Degeneracy applications to partitioning perturbation theory - Part 

Certain, P. R.

Dion, D. R.

Hirschfelder, J. O.

English

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THEORETICAL
UNIVERSITY
AS
I
f=
 J;
ERRATA WIS-TCI-366
NEW-PARTITIONING PERTURBATION THEORY: II.
EXAMP'LE'S OF ALMOST DEGENERACY
Phillip R. Certain, David R. Dion, and Joseph 0. Hirschfelder
3i
i,
f!
ISpage	 !i
	
1	 Eq. (2) interchange w+ and w{
^r
,x
	
2	 Line below (S) interchange w
+ 
and w
,^,	 34
	13	 Six lines from the bottom (which now starts with (RS;Q)).
This line should read: "(RS;ac) gives good values for AE
and fair values for E. The.	 ."
Nk^ -1
NEW PARTITIONING PERTURBATION THEORY:
11. EXAMPLE OF ALMOST DEGENERACY.
by
I	 /	 I
Phillip R. Certain, t David R. Dion, and Joseph 0. Hirschfelder
University of Wisconsin Theoretical Chemistry Institute
Madison, Wisconsin 53706
ABSTRACT
A system of two coupled simple harmonic oscillators where many of
the energy levels are almost degenerate doublets serves as an excellent
"guinea pig" for testing many perturbation techniques including the
Modified Kirtman
.
, Kirtman, and DE-POP-VIM treatments discussed by
Certain and Hirschfelder in a companion paper. These methods are com-
pared with the "usual" approach to almost degenerate problems Which
makes the zeroth order energies of the doublet pair degenerate by a
suitable choice of the zeroth order hamiltonian. In,addition, four-
types of Rayleigh-5chro6dinger expansions are considered. The best
values of the energy are given by the Kirtman treatment. However, the
DE-FOP-VIM and the Modified Kirtman procedures give values almost as
good,, and have the added advantage that they give upper bounds to
the energy of the states which are considered.
fi NSF Predoctoral Fellow, 1965-69; Present address: - Department of
Chemistry, University of Manchester, England.
This work supported by the National Aeronautics and Space
Administration Grant NG1, 50-002-001.
I	 /
run	
(n	 (m	 (k))W-
-1-
I. Introduction
In this paper, the perturbation techniques developed by Certain
and Hirschfeldet in a eompdtiion paper' (which we shall refer to as 1),
as well as verious types of Rayleigh-Schro"dinger expansions, are ap-
plied to a model problem where many of the energy levels are almost
degenerate doublets. We consider the systemz composed of two coupled
simple hatwonic jsu'illators as dL-,,4(ribed by the hamiltonian
;L 2^ 	 1
+	 X '.	 + A 	 (1)
^L
I.
Exact Solution
The Schr'O'dinger Equation for t1jis case is exactly solvable. By,
rotating the coordinates so that % - t, cosh + it sine and
y	 sinO + n cosOo then setting tan(20) 	 X/(2a), the hamil-
tonian becomes
4-	 W	
(2)
4-^
where
y	 I/
E i + oo- t	 +	 21	 (3)
This hamiltonian corresponds to a two dimensional simple harmonic
oscillator with the energy levels
4
44
I,
where the quantum numbers n and m are either zero or positive
integers. The corresponding wave functions are
AP M rm	 ( W4-W )", U M ('g') 1 ,(I-')
where 4' == (w+) ^ and -n	 (w_)	 Letting hn (q) be the n-th
He' mite polynomial,
when a and X are small, the states nm and mn (with nf^n)
are closely spaced doublets. By squaring the sum and difference of co',.
and w and then taking the square root of the result, we obtain
^ B^, b^^h
	
C 7 )
Eq. (7) is convenient for expressing the mean energy of the doub l eL,
(Enm + mn) /2 = ( } (n + m -i- 1) (W+ + wr )	 ($)
and the energy spitting within the doublet
(Enm mn) = (n - m) (w+ - w_)
In the remainder of the paper we consider many different types of
perturbation procedures for determining the mean energy and energy
splitting of the 10,01 doublet. Tables 1 and 2 give the results which
we obtained for a = 0.1 and a varying from zero to 0.35.
2
(5)
(6)
L
and	 4 (11)
3
#I
ti
II. The RaYleiRh-Schr*O* din&er Expansions
The basic assumption of the Rayleigh-Schr lo'dinger theory is that
the exact wave function and the exact energy may be expanded in power
series in the perturbation parameter. For the problem at hand, there
are four types of Rayleigh-Schr'O'dinger expansions which we might
consider:
1) Expansion in Powers of A(RF,
The (RS;X) energies are most easily obtained by expanding the ex-
pressions for the exact energies, Eqs. (7) - (9), in powers of X.
This involves: Urst, the expansion of th y! 	 square root; and
then the expansion of the outer square: 	 in Eq. (7). Thus we
obtain:
(10)
4
(E +E
	 2 = (n+m+l)
nm mn
^f
This series converges provided that
Furthermore,
(12)4
(E 
nm 
-E 
mn	
2 ot (n-m)	++
1 2. ? QL4	 13
	 ^^^^)
R'("Lo - ce)	 j
and the series converges provided that
(13)and
4Thus, the range of convergence for the mean energy series- is much
greater than that for the energy splitting series # For example, if
a w 0.1, then the mean energy series converges if X < 0. 99; whereas
the energy splitting series converges if a < 0.198. Of course, the
first few terms in these series can be used as an asymptotic approxi-
mation to the energy outside of the range of convergence. Thus, in
Table I it is seen that the first two terms in Eq. (10) gave an ex-
cellent approximation to the mean energy for even the largest tabu-
lated value of a (0.35) , whereas in Table 2, the first two terns of
Eq. (12) give a very poor approximation to the energy splitting for
even the smallest values of X.
9
2) Expansion in Powers of a(RS^U)
The (RS;a) energies are also obtained by expanding the inner and
outer square roots in Eq. (7). We shill assume thgt X2 < 1 so that
C = (1	 X 2 ) ^ is real. Then,
(14)
(E um mn+E ) /2 4-(h) (n+m+l)
and the series converges provided that d, 2 < 1 - I X I . Fir thermore,
	
(E -E_ ) (n-m ) au ' C^ 	 } cc (1+ G) 	+	 o."nm mnI 	 2	 oC	 (15)2 X-C
The convergence requirements of this series are that a2 < 1 - 1a1
and that a2 < (-1+4c-3c 2 ) Ac- Z X 2 /4. From T bles 1 and 2, it is ----
apparent that the (RS;a) series give give a good approximation for the
4
Jmean energy but a Pooi approximation for the energy splitting. This
behaviour is similar to thatfor the (RS;A) .
3) Double Power Series Ex ansion in a and A (R.S i a X)
There is no difficulty in obtaining the double power series for
the mean energy by expanding the inner and outer square . hoots of Eq.
(7)
(16)
a
	
(E +E ) /2	 (n+ml-l)nm mn	 ...
This series converges if
	
<	 and	 4'C4 a. _. ^a < Ca
Note that Eq. (16) contains no terms in an since the mean energy of
the doublet is independent of the value of a if A = 0.
However, no double power series expansion can be obtained for the
energy splitting. After expanding the inner square root in Eq. (7),
(17)
	
E -E= n-m d.°t' +	 2.aL°L + ^a — c(4 +° -^ c^.nLa. +	 d.y }
run mn	 (	 A_	 ... I
A further expansion of Eq. (17) would lead to terms involving inverse
^ ►
	
	 powers of either a or A 	 There is a simple theoretical reason
why this double power series cannot occur: 2 Let us resolve the
hamiltonian into three par us,, Ii H( ^) + H (10a) + H (1 ' A) where
6a
(1 .49 C< 	 )AH (19)
(20)
The zeroth order energy of the doublet states is degoacrate,
(0 )	 (0)	 The zeroth order wave functions corresponding11m	 mn
to these energy states are
ti	 (0) cose an (X) um
 
(Y)	 s inO um
 
(X) u n(y)
li
ii
(0) (x) u (Y) + COSO u	 L12	 -sing u	 (X)	 (Y)n	 m	 ni	 n
In order that a double Rayleigh-Schrodinger power series expansion be
possible ,,	 it is necessary that	 (0) and	 2 (0) be
simultaneously the "correct zeroth order" wave functions for the
perturbations
	 H	 ^ and	 H	 Unfortunately, it is impossible
to find a value of	 9	 which permits both sets of conditions to be
satisfied.	 This is easy to see in the special case that In - ml
where the degeneracies are removed in the first order. Here
40)and 2 would be the "correct zeroth order" wave functions for
the two perturbations if
	 C T (0)	 H (1 " ^`^' T 2 (0) > 0	 and<,/, (0)
	 H a	 (0) >	 0.
2
6b
4) Ex pans ion in Potpiers ^af O and 	 )
Finally, as Saul T. Epstein suggested to usplet us consider a
double Rayleigh-Schrodinger power series expansion which exists for
troth the mean energy and the energy spli.ttings, Let us express the
problem in the S and '^' (instead of the x and y) coordinates.
Then the hamiltonian, Eq. (2), can be expressed as the sum of three
ter'nis
H 	`..	 L
 ^.'(0) 	 +	 Z + *^ 
4)
^7
	
H maw	 E	
5z] I
t
1
L
where	 =	 4 0( ^ +	 Then the e-, act energy is given by
Eqs . (3) and (4) with
W4- 
Expanding W+ through terms of the third order ( a< i
 P a where i+j < 3)
1.,	 z	 e	 l
These series converges for (OC 2 -^3) 2 < 1. The advantage of the
(RS;o^ P) treatment i4 that the double power series expansion
exists for both the mean energy and tha energy splitting of a doublet.
III. The Kirtman, Modified Kirtman, and De-FOP-VIM_ Approximations
Next.. let us apply the Kirtman (K), Modified Kirtman (MK), and
DE-FOP-VIM approximations developed in-the companion paper T. Taking
^'	 a
J
x
`a
as the perturbation parameter, the unperturbed hamiltonian is
]y4
21. dx 	, (23)s.
y
4
and the perturbation is
}
r.
p
The eigenvolutions of the unperturbed problem are
and
^►^+/rh (25)
{1
where x'	 (1+e()1/2 x" and y' _ (1i oC) 2y .
G
}q
i	 •
i
i
„
7
R
B^P
XM
Let us apply each of the perturbation techniques to the doublet
pair' 10 and 01 1
Kirtman Approximation
In vector notation> the zeroth order function for the 10 and
01 states is
(0)	 ek^o^
10 ) T 01)
Using the full normalization condition, so that 	 0
t
fl
	
(0)	 4- d,	 0
0
The solution of the first order equa-%-,, ion (1-11)' is
i	 1	
(1) 
= --	
-, ( col ^^.^[a j GO- ) .1	 at
0	 1
0
Li am	 0
0	 1	 )4(1— cL:La ) )
(3)	 0	
1(01 
1 
1 —
Z	
0
(26)      
To this order, the energy of the Kirtman formalism is given by th(-
roots of the secular equation (1-29) which yields
10, W — A.	 4	
+	
co) 17A +
	
CM V'A01
(27)
Modif led Kiftinan &P-ro.x-imation
Alternativoly, the modified Kirtman secular equation (T-14)
N 44 1 reduces for this case to
Aa ow vo
which has the roots
g	 (28)
F06 4 1+^
Since the 10 and 01 states are the lowest energy states with their
symmetry, the roots (28) are upper bounds to the exact energies.,.,
Clearly, the modified Kirtman roots (28) differ from the Kirtman
roots (27) by terms of order X4.
-
DE-FOP-VIM Approximation
For the application of DE-FOP-VIM to this example, the secular
equation to be solved is Eq. (1-28). S^ibstitution of the proper
9L +c(, —E
--A--" ,
a.0 I -00)
,-
i,
I!
O.I)11
10
quantities and transforming to the orthonormal set of functions
C^)	 (0), ^ (0), ^ (0)
  
yields the 4 x 4 secular equation for theltd	 fl 1
	 21	 12
energy
C.Cr^C&)3.mr
	 0	 4`
Note that of the four energy roots, the only tvYo which arc-, significant
approach E(0) and E:QO) in the limit as X ^ 0.
IV. Reduction to Degenerate Perturbation.
As explained in Paper I, the almost degeneracy of the doublet 10
and Ol can be transformed into a true degeneracy by using ttj.e zej:oth
order hamiltonian
	
(0) M (o) + E E w) 	 (d)
	
1Lo	 i s	 1 o 1
+ (	 E (01 / ^ (0) < (off
` 0!	 01	 oL
(30)
1
11
(0)	 (0)where H(0) is the H(0) of Eq. (23). Clearly, ff(o)
^io
and f(0))(0	 (0). The energy e is arbitrary. Thus, it mayC.- Ol
be taken to be the mean unperturbed energy of the doublet pair,
( S (0)  + e ( 0) ) as 2 0 or 7 may be considered to be a variational
10	 01
parameter. Corresponding to H ( 0) is the first order perturbation
potential if(l) = H - -ff(0).
Liaving- reduced the problem to a degenerate perturbation, we
a	 0! (110L:-es u.^ tv the method of solutiun. Wt u- - ; us4,,
the modified Kirtman approach. Then, from Eq. (1-22) and the re-
cursion relations for Hermite polynomials, it follows that
^	
E(D) )
to	 al	 10
C(0)	 (31)
Warning: In general, one should not expect a simple relationship
such as (31) between the^^ 1^and the k 	which results from
the simple harmonic nature of both H (0) and H	 The energy is then
given by
I^^(pIH-EI ^^i)^I =0
Y-1	 + St)
E(` + e)
(32)
I
12
Evaluating the matrix elements leads to the 2 x 2 secular equation,
1
E is + BE rs +Sa.G CO
E(^+ sa l
r(I4-st)
i
where
s	 E (0) Y
t= r,'( - E^^^^
Eq. (32) gives the two energy roots E 10 (c) and E0 1 (E) . These
energies are upper bounds to the exact energy for the states 10 and
01 since: (1) The approximate wave functions for the 10 and 01
states are orthogonal to the 00 ground state wave function; and
(2) the H matrix component linking the approximate 10 (or 01 )
wave function with the 00 is zero. Thus, E can be assigned a reason- µ
able value such as Z 14 ) + E 01)) = 2 or E can be taken as a varies-
tional parameter. The best (or lowest) value of E10 is obtained for
a different value of	 than that which gives the best (or lowest)
value of E01.
13
The degenerate problem can also be treated by the usual Rayleigh-
Schrodinger treatment. Since the degeneracy of the 01 and 10 states
is resolved in the first order, it is not difficult to determine the
wage functions accurate through the first order. These wave functions
are then used to determine the energy correct through the third order.
Numerical results corresponding to e = 2 are given in Tables I and 11
If we had used the DE-FOP-VIM method of approximating the energies
4
of the degenerate perturbation, we would have obtained the secular equa-
tion (29) and the same energies which result from using the DE-FOP-Mi
approximation without making the reduction to the degenerate problem.
However,
-
 this equality is not general,-,but rather it is a result of the
proportionality of the _k l) and the ^ M as given by Eq. (51).
k
V. Numerical Calculationsh
The results of the various perturbation treatments a.re compared
numerically in Tables I, II, and III for a = 0.1 and 0 < X < 0.35.
The mean energy of the doublet, E = II(E10 + E01) is given in Table i;
the energy splitting, Q E = E10 - E 01 , is shown in Table II; and the
optimum values of a for the two states appear in Table III.
First, let us consider the Rayleigh-Schrodinger expansions. The
(RS;a) gives excellent values for E but poor values for AE 	 The
(RS;S) gives excellent values for AE but poor values for E . The
(RS;^.,) gives fair values for E but very poor values for DE	 The
(:;;aa) gives almost the same fair values for E as does (RS;X) , but
the (RS;Xa) perturbation series for AE does not exist.
All of the rest of the perturbation procedures (K, MK, DE-FOP-VIM,
and the reductions to a degenerate problem) give essentially-the same
14
results: good to very good values of the splitting AB, and fair value,_
of the mean energy E	 Although the DE-POP-VIM gives slightly , better
enetgies than the MK (as it must) for bi,)th the 10 and 01 states,
the DE-VOP-VIM values for AE are not as good as the W values. Thus,
as far as this example is concerned, DE-FOP-VIM is not worth the extra
I /
	
	
effort. Vie Kirtman (K) values are slightly better than the Modified 	 .0
Kirwan (IBC) values for both "f and AE . However, the Kirtwaa
value t, io i- 't^iie antergies of the, 10 and of the 01 6tates,
are not up pat bounds. This is the big advantage of thy:
V
treatment. The reduction to a degenerate problem required (_oasidt_-rably
more work and the results ware about the same.
It is difficult to generalize these results to othem 1)rob laws -
However, we are disappointed that no one method shored it ,c'-), elt to be
I
outstandingly better than the others.
Acknowledgments
The authors wish-to thank S. T. Epstein for his helpful comments
and suggestions and we wish i to thank P. J. Fortune for his help in
checking some aspects of the calculations.
15
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16
IL
C) 0 0 0 0 0
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:3	 CY)
440
C) C) 0 0 00
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:0 ci
p 0 C14 m C14 ^l r-I r- m18 -H Z3 m 't ^
cq (:",
H
0
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r- CY) H CD 0 0
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co
0 0 0 0 CD CD
bA
'P C) 01% r", -It M
cd C) 0 Cf) 0 -,1- ^nP4 CD 0 rl r-i %.0 to C\l oll
cn 0 0 C) C) r-I co %10 0
P4 0 0 0 C^ 0 0 C) .a
4J
ri C C14 t31 C%j C) 110 cn r-i
co CD tN 00 h m 1,- 00 r-
x C) .10 Cf) 0 -43- cq -It m
W C) CD in 00 %10
C14 N C14 cq cn
0 0
•
1
0 tn 0 Ln 0 tn 0
c) C'4 C'j C') cn
^r
0
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rl
li •r1
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44
4
ew
0
II 41
0
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rj) * o,
4-4
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4-)
0
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U
CTJ
(1)
lc-
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I	 ip
4
4J
pa
0
H
P4
Table III
The Optimum Values of ^ 1'or a - 0.1 us n8 k^^ 0) of Eck. (30)
followed by MK treatment.
17
a
.05
.10
.15
.20
.25
. 30
.35	 1
tata	 10	 01
2.096	 1.915
2.057
	 1.929
2. 013	 1,949
1.970	 1.972
1.933	 1.996
1.901	 2.022
1.8.73	 2.046
18
footnotes
l.) P. R, Certain and J. 0. Hirsclifelder, J Cham. Phys. (preceding
companion paper; Part 1).
2.) J. 0. Hirsclifelder, W. Byers Brown and S. T. Epstein, Adv
I
	 cant. Chem, 1, (P. 0. Lowdin, Ed., Academic Press, New York,
1964) . See p. 297.
3.) The notation (1-'11) should be read: "Poper T; equation (11)".
4.) See, for example, A. Messiah, Quantum Mechanics, (Wiley, New York,
1964) , Vol. 2, P. 711,


New partitioning perturbation theory. 2 - Example of almost degeneracy

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