AZ-(A+)2)]2= The energy, E,/Aw,
correct
(45)
-$(A4+(~+)4-A2(~+)2-(A+)2~2).
to second order in the perturbational
parameter,
(46)
becomes
5: E,“) + x;‘~EU(~)+ x, Ei2’,
(47)
where E’“)=(o~Ho~u)=u+f,
(48)
E,“’ = (u I HI I u) = 0,
(49)
E,‘2’=(~~H2~~)+(~~HIR,H~~u)=~(u2+u+1)-~(u2+u+~)=
-(u++)‘.
(50)
Hence E,/Ao,-(u-t
+)[I-x,(u+
$)I,
which again represents the exact energy eigenvalue with u = 0, 1, 2,. . . , uM, where Us + 1 = Int (1/2x, wavefunction has the form
(51) spectrum for the bound states of the Morse oscillator, + i). In this instance the first-order correction to the
I~~1’)=f[;(~-3~A31u)Iu-3)-~(u+31(A+)31u)Iu+3) +(u+llA+AA+lu)lu+l)-(u-llAA+Alu)lu-1)] =~[u(u-l)(u-2)]“2~u-3)-~u3’2~u-1) +~(u+1)3’21U+1)-~[(U+1)(U+2)(U+3)]1’21U+3),
(52)
which should be compared with eq. (28), noting that the simple harmonic oscillator states in eq. (52) refer to the variables P and Q, whereas those in eq. (28) refer to variables p and q. As before, the first-order corrections override the zeroth-order term as the vibrational quantum number increases, indicating the increasing importance of the anharmonicity. We note that there are approximately half as many bound sates in zeroth order as in the case of the full hamiltonian. This requires that the perturbational approach become less valid as the vibrational quantum number increases, and an altemative approach is essential if the wavefunctions of the higher-energy levels are to be adequately described in terms of simple harmonic oscillator eigenstates.
5. Variational treatment of the Morse oscillator The fact that the transformed operators whose matrix elements
hamiltonian is now represented as a finite expansion over simple harmonic oscillator eigenstates are known
involving simple allows us to treat
74
I.L. Cooper / Perturbational
the problem such that
variationally.
Hhl/klJ,=A+A
We shall represent
+ 4 + $x;/*(A3+
approach to the Morse oscillator
the hamiltonian
in terms of raising
and lowering
operators
(A+)3-L4++AA+-AA+A)
-~x,(A4+(A+)4-AZ(‘4+)2-((A+)2AZ).
(53)
It is immediately apparent that non-zero matrix elements of this hamiltonian occur between harmonic oscillator eigenstates whose vibrational quantum numbers u and u’ satisfy the relation Au=u-u’=O,
+l,
*3,
simple
+4.
(54)
(Note that non-zero elements of the original necessary matrix elements are as follows:
hamiltonian
occur for all possible
values
of u and u’.) The
(u]H,/ttw,]u)=u+~+~x,(u2+U+1),
(55)
(u 1H&iw,
(56)
1u + 1) = - +x;‘~( u + 1)3’2,
(u]H,,‘Aw,]u+3)=4~~‘*[(u+1)(~+2)(u+3)]~’~,
(57)
(~]H~/Aw,]u+4)=
(58)
-~x,[(u+1)(u+2)(u+3)(u+4)]“2.
This enables US to diagonalize the transformed using the complete set expansion
hamiltonian
in the simple harmonic
basis { I u)}
oscillator
14,) = CC”“< IO VI
(59)
in order to generate approximate eigenvalues and eigenfunctions of the Morse oscillator. If we choose parameter values associated with a typical XH diatomic [2], viz. o, = 3500 cm-’ and *exe = 80 cm-‘, we find x, = 0.0229 and De/Awe = 10.9170. The number of bound states of this Morse oscillator is thus 22, and it is of interest to study the convergence of the expansion as the number of simple harmonic oscillator basis functions is increased. We write
G”(N)=
5
c”,ln>,
(60)
n=O
where N is the highest value of vibrational determine energy differences AE,(N) defined AE,(N)
= (E,(H)
quantum by
number
used
in the expansion,
and
-E,)/&,
(61)
Table 1 Values of A&(N) (as defined in eq. (61)). for selected values of vibrational quantum number v, for the Morse x, = 0.0229, as a function of maximum vibrational quantum number N in the simple harmonic oscillator basis ” 5 10 15 20
N=25 0.0015 0.7394 4.863 14.16
we shall
oscillator
N=50
N=lOO
N=200
N=400
N=WO
10-6 0.0379
10-10 0.0002
10-l’ 10-7
10-13 10-10
;;I::
1.373 5.363
0.2713 2.402
0.0257 1.061
0.0010 0.4644
0.0003 0.3540
with
75
I.L. Cooper / Perturbational approach to the Morse oscillator
where E, represents the exact bound and E,(N) is defined by E,(N)
state eigenvalue
of the Morse corresponding
= C%?/,(N) I HM I ‘kI,(W
to vibrational
level u,
(62)
Table 1 shows the variation of AE,(N) with N for selected values of vibrational quantum number U, for the case x, = 0.0229, where uhl = 21. We observe that the convergence becomes rather slow in the case of the very highest vibrational levels, as expected on the basis of the spatial extent of the corresponding eigenstates. The number of approximate levels which are bound increases with N as follows: 13, 15, 17, 18, 19 and 19 for N = 25, 50, 100, 200, 400 and 500 respectively. No numerical difficulties are encountered with this approach, the basis functions being well behaved in all cases.
6. Conclusions We have shown how the Schrijdinger equation for the Morse oscillator may be transformed to new coordinates which permit a variational solution based upon a complete set expansion in eigenstates of the simple harmonic oscillator. As the size of the basis is increased, the number of approximate bound states increases, and all but the highest states are well represented. This approach offers the possibility of investigating the effect of anharmonicity on the coupling of anharmonic bond modes and this will be discussed elsewhere. Although the exact analytical solutions of the Morse problem are available, the present approach offers the possibility of applying a similar treatment to hamiltonians which represent modifications to the Morse hamiltonian itself and for which no exact solutions are available. In particular, we may apply a variational treatment to the so-called perturbed Morse oscillator model [7-81, where we note that the correction terms, which are powers of the Morse potential itself, now become powers of the new coordinate, for which all matrix elements over harmonic oscillator states are easily obtained. This perturbed Morse oscillator model will be subject of a forthcoming communication.
References [l] [2] [3] [4] [5] [6] [7]
[8]
MS. Child and L. Halonen, Advan. Chem. Phys. 57 (1984) 1. I.M. Mills and A.G. Robiette, Mol. Phys. 56 (1985) 743. P.M. Morse, Phys. Rev. 34 (1929) 57. J.O. Hirschfelder, W. Byers Brown and S.T. Epstein, Advan. Quantum Chem. 1 (1964) 255. M.S. Child and R.T. Lawton, Faraday Discussions Chem. Sot. 71 (1981) 273. D. ter Haar, Phys. Rev. 70 (1946) 222. J.N. Huffaker, J. Math. Phys. 16 (1975) 862. F.M. Femandez and E.A. Castro, J. Mol. Spectry. 94 (1982) 28.
Chemical Physics 112 (1987) 67-75 North-Holland, Amsterdam
PERTURBATIONAL
AND VARIATIONAL
TREATMENTS
OF THJI MORSE OSCILLATOR
I.L. COOPER School of Chemistry,
The University, Newcastle-upon-Tyne
Received
1986
9 October
NEI 7RU, UK
The Morse oscillator hamiltonian is expressed as an infinite expansion in powers of a natural perturbation parameter, the square root of the anharmonicity constant, relative to the simple harmonic oscillator as zeroth-order hamiltonian. A transformation of variables leads to a hamiltonian which involves terms no higher than second order in this natural perturbation parameter. In both cases, the exact bound state eigenvalues of the Morse oscillator are given by second-order perturbation theory. The S&r&linger equation corresponding to the transformed Morse hamiltonian is solved variationally, via a complete set expansion in simple harmonic oscillator eigenstates. Accurate approximations to the exact eigenvalues and eigenfunctions of bound states of the Morse oscillator can be obtained for all but the very highest levels.
1. Introduction There is considerable current interest in the application of the Morse oscillator model to the investigation of local vibrational modes, with particular emphasis on those highly excited vibrational levels which are accessible by laser spectroscopy [l]. In a recent study of the relation between local and normal modes, Mills and Robiette [2] noted that a perturbational analysis of the Schrbdinger equation corresponding to the Morse potential, when based on an expansion in powers of the relative vibrational coordinate, reproduced the exact bound state energy eigenvalues when the potential was truncated at the quartic term and when the quartic and cubic terms were treated by first- and second-order perturbation theory respectively. The present paper is designed to investigate the perturbational approach to the Morse oscillator based upon the simple harmonic oscillator as zeroth-order hamiltonian. It will be shown that the square root of the anharmonicity constant acts as a natural perturbational parameter and that the exact bound state eigenvalues appear exactly at second order in this parameter. This result, which becomes obvious with hindsight, does not appear to be well known. A simple transformation of variable will be shown to lead to yet another form of the Morse hamiltonian, which depends on both position and momentum coordinates in the new variable. This form of the Morse hamiltonian, which does not appear to have been previously identified, is amenable to a similar perturbational expansion in the square root of the anharmonicity and again reproduces the exact bound state eigenvalues at second order. The transformed hamiltonian can also be rewritten in terms of the familiar raising and lowering operators appropriate to the simple harmonic oscillator in the transformed variable. A variational approach to the Morse oscillator problem using a basis of simple harmonic oscillator eigenstates becomes feasible since all necessary matrix elements can be calculated in a straightforward manner using the well-known algebra of the simple harmonic oscillator. This provides a computational approach to the generation of accurate approximations to the exact Morse eigenstates which may prove to be more convenient than the exact solutions for certain applications. 0301-0104/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
I.L. Cooper / Perturbational
68
approach to the Morse oscillator
2. Preliminaries
We shall find it convenient to express the Morse hamiltonian depends solely on the harmonic vibrational frequency and hamiltonian has the form [3]
Hh,f= -
in dimensionless the anharmonicity
coordinates, constant.
such that it The Morse
$ -j$ +D,(l - epux)*,
(1)
where x = R - R, is the displacement from equilibrium, D, is the dissociation energy from equilibrium, (Y is related to the harmonic force constant at equilibrium and /A is the appropriate reduced mass for the vibrational motion of the diatomic molecule represented by the Morse potential. Using the standard definitions, we = (2De,‘#*a
(2)
x, = Ao,/4D,,
(3)
and
we can express
ITM in the form + $1
2P%
Converting
-eeax)*
dx*
to the dimensionless
e coordinate
q,
1.
defined
(4) by
q = (P%/W2X,
(5)
the Morse hamiltonian
becomes -I
H,
= +ho,
- -$
+ &-- (1 - exp[ - (2x,)“*q] e
,‘J,
(6)
where we have used eqs. (2) and (3) to write ax = (,u/2D,)“‘w,( Since x, is generally natural perturbation
&-{le
A/,uc+)“*q
= (h,/2D,)“*q
= (2xe)l’*q,
(7)
small, the exponential in eq. (6) can be expanded in powers of xi/*, which acts as a parameter, to generate an infinite perturbational expansion in the form
exp[ -(2x,)“*q])*
= &[(2x,)“*q-
x,q*
= q* - (2Xey2q3
+ $,(2xe)3’2q3
+ ;x,q4
+ . ..I’
+ ....
(8)
Hence H,-hu.[+(
-$+q*)
-($xe)1’2q3+$xeq4+
. ..I.
where the leading term corresponds to the simple harmonic oscillator Aw,(u + t), where u = 0, 1, 2,. . . . Since the eigenvalues of H, are [3]
(9) hamiltonian,
with
E*=Ao,[(v+f)-x,(u+f)*], we note that only terms quadratic
eigenvalues
(10) in the natural
perturbation
parameter
xi/* can contribute
to the energy.
I. L. Cooper / Perturbational
69
approach to the Morse oscillator
It should be noted at this point that we in eq. (10) denotes the angular harmonic frequency. Customary spectroscopic notation uses w, to denote the harmonic vibrational wavenumber (i.e. Aw, is replaced by hew, in eq. (lo)), while Mills and Robiette [2] use a notation in which tzw, is replaced by hew, and --x,w, by hex,. The remainder of the present paper is devoted to a perturbational and variational analysis of the Schrodinger equation for the Morse oscillator. Section 2 will consider the perturbational analysis based on eq. (9) and section 3 will identify a simple transformation of variables which yields an alternative representation of the Morse hamiltonian, also amenable to a perturbational analysis. Section 4 will treat this transformed hamiltonian variationally using the simple harmonic oscillator eigenstates as basis. The paper will conclude with a discussion of the results, and identification of some possible further applications.
3. Perturbational treatment of the Morse oscillator The Morse hamiltonian, eq. (9) is represented as an infinite perturbational expansion in powers of the hamiltonatural perturbation parameter x,l/2 , based upon the simple harmonic oscillator as zeroth-order nian. The familiar Rayleigh-Schrodinger perturbation theory [4] requires only slight modification to accomodate such an expansion of the hamiltonian, and the approach may be summarised as follows: We express the Schriidinger equations for the Morse oscillator and the simple harmonic oscillator respectively as &l
I 4,) = E” I $“)Y
(11)
H,,Iu)=E,'~'Iu), using an obvious
notation.
(12) The energy
shift A E, for each vibrational
state u is expressible
in the form
AE,~E,-E,c')=(uIH'(\CI,), where H' = HM - Ho and we have assumed
(13) that (u I 4,) = 1. Rewriting
[E,~H,,-H/+(E,-E,Jo))]IJ/,)=o
eq. (11) in the form
04)
yields the identity
I$,)=(E,'"'-Ho)-l(H'-AE,)I$,). Introducing
the projection
operator
(15)
FU by the relation
i” I$“> = I u> and writing
06)
&, = 1 - kU, then
I~“)=(~“+d”hu =tlIu)+
[~u/(E,“‘-Ho)](H’-AE,)I~,)
= I u> + %W’where R,, the reduced R,=
AK)
resolvent
&‘(E,‘“‘-Ho).
I +,>, for state u, is defined
(17) by
(18)
70
I.L Cooper / Perturbational approach to the Morse oscillator
On iteration,
we find
09) and AE,=
f (u(H’[R”(H’-AE,)]“(u). n=O case, with xt/* as the perturbation
In the present H’
cx1/*7/(1)
+
e
v
V(2) +
x
parameter,
we have the following
expansions:
. . .
(21)
e
=X’/*E(‘)+~
AE
(20)
e
e
u
EC*)+
. . .
(22)
”
where ~/cl)/h~~
=
_
2-112~3,
and so on. Hence, substituting parameter, we find
v(*)/~O,
=
hi44
(23)
eqs. (21) and (22) into
eq. (20), and equating
like powers
of the perturbation
E,(')=(vlV('+)=O,
(24)
E,'*' = (uIV(*) Iu)+ (u(V'(')R,V(') Iu) =hw,[~(u*+u+~)-_( u*+u+ g>] = -Ao,(u+ g', where we have utilised order in xi/*, we have
standard
integrals
over simple
harmonic
oscillator
(25) eigenstates.
Hence,
E,=E,'"'+~','2E,"'+xeE;2'=~w,(u+~)[1-x,(u+~)].
to second
(26)
This is the exact energy eigenvalue expression for bound states of the Morse oscillator, and so all higher order terms provide identically vanishing contributions to the energy. (All energy corrections of odd order vanish by symmetry; those of even order higher than second become increasingly complex, although ultimately non-contributing. This can be checked out at fourth order, where it is observed that the disconnected and connected terms identically cancel. This must also be the case for higher-order even terms.) We have thus identified the origin of the result of Mills and Robiette [2], who treated the term in q3 by second-order perturbation theory, and the term in q4 by first-order perturbation theory, while dropping all higher terms in the expansion of the potential. In view of the assumption of the so-called harmonically coupled anharmonic-oscillator model [5] that the coupling between bond modes may be approximated, at least for small values of vibrational quantum number, by interactions between the corresponding harmonic oscillator states, it is of interest to examine the first-order corrections to the wavefunction. From eq. (19) we note that
(27) where the prime on the summation
symbol
denotes
that the term n = u is to be excluded.
I. L. Cooper / Perturbational
Since v(l) a q3, only terms involving
71
approach to the Morse oscillator
n = u * 1 and n = u _+ 3 are non-vanishing,
and we find that
~,@)=~[(~+1)(~+2)(~1+3)]“~~~+3)+~(u+l)~’~~~+l) -~~“‘~u-1)-~[~(~-1)(u-2)]“~~~-33).
(28)
If we choose x, to be 0.023, which is a typical value for XH diatomic approximated to first order in the perturbation parameter as I$,)
molecules,
then the lowest states are
= IO) + 0.114 11) + 0.03 13),
I I//,) = -0.114 I$,)=
IO) + 11) + 0.322 12) + 0.062 14),
-0.32211)+
~2)+d.591~3)+0.098~5),
14,) = -0.031
IO) - 0.59112)
I+,)
11) - 0.910 13) + 14) + 1.272 IS) + 0.18317).
= -0.062
+ 13) + 0.910 14) + 0.138 16), (29)
Note that the first-order correction overrides the zeroth-order term when u = 4, indicating that the harmonic coupling of anharmonic bond modes will only model the behaviour of the very lowest eigenstates, and fails to provide an adequate description of the more highly anharmonic states. This matter will be discussed further elsewhere.
4. Transformation
of variables
The Morse hamiltonian, represented by eq. (6), has been analysed using the expanded form displayed in eq. (9). This infinite expansion is amenable to a perturbational treatment based on the simple harmonic oscillator as zeroth-order hamiltonian, and the exact energy eigenvalues are obtained when the energy is evaluated to second order in the square root of the anharmonicity parameter. The fact that this hamiltonian expansion is an infinite one is unsatisfactory, and it is of interest to seek a transformation of variables such that the resultant transformed hamiltonian contains only a finite number of terms. This objective may be achieved in the following manner. We define a new variable, Q, by the relation Q = (2~,)-“~(
1 - exp[ - (2x,)“2q]
where we note that Q + q in the harmonic
), limit,
(30) x, + 0. The conjugate
momentum,
d p= is related 2
by
(31)
-% to the original
P, defined
momentum,
= exp[ - (2x,)“2q]
p = -id/dq,
as follows:
Since (32)
= 1 - (~x~)~‘~Q,
we have
=f[P(1-t2~,)“~Q)+(1-(2x,)“~Q)P]
=P-($Xe)1’2[Q,
p]+,
(33)
where
IQ,
PI+= QP+PQ
(34)
I.L. Cooper / Perturbational approach io the Morse oscillator
12
represents momentum
the anticommutator of P and Q, and the symmetrisation operator retains its hermitian character. Since
f’,[Q, PI+ ] + + b,[Q,
P* = P* - (b,)“‘[
ensures
that
PI:
the
transformed
(35)
we have H&w,
= 4 ( P*
+Q~)=~(~~+Q~)-t(t~,)~‘~[~~[Q,
p]+]++ix,[Q,
PI:.
(36)
Eq. (36) represents the desired expansion, in which the transformed Morse hamiltonian now contains terms no higher than quadratic in the natural perturbation parameter, xi/*, and the correction terms to the zeroth-order hamiltonian now depend on both position and momentum. It should be noted at this point that we have been treating s-state solutions of the full three-dimensional Morse oscillator. The coordinate q thus ranges from 0 to infinity, and this implies, from eq. (30) that Q if aR, = In 2, ranges from - (2xe)-l/*(eaRe - 1) to +(2x )-l/*. This range will only be symmetrical whereas in general we must have aR, < In 5 so that the potential function becomes repulsive at small internuclear separations. Since we are only concerned here with bound state solutions we may consider Q to lie in the symmetrical range -(2x,)-‘/* G Q G (2x,)-‘/*. with respect to the harmonic potential given by V(Q)/Ao,
= +Q’.
(37)
As pointed out by ter Haar [6], the analytical solutions obtained by Morse [3] are only approximate since they fail to exhibit the correct boundary condition at q = 0 (viz. the wavefunctions must vanish there). The error ought to be negligible since it corresponds physically to penetration of the (finite) potential barrier at small internuclear separations, and the resulting wavefunctions should decay rapidly to zero. We can investigate the validity of the perturbational approach by noting that the number of harmonic oscillator states with energy less than the dissociation limit is given by u0 + 1 where u0 + $5 1/4x,, i.e. the number of bound harmonic levels is Int(l/4x, + i). No bound states exist if 1/2x, 2 1, corresponding physically to the relation D,/( iho,)
5 I
(38)
and is equivalent to one of the conditions quoted solutions become appreciable. Finally, we note that Us + 1, where the vibrational quantum number Us is Int (1/2x, + 3) and no bound state exists if xi’ D,/( $Aw,) 5 I.
by ter Haar [6] at which deviations from the rigorous the number of bound Morse state solutions is given by satisfies uh? + f ,< 1/2x,. The number of bound states 5 1, i.e. (39)
We note that the number of bound states in zeroth order is approximately half the number of bound states supported by the Morse potential and this confirms that the perturbational approach must rapidly break down with increasing value of the vibrational quantum number. Although the perturbational treatment of the transformed Morse hamiltonian, eq. (36), may be carried out in terms of the variables P and Q, we shall find it convenient to transform to the familiar raising and lowering operators of the simple harmonic oscillator, i.e. A = 2_l/*(Q A+=
2_l/*(Q
+ iP),
(40)
- iP),
(41)
such that [A, A+] =/t/I+-A+A
= 1.
(42)
I. L. Cooper / Perturbational
(Note that the ground exp( - Q ‘/2).)
state
of this harmonic
73
approach io the Morse oscillator
oscillator
satisfies
the relation
A IO) = 0 so that
10) a
Hence
H,/liw,
= Ho + x;“~H~ + x, H2,
(43)
where Ho= +(P’+ Hi=
Q’) =A+A+
-2-3/2[P,[Q,
+,
(44)
P]+]+=f(A3+(~+)3-A+~A+-~~+~),
H,=+[Q, P]:=$[-i(AZ-(A+)2)]2= The energy, E,/Aw,
correct
(45)
-$(A4+(~+)4-A2(~+)2-(A+)2~2).
to second order in the perturbational
parameter,
(46)
becomes
5: E,“) + x;‘~EU(~)+ x, Ei2’,
(47)
where E’“)=(o~Ho~u)=u+f,
(48)
E,“’ = (u I HI I u) = 0,
(49)
E,‘2’=(~~H2~~)+(~~HIR,H~~u)=~(u2+u+1)-~(u2+u+~)=
-(u++)‘.
(50)
Hence E,/Ao,-(u-t
+)[I-x,(u+
$)I,
which again represents the exact energy eigenvalue with u = 0, 1, 2,. . . , uM, where Us + 1 = Int (1/2x, wavefunction has the form
(51) spectrum for the bound states of the Morse oscillator, + i). In this instance the first-order correction to the
I~~1’)=f[;(~-3~A31u)Iu-3)-~(u+31(A+)31u)Iu+3) +(u+llA+AA+lu)lu+l)-(u-llAA+Alu)lu-1)] =~[u(u-l)(u-2)]“2~u-3)-~u3’2~u-1) +~(u+1)3’21U+1)-~[(U+1)(U+2)(U+3)]1’21U+3),
(52)
which should be compared with eq. (28), noting that the simple harmonic oscillator states in eq. (52) refer to the variables P and Q, whereas those in eq. (28) refer to variables p and q. As before, the first-order corrections override the zeroth-order term as the vibrational quantum number increases, indicating the increasing importance of the anharmonicity. We note that there are approximately half as many bound sates in zeroth order as in the case of the full hamiltonian. This requires that the perturbational approach become less valid as the vibrational quantum number increases, and an altemative approach is essential if the wavefunctions of the higher-energy levels are to be adequately described in terms of simple harmonic oscillator eigenstates.
5. Variational treatment of the Morse oscillator The fact that the transformed operators whose matrix elements
hamiltonian is now represented as a finite expansion over simple harmonic oscillator eigenstates are known
involving simple allows us to treat
74
I.L. Cooper / Perturbational
the problem such that
variationally.
Hhl/klJ,=A+A
We shall represent
+ 4 + $x;/*(A3+
approach to the Morse oscillator
the hamiltonian
in terms of raising
and lowering
operators
(A+)3-L4++AA+-AA+A)
-~x,(A4+(A+)4-AZ(‘4+)2-((A+)2AZ).
(53)
It is immediately apparent that non-zero matrix elements of this hamiltonian occur between harmonic oscillator eigenstates whose vibrational quantum numbers u and u’ satisfy the relation Au=u-u’=O,
+l,
*3,
simple
+4.
(54)
(Note that non-zero elements of the original necessary matrix elements are as follows:
hamiltonian
occur for all possible
values
of u and u’.) The
(u]H,/ttw,]u)=u+~+~x,(u2+U+1),
(55)
(u 1H&iw,
(56)
1u + 1) = - +x;‘~( u + 1)3’2,
(u]H,,‘Aw,]u+3)=4~~‘*[(u+1)(~+2)(u+3)]~’~,
(57)
(~]H~/Aw,]u+4)=
(58)
-~x,[(u+1)(u+2)(u+3)(u+4)]“2.
This enables US to diagonalize the transformed using the complete set expansion
hamiltonian
in the simple harmonic
basis { I u)}
oscillator
14,) = CC”“< IO VI
(59)
in order to generate approximate eigenvalues and eigenfunctions of the Morse oscillator. If we choose parameter values associated with a typical XH diatomic [2], viz. o, = 3500 cm-’ and *exe = 80 cm-‘, we find x, = 0.0229 and De/Awe = 10.9170. The number of bound states of this Morse oscillator is thus 22, and it is of interest to study the convergence of the expansion as the number of simple harmonic oscillator basis functions is increased. We write
G”(N)=
5
c”,ln>,
(60)
n=O
where N is the highest value of vibrational determine energy differences AE,(N) defined AE,(N)
= (E,(H)
quantum by
number
used
in the expansion,
and
-E,)/&,
(61)
Table 1 Values of A&(N) (as defined in eq. (61)). for selected values of vibrational quantum number v, for the Morse x, = 0.0229, as a function of maximum vibrational quantum number N in the simple harmonic oscillator basis ” 5 10 15 20
N=25 0.0015 0.7394 4.863 14.16
we shall
oscillator
N=50
N=lOO
N=200
N=400
N=WO
10-6 0.0379
10-10 0.0002
10-l’ 10-7
10-13 10-10
;;I::
1.373 5.363
0.2713 2.402
0.0257 1.061
0.0010 0.4644
0.0003 0.3540
with
75
I.L. Cooper / Perturbational approach to the Morse oscillator
where E, represents the exact bound and E,(N) is defined by E,(N)
state eigenvalue
of the Morse corresponding
= C%?/,(N) I HM I ‘kI,(W
to vibrational
level u,
(62)
Table 1 shows the variation of AE,(N) with N for selected values of vibrational quantum number U, for the case x, = 0.0229, where uhl = 21. We observe that the convergence becomes rather slow in the case of the very highest vibrational levels, as expected on the basis of the spatial extent of the corresponding eigenstates. The number of approximate levels which are bound increases with N as follows: 13, 15, 17, 18, 19 and 19 for N = 25, 50, 100, 200, 400 and 500 respectively. No numerical difficulties are encountered with this approach, the basis functions being well behaved in all cases.
6. Conclusions We have shown how the Schrijdinger equation for the Morse oscillator may be transformed to new coordinates which permit a variational solution based upon a complete set expansion in eigenstates of the simple harmonic oscillator. As the size of the basis is increased, the number of approximate bound states increases, and all but the highest states are well represented. This approach offers the possibility of investigating the effect of anharmonicity on the coupling of anharmonic bond modes and this will be discussed elsewhere. Although the exact analytical solutions of the Morse problem are available, the present approach offers the possibility of applying a similar treatment to hamiltonians which represent modifications to the Morse hamiltonian itself and for which no exact solutions are available. In particular, we may apply a variational treatment to the so-called perturbed Morse oscillator model [7-81, where we note that the correction terms, which are powers of the Morse potential itself, now become powers of the new coordinate, for which all matrix elements over harmonic oscillator states are easily obtained. This perturbed Morse oscillator model will be subject of a forthcoming communication.
References [l] [2] [3] [4] [5] [6] [7]
[8]
MS. Child and L. Halonen, Advan. Chem. Phys. 57 (1984) 1. I.M. Mills and A.G. Robiette, Mol. Phys. 56 (1985) 743. P.M. Morse, Phys. Rev. 34 (1929) 57. J.O. Hirschfelder, W. Byers Brown and S.T. Epstein, Advan. Quantum Chem. 1 (1964) 255. M.S. Child and R.T. Lawton, Faraday Discussions Chem. Sot. 71 (1981) 273. D. ter Haar, Phys. Rev. 70 (1946) 222. J.N. Huffaker, J. Math. Phys. 16 (1975) 862. F.M. Femandez and E.A. Castro, J. Mol. Spectry. 94 (1982) 28.