Citation

## Material Information

Title:
Lower bounds to eigenvalues of the Schrödinger equation
Creator:
Wilson, Timothy Michael, 1938-
Publication Date:
1966
Language:
English
Physical Description:
v, 56 leaves. : illus. ; 28 cm.

## Subjects

Subjects / Keywords:
Asymptotic value ( jstor )
Eigenfunctions ( jstor )
Eigenvalues ( jstor )
Eigenvectors ( jstor )
Evaluation points ( jstor )
Graduates ( jstor )
Mathematical variables ( jstor )
Spectral theory ( jstor )
Symmetry ( jstor )
Vertical asymptotes ( jstor )
Boundary value problems ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Eigenvalues of the Schrödinger equation ( lcsh )
Quantum theory ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

## Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 54-55.
Also available on World Wide Web
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text

LOWER BOUNDS TO EIGENVALUES OF

THE SCHRODINGER EQUATION

By

TIMOTHY MICHAEL WILSON

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

August, 1966

ACKNOWLEDGMENTS

It gives me great pleasure to express my appreciation for the

assistance, both direct and indirect, provided by my supervisory

committee and by numerous other members of the Chemistry and Physics

departments faculties and the staff of the Quantum Theory Project.

In particular, I would like to express the gratitude I feel

toward the chairman of my committee, Professor Charles E. Reid, for

both the time and advice he has so frequently given me over the years

I have known him. His encouragement and suggestions have been

invaluable.

I am particularly indebted to Professor Per-Olov L8wdin for

first introducing the subject of lower bounds theory to me and for his

stimulating lectures and pioneering work in this area.

To Dr. Jong Hyuck Choi, I am extremely grateful for the many

helpful discussions and the expert assistance in writing some of the

programs used in the lower bound calculations.

It is with pleasure that I acknowledge the excellent assistance

of Mrs. Philamena Pearl in the preparation of this dissertation.

The financial support of the National Aeronautics and Space

Administration and the grant of considerable computer time by the

University of Florida Computing Center is gratefully acknowledged.

ACKNOWLEDGMENTS . . . . . . . .

LIST OF TABLES .. . . . . . .

LIST OF FIGURES . . . . . .

Chapter

I. INTRODUCTION . . . . . .

II. THE PARTITIONING TECHNIQUE AND THE

. . . .

. . .

BRACKETING THEOREM. . . . .. ...

III. LOWER BOUNDS TO ENERGY STATES . .

IV. LOWER BOUNDS TO THE REACTION OPERATE(

V. LOWER BOUNDS TO THE S STATES OF He

APPENDIX . . . . . . . . . .

REFERENCES . . . . . . .......

BIOGRAPHY . . . . . . .

)R . .

and Li+

Page

ii

iv

V

3

. . . 16
. . . 324
S . . 16

. . 524

S . . 35

* * 50

* * * 54

. .. 56

' '

LIST OF TABLES

Table Page

1. Optimum lower bounds to some of the S states of
He and Li+ for NINT = 40 ............. 40

LIST OF FIGURES

Figure Page

1. Typical behavior of the multivalued bracketing function
E, = )for g = 3 ......... ....... 14

2. Typical behavior of the multivalued bracketing function
for g = 4 and k = 2 . . . . . . 23

3. Lower bound curves of He for g = 4, NINT = 40
and I = 1.2 . . . . . . . . ... .. . 43

4. Lower bound curves of Li+ for g = 4, NINT = 40
and I = 2.1 . . . . . . . . ... ... 44

5. Lower bound curves of He for g = 4, NINT = 20
and 1 = 1.2 . . . . . . . . ... .. 45

6. Lower bound curves of He for g = 4, NINT = 10
and 7 = 1.2 . . . . . . . . . .... 46

7. Lower bound curves of He for g = 4, NINT = 10
and = = 1.35 . . . . . . . . . . 48

CHAPTER I

INTRODUCTION

One of the basic problems in quantum theory is the solution

of the time-independent Schrodinger equation

>Mt-'= F(1.1)
where the Hamiltonian J is bounded below and has associated with it

a set of solutions (E ) corresponding to the discrete levels lying

below the continuum.

This equation cannot be solved exactly except for a few cases

corresponding to very simple systems. Instead the problem is solved

by adopting a suitable method for obtaining an approximate solution.

One particularly useful method for treating this problem is the parti-

tioning technique.6 This approach has the interesting feature that

it contains many of the more conventional methods such as the variational

principle and perturbation theory as special cases.

Using the partitioning technique one may construct a function

, = f(r), where is a real variable, having real solutions I11'

E 12'613' "',,g with the property that each pair (E,6li) will
bracket at least one eigenvalue E of the Hamiltonian .. This means

that if E is chosen as an upper bound to an eigenvalue E, at least

one of the solutions P ii will be a lower bound to E. The problem of

constructing this function and a way of determining which of the

solutions S li is the lower bound to E will be studied. The lower

bound to E is taken to be the largest of the solutions Cli' (i=1,2,

...,k,k g), providing a lower bound to E. Furthermore, it will be

shown that lower bounds to the solutions of E1 may be constructed by

making an inner projection of the reaction operator tg with respect

to a finite basis, where the reaction operator is associated with the

separation of the Hamiltonian f = o + V. Several types of inner

projections are discussed with particular emphasis on the "Lowdin

Projection."5

An application of the method is made to a calculation of

lower bounds to some of the S states of He and Li using the "LBwdin

Projection."

CHAPTER II

THE PARTITIONING TECHNIQUE AND THE BRACKETING THEOREM

Consider the problem of solving the SchrSdinger equation
\$i = E*, where the Hamiltonian R is assumed to be hermitian, i.e.,

a = ( and bounded below. Let 2 = { i be a complete ortho-
normal set of functions. As an immediate consequence there is a
resolution of the identity,

(2.1)
I = < ^><9. .

where the sum over the index i implies summation over the discrete
indices and integration over the continuous ones.
Let us define a self-adjoint projection operator O such that
it projects out of the space spanned by 9 a subspace Ta of order g:

(2.2)

From this definition it follows that satisfies the relations

J (2.3)

Let P = 1 I be a projection operator associated with the subspace

b', the orthogonal complement to the subspace T :

f^\$-^

It then follows that

OP= P&= o.

The function V can be projected onto the complete set

P = ( +PJ

Y = 0yP+ PV)

and the Schrodinger equation can be rewritten as

jI = (+ P) (s6+P) 2
= [OR&+ PWeS>J9 +[P9P+0+ fP]Py

= E (6 + P W.
Operating from the left first by 6, then by P, we obtain the set of
equations

( s5) ) + 6o) P) =

(PR + P )?PP-)
From (2.7b) we find that

(2.7a)

(2.7b)

(P9 ) P =

(PS)OGP = (EP

(2.8)

= (E-n a n er )P P
From (2.4) we see that, for an arbitrary number a,

P) = ( PG) =

P (C6) = 0.

Then (2.8) can be rewritten in the more general form

(Pts)P= [Co-9+ P(CE- )Pj pv.

(2.4)

(2.5)

(2.6)

(2.9)

s"= O

EO .
E PV)

(<9 )p =

- pPMP)V7

Solving (2.9) for P*, we obtain the expression

Pip= [9+p+(E-iP)P]. (PRGO) .
Operating on (2.10) from the left by P gives

p?= fP f[.- 4 PE-)Pj-TPF ] C).
Substituting (2.11) for P* in equation (2.7a), we obtain

( S)sW V ((SVP) P? = E02P

=BP {(9 (B P) Pr((ES + P(E- P 90

S{r + P L.+ PCE-s)P3''P?]&}
Let us introduce the reduced resolvent TE:

T, P[aI +P(E -~4)P P
Equation (2.12) then becomes
Equation (2.12) then becomes

TE [)J0 =

)s &7.

Introducing (2.11) and (2.13). into (2.5) we obtain the expression

S= (I 4-c iF )0;.
P
Let us introduce the symbolic notation T = It is easily
E E -
seen that TE satisfies the relations P(E -4?)TE = P,

O--i&. 0,

(2.16)

PT- -= EP = iF.

(2.10)

(2.11)

(2.12)
6>4>

(2.13)

(2.14)

(2.15)

rvl + V

Since als= R> C)( >C 9j multiplying
(2.14) from the left by each of the 9 k a' (k=1,2,...,g) and
integrating we obtain a set of g simultaneous equations

I (k=1,2, .. .,g).

This set of equations can be rewritten as a matrix equation

H ( E-) (T4 = E' Ca ) (2.18)
where ).(E) <+ RTE ) a (2.19)

and a is a column vector composed of the coefficients appearing in
the expansion of Orin terms of the functions Pi a '

The solutions to this set of equations are found by setting
the determinant of the coefficients of in the matrix equation
C O (E) Ejiaj ) =C 0 equal to zero:

I OO)- Eij = 0. (2.20)

The solutions of the secular equation (2.20) are eigenvalues of
associated with states not orthogonal to the subspace a. This point
will be discussed in greater detail later.
One way of solving equation (2.20) is by means of some iterative
procedure.1 Let E and E1 be real variables. By replacing the
variable E in 0a with the elements of the matrix "0 become
aa aa
functions of 4 and the secular equation (2.20) has a set of g solu-
tions [ li3 for each value of corresponding to those values of

the variable e1 that satisfy the equation

E) I 0 (2.21)

Hence, the variable E1 is a multivalued function of
having the same multiplicity as the order of the subspace ta and
this is denoted by the expression 1 = ()

Corresponding to the set of g solutions fli obtained by
solving (2.21) for a fixed value of ( there will be a set of g
vectors CQ a. .
Making the substitution of for E in (2.19) we obtain the
expression

=

where + (2.22)

(2.23)

Then, for arbitrary each of the solutions Eli (i=1,2,...,g) may
be expressed as a function of i.e., Eli = f.( ), having the form:

t

CA"'i (Cci
I-

cti

= (E) .

(2.24)

Taking the first derivative of (2.24) with respect to and evaluating
it at the same value of E for which the ai were evaluated, we have7
at

-P dip& ^ M
CIL = '44 OC4 l

= c^~~ot
Cai Mah~

d ib E)

(2.25)

0Sa

We see from (2.23) that

d1/jE)
dE

- j! a=')C
C-A Q b

We then see from (2.16) and (2.23) that

- (~bflWJ ~b)

--<0'\T

= 76E)

from which it follows that

c~E74~ ce)

aCE)

(2.27)

0

dEli
2T

T 1 )10. (2.26)

Ei=

dTbb(S)
Replacing in (2.25) by the right hand side of (2.27),
dE
we obtain the result

dd- H P ) Zabo. &ai
ae ~iCat

=* % A-& f(O Ca.i) (2.28)

< ) (i=1,2,...,g)

Therefore, each of the solutions li is a function of having a
negative slope for all E .
Let e = E + e where E is the difference between the value
of E and some eigenvalue E of Let one of the solutions given
by (2.24) be represented as li = E + li. If G is a small quan-
tity, we can use the Lagrange mean-value theorem to obtain the result
that

El = i+6^ = E+e?)

= {.CE) -+ I( E* 9e)

where 0 4 9 1. Using the identity E = f.(E) (see (2.18)) we
obtain the relation

6, = Ef ;(E + e) (2.29)

From (2.28) we know that the first derivative f' is negative; hence,
1i

the quantities C and eli have different signs. This leads to the
bracketing theorem which says that the pair (( ,8li) bracket at
least one true eigenvalue of X Then the g intervals ( li) will
each contain at least one true eigenvalue E such that if E < E,
then Eli E and vice versa.
Let us examine the behavior of aa(E) in the limit of
E + oo. We see from (2.22) that

-=+ =D'lIJ E (2.30)

Hence, the set of g solutions fi j have as horizontal asymptotes
the eigenvalues of the matrix aa

+ A-L (2.31)

Let us define the "outer projection" \$( by the relation

w = P R P.
Since PO = 0P = O, it follows that all the functions ) i a a
are eigenfunctions of S( associated with the eigenvalue 0.

Let us consider the normalized eigenfunctions Ti of -

associated with the eigenvalues E. that are situated in the subspace

b. We then have the spectral resolutions

(2.32)

It is obvious that the reaction operator has vertical asymptotes for
the values E = E., (i=l,2,...). It then follows that the bracketing
function 1 has vertical asymptotes for the values = E., provided
E. is not simultaneously an eigenvalue of
Let .i be a function satisfying the Schr8dinger equation

= E.i~.. From (2.5), we see that i = iri + Pi.. Therefore, if

)i = 0 and Pii # 0, it follows that

Sp ) = 7 (2.33)

Then, from (2.7a) and (2.7b) we find that

(~P)4 2i =(2.34)

Fa = (P P) Vj EE;

That is, for those values & = E. where E. is an eigenvalue of (
associated with a state i. that is orthogonal to the reference mani-
fold 0a' i, is also an eigenstate of ; having the same eigenvalue,
i.e., E. = E..
1 1
Let us make the separation
P P + P (2.35)

where P' = ) EiUl
a

oI = r a such that =0, = 0 and

P" I TOlE) I

for all the Jr

such that ()Da # 0.

The reaction operator T

-E I E,
S= 4 C- E,

may then be separated:

+ E

(2. 36)

-F II

It then follows from (2.22) that

^<= Jt)- <'+ FE

(2.37)

+z + E- E
Since c c = E,* and C = O, we see that
E = 0 for all (k .e a Hence the second term in (2.37)
drops out and we obtain the final result

H~b=) Hy +

where

(2.38)

From (2.38) it is easily seen that for
.From (2.38) it is easily seen that for

E = E., where E.
1l

solution of i = Ei, and O0i = O, the bracketing function1

is a

for all the

= I,

*- E.<

= Haa

is E E8

has no

vertical asymptote. Further, we see that 1 has vertical asymptotes

only for those values of a corresponding to eigenvalues of P R P not

simultaneously eigenvalues of Jf, i.e., only for those cases where Q

Ji o.

It follows from the variation principle that the eigenvalues

of are upper bounds to the eigenvalues of R in order,

S> (2.39)

Hence the vertical asymptotes appear in the bracketing function E 1

for values of E that are upper bounds to eigenvalues of R corresponding

to the states *i satisfying the criterion that C)i # 0.

A plot of the g-valued function 1 as a function of the vari-

able E can be made by imposing the non-crossing rule, which is valid

for states having the same symmetry. The values of the g curves will

satisfy the relation

S E, (i=1,2,...,g), (2.40)

for all E < E 1. If Ell is taken to be the lowest value of the set

of solutions li then for chosen equal to El we find that

I 1 = E1 provided that i # 0. Then for E ) E1, we see from
(2.29) that El < El and the C11 curve will undergo a discontin-

uous change from oo to + oo as passes through ET. In general,

we find that the g curves have the behavior illustrated in Fig. 1,

which is for g = 3. We see that each branch of the curves crosses the

1~ = C line at an eigenvalue of R associated with a state .i not
orthogonal to the reference manifold a'

E1 E2 3 \ \

EI I
II I
h3 I I I
I I

E E EI

E E2 3 4

Fig. 1. Typical behavior of the multivalued bracketing
function El = f(S) for g = 3.

15

It is apparent from this discussion that for Jri # 0 and
Ei < E Ei, one of the eigenvalues of O aa(), E ik' is the lower
bound to E., i.e., E. lies in the interval (elk,6) and for = Ei,

Eik= E..

CHAPTER III

LOWER BOUNDS TO ENERGY STATES

Assume that the Hamiltonian 4 of a system may be separated

into two parts; a zeroth order Hamiltonian or unperturbed part o and

the interaction part or perturbation V. Let there be associated with

the Hamiltonian Yoa complete orthonormal set of eigenfunctionst(p

and a known eigenvalue spectrum Eo, (i=,2,...), such that
2.

(3.1)

Let it be further assumed that the perturbation V is positive definite:

(3.2)

The space 0o

into two subspaces a 0
a

spanned by the set ( ?j can be partitioned

and b by defining the projection operators
b"~

(3.3)

rr1

=~Pe
~0q+ p

such that
such that

oD 0 0- (3.4)

The operators 0 and P satisfy the relations given in (2.3) and (2.4).

From (3.1) we also see that

> 0

t :: 0

s

j-o=F I0><91

a R, -. ;f ) =e"ix
and using (2.16) we have the result

UW9,1 = go 0 .
Equation (2.19) may be rewritten as

^ ,(E)= <^'/ 69,0 + v 4 V Ab 0>

Let us introduce a reduced resolvent T associated with the unperturbed
Hamiltonian o'
o
T = (3.6)
-o o +1 E E

where again the sum implies summation over the discrete states and
integration over the continuous part of the spectrum.
For any inverse operator of the type (A-B)-1 there are the
following identities:

(A-8)'' = -"'+ ,?"8( -B)"/,
(3.7)
= "+ ()F-Y'- ",

provided the inverse operators involved actually exist. Using the first
of the identities given in (3.7) we find that

Tr = T + 7TV V7F-
(3.8)
T V= T7(v + v-rv) = T, ,

where 4 V + V 7 V - V + V 7 4te (3-9)

is the reaction operator.2 The last relation corresponds to the
Lippmann-Schwinger integral equation in scattering theory.
Provided V-I exists, we find from (3.9) that (V--T )te = 1;
hence,
-I
S= (V'-T) (3-10)

is ordered,...
Assuming the spectrum of is ordered, E < E
it is easily seen from (3.6) that for V < Eo T < 0. Since it
g+- o
is assumed that V > O, we see that V- > 0 and hence t, > 0.
Introducing (3.9) into equation (3.5) we obtain

To-(E-= I R. + ; I f .) = E, (3.11)
Hence, we see that the choice of O as given in (3.3) leads to a
multi-valued bracketing function ~ From Chapter II, it follows
that this bracketing function will have vertical asymptotes only for
those states *k of X that satisfy the relation ()k # 0. That is,
only those states not orthogonal to the reference manifold (
spanned by the set of g eigenvectors of 0, will cause vertical
asymptotes to appear in El The function 1 has the important
property that the first vertical asymptote occurs for a > E
g+l
Hence, if P J P has the ordered spectrum El < E2 < ..., then E > Eg+
To prove this let us consider two self-adjoint operators A and B
.atisfying the relation

ft > (3.12)

Then, for any functions < t > O B *>

Let P be an arbitrary linear operator. Then replacing q) by PC we
find that <9 JP+API ~)>> IP+BPI ) i.e.,

P R/ P > P B. P.13)
Since V 0> we have the inequality = @o + V> 40, where + =

Sand? Then from (3.13) we find that for the self-adjoint
projection operator P defined in (3.3)

P iP > P P (3.14)

Hence, <( jPgP P )>>< lPJfPI) for any function Let I= ~'
where Vl is the lowest state of P9P not orthogonal to ( a. Since -
the ground state of P&P is Eg+1, it follows from the variation princi-
ple that

=, <-P^PI^ > >< I(PJ 0PI4)I P p.(3.15)

It follows at once that the first vertical asymptote of ( appears
O
for > Eg+. In general, the eigenvalues of a self-adjoint operator A
are greater than or equal to the eigenvalues of a self-adjoint operator
.B in crder if A > 3.2
It is an immediate consequence of this that for E < E0 t-
2,9
i sts and satisfies the inequality'

S : -T > V- ? c. (3.16)

Hence V t> 0 and o + V >/f + to > f. Applying (3.13) we
find

QC ^ k U 0+> 1'+0 + ;4

and therefore,

S>(i,2,...,g) (3.17)
L 7IL L)
This says that the eigenvalues of 1 are bounded above by the average
values of with respect to the basis o and bounded below by the
eigenvalues of in order for < EO
S~o -g+l1
For each value of , E will satisfy the equation

( (i=1,2,...,g). (3.18)

From this it follows that for any P the curves i. (i=l,2,...,g),
have negative definite slopes. From (3.10) and (3.11) we have

(3.19)

-- ( 1 (v-'"(- !-) V) "-z)"I ) .

dT0 2
We see from (2.26) that -- = T Making this substitution in
(3.19) we obtain

/= -

Each of the curves has a slope at the point E for which ii and (i
were evaluated given by

I (ie d I, (3.21)
( s,2 g

Hence, the curves have negative definite slopes and the bracketing

theorem holds for each of the g eigenvalues of 1. From (3.21) and

the inequality (3.15) it follows that for every Ek< Eg+, where

Ek is the k th lowest eigenvalue of the Hamiltonian associated with

a state *k not orthogonal to the subspace there is at least one

solution to (3.18) that will provide a lower bound to Ek. In fact, it

is now easy to prove that under these conditions the k th lowest

solution of (3.18) is the lower bound (or highest lower bound) to Ek.

From the discussion associated with Fig. 1, we observe that the

curves of E1 cross the =l = 2 line at the eigenvalues of 9 asso-

ciated with states *i having the property that (Di # 0. From (3.15)

we found that the first vertical asymptote was situated to the right of

Eg0, i.e., E > Eg+1. Let there be k eigenvalues of f having the
g+,, g+i1 1
ordering E < E2 < ... < Ek Eg+1, associated with states satisfying

the above property. Let the eigenvalues of 1i also be ordered, i.e.,

11 s 12 < <"' < ig. Then, E is the lower bound to El for
E1 <
let us assume there exists for some j > 1 a solution such that

Cij = El for e = E1. Then, from the non-crossing rule and (3.21),
S11 must be < E1 and hence crosses the 1 = e line for some value of
S< E1. This contradicts the assumption that El is the lowest eigen-

value of associated with a state satisfying the requirement that

O)r # 0. Similarly, E12 provides the lower bound (or highest lower
bound) to E for E < (Eg+. If for some j > 2 there exists a

solution lj such that lj = E2 for E = E2, then by once again

22

imposing the non-crossing rule and (3.21) we find that 12 must cross

the E =E line for a value E < < E2 which leads to a contra-

diction. By induction, we see that, for any Ei ( ( Eg+l,' li is

the lower bound to E., (i=l,2,...,k).

Figure 2 illustrates the typical behavior of the multivalued

bracketing function (3.11) for g = 4 and k = 2.

E,
_ _ I

E E2
1 2

Fig. 2. Typical behavior of
function 1 for g

the multivalued bracketing
= 4 and k = 2.

CHAPTER IV

LOWER BOUNDS TO THE REACTION OPERATOR

In principle, equation (3.11) can be solved exactly and provides

lower bounds to those states of the Hamiltonian i satisfying the

criteria that .i # 0 and < E .g+ In fact, there are few prob-

lems for which (3.11) can be solved exactly.

From (3.10) we see that the reduced resolvent T must first be

obtained in order to calculate the reaction operator t. From (3.6)

we know that T < 0 for < Eo. If T cannot be evaluated exactly,
o g+1 o
one approach would be to estimate it by the operator To(p):

19 1I L
(4.1)
0
P P.

/ r d .p+-

where it is assumed that the eigenvalues are arranged in the order

E < E2 < .... T (p) is an upper bound to T Hence, from (3.10) we

see that

and snc it is a d tt V > (4.2)

and since it is assumed that V > 0,

-1 -1
Since tg (p) and tg are positive definite, the inverse operators t&(p)

and t. exist, are positive definite2'9 and satisfy the inequality

;t > CLf) P0. (4.4)

Better lower bounds to the reaction operator maybe gotten by

introducing the idea of inner projections.0

Let f = (fl,f2,...,f ) be a set of n linearly independent

vectors spanning a subspace in Hilbert space having the metric A =
Then the projection operator associated with this linear manifold is

given by

Let P = 1 Q be the projection operator associated with the orthogonal

complement to f.

The projection operator Q is positive definite since

(4.6)

Hence, P = 1 Q > 0 since P also is a projection operator. Therefore,

Q satisfies the inequalities

& s0 (47)

The reaction operator tL is positive definite for E + 1 11
adjoint: t = tg Hence, t2 exists and is positive definite. The

inner projection of t- with respect to the subspace f is given by

(4.8)

From (4.7) we find that t' satisfies the inequalities

^ >0 F, > 0 (4.9)

Let us define two new manifolds and where

S, 4-(4.10)

Substitution of these expressions into (4.8) results in the following

relations:

.A"> 4, AE-<^^g 4(4.11)

and

_ I^ "A / = < P-/ >. (4.12)

The inner projection given by (4.11) will be called the Aronszajn projec-

tion and the space = (gl,g2,...,g ), the Aronszajn space. That given

by (4.12) will be called the Bazley projection and the associated space

S= (hl,h2,...,hn), the Bazley space.

LBwdin has shown that a lower bound to an energy level E of the

Hamiltonian O( may be obtained when there are g eigenvalues of Ro

less than E, if the Aronszajn space contains at least all the
Associated with the g unperturbed levels
functions 1 2' ... associated with the g unperturbed levels

and the reference function P is constructed by taking a linear combi-
nation of these g functions.
Replacing the reaction operator tE in (3.11) with the Bazley

projection defined by (4.12), we have a new bracketing function

^=. +(4.13)

From (4.9) we see that for E < E g
g+l'

go+.te >' ++te > (O. )

and therefore

+ (4.15)

From this we may conclude that the eigenvalues of i are bounded above
by the eigenvalues of i and bounded below by the eigenvalues of a
in order;

El, > (i=1,2,...,g) (4.16)

From the bracketing theorem, Eli is a lower bound to E., where

E is the i th lowest eigenvalue of f satisfying the criteria E +1) E >E
and 0 i i 0. Therefore, '1i is also a lower bound to the same eigen-

value. From (3.17) we observe that the eigenvalues of~ are bounded

above by the eigenvalues of R with respect to the basis a in order.
It follows at once that the latter also provide upper limits to the

eigenvalues of i in order.

More favorable upper bounds to the eigenvalues of E1 may be
obtained by introducing the inequality (3.16) into the expression for
tg which gives the result

Hence, we have that

V ">, 0 (4.17)

I I I
where Lim t = V and V is simply a Bazley projection of V onto
F ,-* 66' E
the space spanned by the manifold R, It follows at once that

V >,/ V > 0 (4.18)

From (4:17) we see that

V V ( + >S4

hence,

&( ^+ v')
The eigenvalues of 1 are bounded above by the eigenvalues of 4o + V
and bounded below by the eigenvalues of o:

h' E (i=1,2,...,g), (4.20)

where h. is the i th lowest eigenvalue of with respect to the
bass (40 and
basis o. The inequalities given in (4.20) hold for all 4 Eg and
tell us at once that has no vertical asymptotes in this region for

any Bazley projection. The equality li = E occurs whenever. = 0:
when the Bazley space is chosen orthogonal to the reference manifold
(.
a
The inner projected bracketing function given in (4.13) can be
computed exactly in only a few cases involving particular choices of the
Bazley space; 21314 otherwise, T is truncated according to (4.1),
which results in an operator t. (p) that, for E < Eg+1 satisfies the
inequalities

E V '; V C0 (4.21)

To remove the problem of dealing with the reduced resolvent To,
LBwdin has suggested the substitution A = (E- S o) be used in
making the inner projection.2 Substituting this into (4.12) we find that

1 (E-A0 (4.22)

where

K~J(E- K) v_'-) -

since, from (2.16), ( Vo)To = ( E )PT = P( o )To = P.
We shall call tE defined by (4.22) the LBwdin projection and the asso-
ciated space = (j,j2,...,jn), the LBwdin space. This projection
has the interesting property that the reduced resolvent T has been
O

eliminated from the calculation. It also leads to some very interesting

behavior not observed wnen using the Bazley or Aronszajn projections.

This behavior can be explained by the fact that the linear manifold onto

which the projection of te is made is itself a function of the variable

Sand varies with changes in .

An immediate consequence of the C dependence of the Lbwdin

space is that the slopes of the eigenvalue curves of El are no longer

negative definite. In fact, minima occur in these curves for values

of = E, (i=1,2,...,g). That is, one of the eigenvalues of l for

E = E < Eg+ and e is = E and Eo will be a minimum
k k g+l k V a' k
value of the function C '

This is easily seen when we write down the expression for the

iU th element of 1 using the LBwdin projection (4.22).

V- (4.23)

/

The only non-vanishing term in the k th row or k th column of 1 when

= E is the kk element, ( )k = Eo. We can write the expression
for (413) evaluated at the point E
for (4.13) evaluated at the point jE Eo

tlk-l
0-

0

S+tk+l,k-l

tgk-
-g,k-1

0 ck+

tk-l, k+I
0

Ek+l +t'+l, k
k+1 k+1,k+i1

0 tg' ,k
g,k+l

1 11
*I tli* "

*.

Obviously, E0 is an eigenvalue of for = E. That this is also a

minimum in some eigenvalue curve, e.g., of I is seen by looking

at the first and second derivatives of i with respect to E evaluated

at E. If E = Ek = then

= o > 0

are sufficient criteria for the point

( I, curve. From (3.21) we know that

S= Ek to
k

be a minimum in the

(4.25)

tk-l,
0

k+l, 1

\t.g,1

1 =
i -^

(4.24)

.. t' \

K l,g

0

k+l, g
*

... E+t'
'g gg

cdl./ =-
dE

.\$~
CeE

where ( is the)normalized eigenvector of i associated with the
eigenvalue Ek. Since
0
0
C1 10
ok
0
0

is the eigenvector for

d e ,
d4

= 8, it follows from (4.25) that

(0 ... 01kO ...

- d E E

(4.26)

From (4.23) and the
Swe obtain the express

A-I

identity A'
s5ion
vision

(n

-I

<^>-
1

from which it follows that

(4.28)

= 0.

(4.27)

c60

= A-' dA
d

Ec')

= si: E\

0) (0 E
do.

C1f

= C[E-

I-'4A

The expression for the second derivative is given by

In general, the curve 1 can have minima and cross the
= g It fol s tt

( l
X F (l> 4") <-i<

c < <> (4.29)

Hence, from (4.28) and (4.29) it is obvious that the g curves of^-
using the LBwdin projection are no longer monotonically descending.
In general, the curve 8i can have minima and cross the 4 =E
line at several of the points E = E?. It may also cross the E =
line at values of 6 not eigenvalues of 0. From (4.20) we have the
relation h > E for any a < Eog It follows that

E / h. These features of the LBwdin projection are developed further
in the application of this theory to He and Li.
In this section, we have developed a procedure for calculating
lower bounds to the eigenvalues of a Hamiltonian R = ?o + V,where

V > 0, and L has a set of known eigenfunctions. The subspace o

is chosen such that C .i O, where *i is the eigenfunction of

associated with the i th lowest eigenvalue E. having a particular
1
symmetry and such that it contains all the eigenfunctions associated

with eigenvalues of qo less than E. having the same symmetry as r.

The L'wdin space must be chosen such that the overlap integrals

< 01jk> with the functions of the reference manifold O are non-

zero. It is then a rather straightforward matter to calculate the

matrix elements given in (4.23). If Tr() = g, then by choosing the

parameter > E. and <(Eg+ the i th lowest solution of the secular

ccuation

S- i (0.30)

provides a lower bound to E..

CHAPTER V

LOWER BOUNDS TO THE 3S STATES OF He AND Li

The procedure-discussed in Chapter IV is used to calculate
3 +
lower bounds to some of the low-lying S states of He and Li For the
two-electron series, the nonrelativistic Hamiltonian in atomic units
is given by

1-.- (5.1)

where z is the nuclear charge, 7r and r2 are the position vectors of
the electrons, r. = 7.i i = 1, 2 and r12 = Ir1- T21
We will take advantage of the symmetry properties of the
Hamiltonian (5.1) and restrict ourselves to states having 3S symmetry.
For these states, the eigenfunctions of (5.1) depend only on rI, r2 and
r12 and are antisymmetric in the spatial coordinates of the two elec-
trons,that is

P --, ) P1 (5.2)

where PI2 is the permutation operator acting on the coordinates rl and r2.
The Hamiltonian (5.1) may be separated into an unperturbed part

Sand a positive definite perturbation V, where

o. Y2 .p (5.3)

V- (>3)>
V--L->0
v- I;.?-

The eigenfunctions of I. are well known and for the S
syetry are given by
symmetry are given by

where the R n are normalized hydrogen radial
n"i
the a th normalized Lengendre polynomial and

the radial vectors r1 and r2. The C n are
o 2
the >) n2 are normalized to t ,
b 12

wave functions, P is

912 is the angle between

chosen in such a way that

( ) = 7T (

The eigenvalue spectrum for the 3S levels of is given by

Th e I

5.5)

o 4
MIMI 4 (5

For the lowest ordered S states of J { one electron is in the n = 1

level and the angular momentum L2 has the eigenvalue zero, 0 = 0. The

eigenfunctions and eigenvalues of o are then given by

o 10
= i-[ (5.

--(5
S r

By including in the reference manifold (') only those functions ^O

having S symmetry, we obtain from (4.30) lower bounds only to those

states of \ having this same symmetry since r = 0 for any state V

of not having S symmetry. Hence the solutions of (4.30) will be in

order lower bounds to the 23S, 33S, 43S, ..., jS states of i.e.,

6)

7)

for <(E+1 and >E3S,

< < ^ .s 5) < E53 i> I iS

With the solutions of Ro available for the construction of the

reference manifold &0, we must next select the functions for the
Lbwdin space It has already been pointed out in Chapter IV that the
choice of functions for the 4 manifold should be made subject to the

condition that the overlap integrals satisfying the criteria that

4 a. (5.8)

2 0
where L is the angular momentum operator, can satisfy these conditions.
The functions were selected on the basis of those introduced by
Hylleraas15 and are given by

Sc' Ur/ (5.9)

where Y is an arbitrary scaling parameter and

S= + = t; (5.10)

The Ck.m. are normalization constants selected in such a way that the

functions given by (5.9) are normalized to l6ir2. This choice of normal-
ization proves to be convenient in the calculation. It is easily seen
that the functions given by (5.9) satisfy the criteria (5.8).

The minimum size of the reference manifold is determined by the

number of states of o having the same symmetry as and eigenvalues

less than those of the state of S to which a lower bound is sought.

For the S states of He the ordering in atomic units (a.u.) is

given by
o o 16,17
E = -2.500 ... E = -2.22 ... < E -2.175229 16,17<
12 23S

o = -2.1250 ... ( = -2.080 ... E = -2.0687 18<
3 4 33s

Eo = -2.05050 ... (5.11)

and for Li ,

El = -5.6250 ... <(E3 = -5.110727 19 2 = -5.00 ...
1 23s 2

E = -4.781250 ... ( 3 33S 4

E o= -63718 .<.E= -4.6250 ... (5.12)
43s 5
From (5.11) it follows that, to calculate a lower bound using (4.30) to

the 23S state of He, we must at least include 01 and 0o in the

reference manifold a' i.e., g = 2 and for a lower bound to the 33S

state we must choose g = 4, i.e., (a 0 2' 3 From

(5.12) we see that, to obtain lower bounds to the 23S, 3 S and 43S levels

of Li it is necessary to have g = 1, 3 and 4 respectively. Of course,

as many additional functions as desired may be included but these values

of g give the minimum sizes of the reference manifolds necessary in

order to calculate lower bounds by this procedure to the levels indicated.

The size of the Lbwdin space, which shall be denoted by NINT,

can vary from one to all the functions of a complete set. If the set

is complete, then the exact eigenvalues of S are found to be those

values of E at which the curves cross the = line. In

general, however, this set is finite and the curves cross the E =

line at the eigenvalues of o associated with functions in the

reference manifold. Crossing may also occur for values of f E,

i g + 1.

Once the reference and internal projection manifolds are chosen,

the matrix elements of EI, given by (4.23), may be found and the

secular equation (4.30) solved. Both of the parameters I and 1 are

varied to give a maximum in each of the g lower bound curves, where the

variation of El with respect to E is subject to the condition that

be greater than the energy of the state to which a lower bound is

desired and less than E The computation procedures involved will be
g+1
discussed in the Appendix.

We see in Table 1 the lower bounds obtained by this procedure

using 40 functions of the type given in (5.8) to construct the Lbwdin

projection, where the restrictions k. = 0, 1, 2, 1. = 0, 1, 2, 3, 4,
1 1
m. = 0, 1, 2 and k. + m. ( 3 were imposed on the powers of the variables

s, t and u respectively.

Lower bounds were calculated for the 23S level of He by taking

the size of the reference manifold to be g = 2 and g = 4, and for the

23S level of Li by taking g = 1 and g = 4. Since the size of the

reference manifold determines the size of the secular equation, a fourth

order equation is the largest to be solved. From the results in

Table 1 we observe that, so long as all the eigenfunctions of 9o

associated with eigenvalues less than the 23S level of R are included

0 0 3
*d C O

4C) CM Z -4

o S
0

o *

i- (

iI

Z
z
z

0
erX
+

w
0

E-4

CLI
r-4 H

i-. C/)

E---

0c
0

Ci

Z

3

H

E-r
CL4
0

0 0 0 0
o 0 co 0 0 nt\ -
0 0 -1 0 0 0 n-
0 0 0 0 0 0 0

t- O \O 0
CM CMJ 1 -- 1 -- 1l-
UO. I \O 0 0 ,-
C ) o -I -4 I\ ON
,-4 ,-4 0 ,-4 ,-4 t- \O
C CM CM -
I I I .I I I I

0 CM 0 0

m C- L'--- L'-- -(
on 01 0 0O
S CM CM CM -4

CM CM I I- I-
Lr\ U \ 0 0 0,
n 1- 4\ -- 2 \B CM
-- 0 -1 '- \ 10

CM CM CM I \ -

c) co

n n cO 0 0 CM C-
-4 -4 0 r-4 r- t 0

CM CM CM U\ -

CM CM CM (1 ) (1 y yM

_-t CM -_: -_t -_ --t

M M O / /) M M
CM CM f1 CM CM on -t

o, v

N

1-4
+1

.r

u
C:

,-4

01
0
LW

-'-I
-4
C

4

0
0

0

0)
1-4

co *l 0 C

L) (L) L) 0)
(B 0 '

r-4
a) 0
ca rz
CO O
Crih

in )o, little improvement is found when more reference functions are
dja
added. The magnitudes of the errors in the bounds to the higher states

are seen to be much greater than that of the 23S state in both the He

and Li calculations. This is, however, not an unexpected result. It

has been pointed out by Lbwdin7 that a truncated set which works well

enough for the ground state may give a surprisingly poor result for

even the first excited state of the same symmetry.
17
Quite good lower bounds have been calculated by Pekeris1 to the
S20
2-S state of He using Temple's method. By solving a determinant of

order 252, he obtained a: lower bound of -2.175239 a.u. The lower bound

to this state, for g = 4, which is given in Table 1, is very close to

this value. In order to obtain a lower bound to an energy level of R

using Temple's formula, it is necessary to calculate matrix elements of
2
RP and to have a knowledge of the next higher level of the system.

It is apparent from the work of Pekeris that in order to obtain a good

lower bound using Temple's formula, the size of the secular equation

must be very large.
21
Goscinski21 has shown in his dissertation that Temple's formula

can be derived from (2.19) for g = 1.
22 3
Miller22 has obtained lower bounds to the low-lying S states

of He using the method of intermediate Hamiltonians.23 Solving a

secular equation of order 9, he calculated lower bounds to the 23S and

3 3S levels of -2.1802 a.u. and -2.0704 a.u. respectively. His result

for the 33S state is somewhat better than that given in Table 1.
Goscinski21 has shown that Miller's equation can be derived from an
Goscinski has shown that Miller's equation can be derived from an

expression for the bracketing function r using a truncated resolvent

To(p) as is given in (4.1) and that the points = elk correspond to

its solutions. By making this truncation, however, one reduces the

contribution from the continuum, a problem that is eliminated in using

the LBwdin projection.

A study was made to determine the behavior of the lower bound

curves when the scaling parameter 9 is fixed and is permitted to

vary over a wide range of values. For both He and Li g was taken to

be 4 and NINT to be 40. In Chapter IV we observed that the LBwdin pro-

jection greatly changes the character of the lower bound curves from that

shown in Fig. 2 for example. We found that, using this projection, a

minimum appears in one of the lower bound curves at each of the points

where 9 equals an eigenvalue of associated with an eigenfunction

in the reference manifold. This behavior is shown in Figs. 3 and 4.

These figures demonstrate the effect the L8wdin projection has on the

eigenvalue curves of ".

It is interesting to examine the effect that changing the number

of internal projection functions in the LBwdin space has on these curves.

Of course, using fewer functions in constructing the projection t&

will result in a poorer approximation to the reaction matrix t

Keeping ? = 1.2, the results of using NINT = 20 and NINT = 10 are

shown in Figs. 5 and 6 respectively.

We observe in Fig. 5 that the lowest bracketing curve 11 does

not recross the 1 = ( line after the minimum at = E2, as it does

in Fig. 3 and the minimum at = E occurs in the 13 curve and not
3 E13

0 0 0 0 0 0
O O O O O O
OM M CM C M
j aj av o o

in
* z
-I ,

0
-r\

C C Z
o* z
C Z

1 1
II

o

Co
0

0
II

,

0
COM
'0

cn

^1

if\
-i-
C .
OQ

o0 0 0 0
ON c-, U-

-~~ -rt\U- -

C

0

z
0 H
o z

I -

4-
0

4-4

o 0
- CM

C1C)

0
C,
o 0
4

CO

0
0

U-

0 0 0 0 0 0
0 -u C'] -
C' CU c']C '

I
0

UJ
c-

O
C'.] 0
,-4

II

0
II

.0
I 4

o
~ O

0
,-4

I

rr4

3
C']
1P

I

r-e

CM
I HC

10

II

O
Z

0
C-)

CO

0
0
C,
3

U-

OJ

o 0
0 *-

I I

in the ~ 2 curve as it does in Fig. 3. In Fig. 6, we find that using

only 10 functions to construct the LBwdin projection results in the

minimum at E = E2 being shifted from the E1 to the 2 curve.

It is also apparent that the ' 13 and curves vary only
12' 134
slightly from the unperturbed eigenvalues E2, E and E respectively,

over the region under consideration. Only the E curve is appreciably

affected by tz

For NINT = 10, we find, by permitting 7 to vary, that the mini-

mum for = E is shifted back to the 11 curve for ) = 1.35. For

; = 1.35 and E = -2.06, the lower bound to the 23S state of He is

-2.216 a.u. These results are shown in Fig. 7,

We conclude this study of the lower bound curves by noting that,

if the LBwdin projection provides an adequate approximation to the

reaction operator tE the i th lower bound curve will have minimums at

each point = E where E. j- j 1
line for the last time for some value of 9 not an eigenvalue of

If the set onto which the projection is being made is complete, this

latter crossing will occur at the eigenvalue E. of

This behavior is observed in Fig. 3 where the Ii curve has

minima for = E and = Eo and crosses the 1 = J line finally
1 E2
for = -2.1753. From (5.11) E < Eo < E < E ; hence, the 12
23s 3 4 33S E 12
curve should have minima for = E3, Eo if this projection adequately

represents the reaction operator. We see in Fig. 3 that the 1 curve

crosses the = line only once, at the minimum 2 = = E9.

The minimum for = E appears instead in the 13 curve. From the

results in Table 1, we see that 2 provides a much poorer lower bound
to the 3 S state than 11 does to the 2 S state of He.

0 0 0 0 0
(Mc***cyc

CM O
1' 1

,-
II

0

C)
O
'-4

I

0.
ci

11
a,

-0

w

iCi

49

A similar analysis can be made of the Li curves shown in Fig. 4

and the He curves shown in Figs. 5, 6, and 7.

Although the choice of functions of the type (5.9) used to con-

struct the Lbwdin projection is perhaps not the best possible selection,

the bracketing curves obtained using this choice illustrate the general

behavior we may expect when using this projection and the material

discussed in Chapter IV.

APPENDIX

This is an outline of the procedures used in the calculations
of Chapter V.
The calculation of the LBwdin projection (4.22) and the matrix
'; (4.13) require the evaluation of the quantities Voji, i=1,2,...,
40, and the a matrix whose matrix elements are given by

<'> + 'V

^<> (<^I^}J +

OIJ) ,

where g = 4.
The Hamiltonian R and the inverse of the perturbation V ,
defined by (5.3), may be expressed in terms of the corrdinates S, t
S 24
and u:

(A.2)

,

D tUDS' SS-tL 7Yb < S 2i(&-t b b2At Sp-ta

7t -'+ + t .Z+

50

V-'/= t

From (A.3) and (5.9) we see that \$o operating on the functions
ji gives a linear combination of terms having the form

(SW-tz)

s ,22o'-l
St

where n = -1, O, k > 0, L > 0 and m >, -1.
From this we can see that the matrix elements of (A.1) involving
only functions of the LBwdin space are simply linear combinations of

integrals of the type

(m, ) 0 0
00 0

(A.5)

where n = -1, 0, 1, k ) O, 1 >/ 0 and m > 0. These integrals may all be
evaluated in closed form:25

1
- + 2t +an+3

(A.6)

-a. ( 4e hrm th >
a +

AT'
(1Y I /3(-1 im -t
I
(Rtw3 ) (X7wf^mt4)S '

L( +r U
.2r 21 ) i +\ S It+frMW

f U +ro

(A.7)

I I-+- (--). A8)
V ^ (A.8)

*(t- )(-I),)2U...-

e-7

(A.4)

(o, p,rr"") =

(2ztti)(22tm7tZ)

(-, u M) =

( 4=0 -,

The integrals involving functions of the reference manifold and
the LBwdin space <( .jk) can be expressed as a linear combination of
integrals of the type

(S-t') tQ e-t
(A.9)

where a > 0, k / 0 and m >0. The solution to this type of integral is
also known in closed form3 and is given by

(4, u, 4,4, )= ^2 5-
ff0

(g-b)e' f' ,

,H ,.2
L t+M +^

(4 1,)

.b)2^ -/

(a)b, /)= b)

L ( -)'

r 4ch+;)Q +/)
( 4-b .

b. 4
b Q~rz'

( z a 4' I

(^/M- /)! (z2) /
6j

Vim .
frn^

dj 4 6 ~ 'hl/

+' 1

(ptnvw \;
n

I+

(A.10)

~P i

Ipf-rm~!
;P!

Cet~) (a~-~ 'IC

(4t~)(P+I) ii-

I' c-lr~

The number of integrals of this type and of the type given by

(A.6), (A.7) and (A.8) was quite large, so a program was written for
evaluating them on the IBM 709 computer.

Once the integrals were solved, the matrix elements 4kn were

calculated and the A matrix constructed. The inverse of this matrix,

-1, was obtained using the method of successive partitioning.26 It
was then a rather easy task to determine the matrix elements of 1'
which are given by the formula

1 E?) (E- E~- <1
(A.11)

The secular equation (4.30) was solved using the subroutine
JACFUL, that generates the eigenvalues of a real symmetric matrix by
the Jacobi method.27

REFERENCES

1. P. O. Lbwdin, J. Mol. Spectry. 10, 12 (1963).

2. .P. O. Lbwdin, Phys. Rev. 139, A357 (1965).

3. P. 0. Lbwdin, J. Chem. Phys. 43, S175 (1965).

4. P. O. Lbwdin, Proceedings of the Madison Symposium on Perturbation
Theory, 1965 (J. Wiley and Sons, Inc., New York, to be
published ).

5. J. H. Choi, "Lower Bounds to Energy Eigenvalues by the Partitioning
Technique," Ph.D. Dissertation, Department of Chemistry,
University of Florida (1965).

6. P. O. Lbwdin, J. Math. Phys. 3, 969 (1962).

7. P. 0. L8wdin, Advances in Chemical Physics (Interscience Publishers,
Inc., New York, 1959), Vol. II, P. 207.

3. E. A. Hylleraas and B. Undheim, Z. Physik 6L, 759 (1930). J. K. L.
MacDonald, Phys. Rev. 43, 830 (1933).

9. P. R. Halmos, Introduction to Hilbert Space (Chelsea, New York,
1957).

10. N. Aronszajn, Proceedings of the Oklahoma Symposium on Spectral
Theory and Differential Problems, 1959 (unpublished).

11. P. R. Halmos, Finite Dimensional Vector Spaces (D. Van Nostrand,
Princeton, 1958).

12. C. E. Reid, J. Chem. Phys. 43, S186 (1965).

13. J. H. Choi and D. W. Smith, J. Chem. Phys.,.43, S186 (1965).

14. C. F. Bunge and A. Bunge, J..Chem. Phys., 43, S189 (1965).

15. E. A. Hylleraas, Z. Physik 54, 347 (1929).

16. C. L. Pekeris, Phys. Rev. 115, 1216 (1959).

17. C. L. Pekeris, Phys. Rev. 126, 1470 (1962).

18. C. E. Moore, Atomic Energy Levels, National Bureau of Standards
Circular No. 461 (U. S. Government Printing Office, Washington,
D. C., 1959), Vol. I.

19. C. L. Pekeris, Phys. Rev. 126, 143 (1962).

20. G. Temple, Proc. Roy. Soc. (London) A119, 276 (1928).

21. S. O. Goscinski, "Upper and Lower Bounds to Eigenvalues by the
Partitioning-Technique,'' Ph. D. Dissertation, Department
of Chemistry, University of Florida (1966).

22. W. H. Miller, J..Chem. Phys. 42, 4305 (1965).

23. N. W. Bazley and D. W. Fox, Phys. Rev. 120, 144 (1960).

24. L. Wilets and I. J. Cherry, Phys. Rev. 103, 112 (1956).

25. J. G. Gay, "Lower Bounds to the Eigenvalues of Hamiltonians by
Intermediate Problems," Ph. D. Dissertation, Department of
Physics, University of Florida (1963).

26. R. A. Frazer, W. J. Duncan and A. R. Collar, Elementary Matrices
(Cambridge, London, 1938).

27. F. Prosser, "Matrix Diagonalization by Jacobi's Method," QCPE 4
(1963).

BIOGRAPHY

Timothy Michael Wilson was born August 3, 1938 in Columbus,

Ohio. He moved to Saint Petersburg, Floridain 1950. He was gradu-

ated from Saint Paul's High School in June, 1956. In June, 1961, he

received the degree of Bachelor of Science in Chemistry from the

University of Florida. Since that time he has been pursuing work toward

the degree of Doctor of Philosophy at the University of Florida. He

was awarded the Kopper's Summer Fellowship in July, 1963, and in

September, 1965, he received the DuPont Post-graduate Teaching Assis-

tantship award in Chemistry for the academic year 1965-66.

Timothy Michael Wilson is married to the former Iris V. Barron

and is the father of one child. He is a member of the American

Physical Society and Lambda Chi Alpha.

This dissertation was prepared under the direction of the

chairman of the candidate's supervisory committee and has been

approved by all members of that committee. It was submitted to the

Dean of the College of Arts and Sciences and to the Graduate Council,

and was approved as partial fulfillment of the requirements for the

degree of Doctor of Philosophy

August 13, 1966

Dean, Colle f ,
Art and Sciences

Supervisory Committee:

Chairman

/ J ^

Full Text

PAGE 1

LOWER BOUNDS TO EIGENVALUES OF THE SCHRODINGER EQUATION By TIMOTHY MICHAEL WILSON A DISSERTATION PRESENTED TO THE GRADUATE COUNOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1966

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UNIVERSITY OF FLORIDA 3 1262 08552 2539

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ACKNOWLEDGMENTS It gives me great pleasure to express my appreciation for the assistance, both direct and indirect, provided by my supervisory committee and by numerous other members of the Chemistry and Physics departments faculties and the staff of the Quantum Theory Project. In particular, I would like to express the gratitude I feel toward the chairman of my committee, Professor Charles E. Reid, for both the time and advice he has so frequently given me over the years I have known him. His encouragement and suggestions have been invaluable. I am particularly indebted to Professor Per-Olov LUwdin for first introducing the subject of lower bounds theory to me and for his stimulating lectures and pioneering work in this area. To Dr. Jong Hyuck Choi, I am extremely grateful for the many helpful discussions and the expert assistance in writing some of the programs used in the lower bound calculations. It is with pleasure that I acknowledge the excellent assistance of Mrs. Philamena Pearl in the preparation of this dissertation. The financial support of the National Aeronautics and Space Administration and the grant of considerable computor time by the University of Florida Computing Center is gratefully acknowledged. 11

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES iv LIST OF FIGURES v Chapter I. INTRODUCTION 1 II. THE PARTITIONING TECHNIQUE AND THE BRACKETING THEOREM 3 III. LOWER BOUNDS TO ENERrv STATES 16 IV. LOWER BOUNDS TO THE REACTION OPERATOR . 24 V. LOWER BOUNDS TO THE -^S STATES OF He and Li 35 APPENDIX 50 REFERENCES 5^ BIOGRAPHY 56 iii

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LIST OF TABLES Table Page 1. Optimum lower bounds to some of the S states of He and Li"*" for NINT = UO kO tv

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LIST OF FIGURES Figure Page 1. Typical behavior of the multivalued bracketing function S", = ^CÂ£)for g = 3 Ik 2. Typical behavior of the multivalued bracketing function f for g = i^ and k = 2 23 3. Lower bound curves of He for g = k, NINT = t^O and ^ = 1.2 1+3 k. Lower bound curves of Li"*" for g = U, NINT = kO and )7 = 2.1 kk 5. Lower bound curves of He for g = k, NINT = 20 and n = 1.2 U5 6. Lower bound curves of He for g = k, NINT = 10 and 7^ = 1.2 U6 7. Lower bound curves of He for g = k, NINT = 10 and T^ = I.35 kQ

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CHAPTER I INTRODUCTION One of the basic problems in quantum theory is the solution of the time-independent SchrBdinger equation where the Hamiltonian ^ is bounded below and has associated with it a set of solutions (E,) corresponding to the discrete levels lying below the continuum. This equation cannot be solved exactly except for a few cases corresponding to very simple systems. Instead the problem is solved by adopting a suitable method for obtaining an approximate solution. One particularly useful method for treating this problem is the parti1-6 tioning technique. This approach has the interesting feature that it contains many of the more conventional methods such as the variational principle and perturbation theory as special cases. Using the partitioning technique one may construct a function C, = f(C)> where Â£ is a real variable, having real solutions Â€,-,Â» Sl2*^lV"''^l ^^^^ ^^^ property that each pair {Â£,Â£,.) will bracket at least one eigenvalue E of the Hamiltonian Ji. . This means that if Â£ is chosen as an upper bound to an eigenvalue E, at least one of the solutions g .. will be a lower bound to E. The problem of constructing this function and a way of determining which of the

PAGE 8

solutions Â€ , . is the lower bound to E will be studied. The lower bound to E is taken to be the largest of the solutions Ci-' (^~^'^Â» ...,k,k^g), providing a lower bound to E. Furthermore, it will be shown that lower bounds to the solutions of C, may be constructed by making an inner projection of the reaction operator twith respect to a finite basis, where the reaction operator is associated with the separation of the Hamiltonian Jf ~ >f + V. Several types of inner projections are discussed with particular emphasis on the "LBwdin Projection." An application of the method is made to a calculation of lower bounds to some of the S states of He and Li using the "LBwdin Projection."

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CHAPTER II THE PARTITIONING TECHNIQUE AND THE BRACKETING THEOREM Consider the problem of solving the SchrBdinger equation J>i|^ = E\lf, where the Hamiltonian }\ is assumed to be hermitian, i.e., 0( = J^ and bounded below. Let (jD = [ ^^ J be a complete orthonormal set of functions. As an immediate consequence there is a resolution of the identity. I = E-l9><9/ . From this definition it follows that ^ satisfies the relations Let P = 1 ^ be a projection operator associated with the subspace (jO^^f the orthogonal complement to the subspace (Â© : (2.2) (2.3) f.=f-S Q. Â•

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It then follows that p*= p ^ Â©''= iS) , OP^PÂ©=^o. (2.4) (2.6) The function y can be projected onto the complete set CXr and the SchrBdinger equation can be rewritten as Operating from the left first by C?, then by P, we obtain the set of equations (Â©i8Â© + (95^P)iP= FOV^, (2.7a) (PJ?0 + PSÂ«P)'!P= Â£PV^. (2-7^) From (2.7b) we find that ^Lpce-k)p1p^^'-'^ From {2,k) we see that, for an arbitrary number a, Then (2.8) can be rewritten in the more general form (PKÂ©)Â©^= t'+ P(E-Â«)Pj P"^, (2-9)

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Solving (2.9) for P\|/, we obtain the expression Pf=Z^-0iPCE-5?)PTlPifS>)(^V. (^-1Â°) Operating on (2.10) from the left by P gives (2.12) Substituting (2.11) for P\|; in equation (2.7a), we obtain (Oii)&Pi-CS>iliP)P'iP =^ BS>P (2.1 = f (9sf (s> f (fiÂ»i?p; prÂ«<<5 + p(Â£Â»)pr wÂ©] Â©v^ Let us introduce the reduced resolvent TÂ„ -]j = prw.(p+p(Â£->f)pj'p . (2-13) Equation (2.12) then becomes Introducing (2.11) and (2.I3). into (2.5) we obtain the expression P Let us introduce the symbolic notation TÂ„ = = rr . It is easily seen that T satisfies the relations P(E -i^)Tp = P,

PAGE 12

Since Orp^ ^'9!K5>JV^ ^ ^i J^i # multiplying (2.14) from the left by each of the m ^Â€i C^^, (k=l,2, . . . ,g) and integrating we obtain a set of g simultaneous equations (k=l,2,...,g). (2.17) This set of equations can be rewritten as a matrix equation and {C is a column vector composed of the coefficients appearing in the expansion of C'r in terms of the functions Â©.Â€Â• ^P . The solutions to this set of equations are found by setting the determinant of the coefficients of (L in the matrix equation Â— Â» a C W^fc^^^ * EJ^ai ^a ~ ^ equal to zero: The solutions of the secular equation (2.20) are eigenvalues of ^ associated with states not orthogonal to the subspace dp . This point will be discussed in greater detail later. One way of solving equation (2.20) is by means of some iterative procedure. Let Â£ and c, be real variables. By replacing the variable E in QrO with f , the elements of the matrix f/y become functions of ^ and the secular equation (2.20) has a set of g solutions I C 1 Â• f ^Â°^ each value of ^ , corresponding to those values of

PAGE 13

the variable Â£, that satisfy the equation Hence, the variable Â£ ^ is a multivalued function of Â£ having the same multiplicity as the order of the subspace (TD and this is denoted by the expression C , = jT ( Â£ ) Â• Corresponding to the set of g solutions j wi-j obtained by solving (2.21) for a fixed value of ^ , there will be a set of g vectors JsZ a.^ . Making the substitution of Â£ for E in (2.I9) we obtain the expression where (2.23) Then, for arbitrary Â£ , each of the solutions ^, . ( i=l,2, . . . ,g) may be expressed as a function ofÂ£, i.e., Â£^. =f.(C), having the form:

PAGE 14

8 Hi V Â£ . Â— = Ti ^^) Â• ^'-^'^ Taking the first derivative of (2.2U) with respect to Â£. and evaluating it at the same value of Â£ for which the (S . were evaluated, we have' ai fiTlk'^^ AÂ„ rf. . (2.25) We see from (2.23) that ^'=
PAGE 15

Replacing Â— Â— in (2.25) by the right hand side of (2.27). dÂ£ we obtain the result < , (i=l,2,...,g) Therefore, each of the solutions Â£ , . is a function of g having a negative slope for all Â£ . Let ^ = E + 6 , where e is the difference between the value of Â£ and some eigenvalue E of V^ . Let one of the solutions given by (2.2U) be represented as Â£ = E + 6,.. If fe is a small quantity, we can use the Lagrange mean-value theorem to obtain the result that = -^-(.B) )& ^'te-tfie) ^ where <, ^ 1. Using the identity E = f.(E) (see (2.18)) we obtain the relation From (2.28) we know that the first derivative f . is negative; hence,

PAGE 16

10 the quantities Â£ and S^. have different signs. This leads to the bracketing theorem which says that the pair (f Â»Â£Â•,.) bracket at least one true eigenvalue of J^ . Then the g intervals (Â£, Â£,-,.) will each contain at least one true eigenvalue E such that if Â£ Â• < E, then E 1 y E and vice versa. ** 1l ' ____^ Let us examine the behavior of (rO (Â£) in the limit of 3d Â£ -* + GO. We see from (2.22) that = ^.Â» + MÂ«.( Â£^,. <3>J i?Â« If,)} ^6a (-30) = W)*a Â• Hence, the set of g solutions { fi-j have as horizontal asymptotes the eigenvalues of the matrix 0^ : Let us define the "outer projection" j/f by the relation k= PHP. Since PÂ© = Â©P = 0, it follows that all the functions (D . Â€ (D . are eigenfunctions of S^ associated with the eigenvalue 0. (2.31)

PAGE 17

11 Let us consider the normalized eigenfunctions ^. of [^ associated with the eigenvalues E. that are situated in the subspace (S) . We then have the spectral resolutions p= Eivjx^i . *(2.32) It is obvious that the reaction operator has vertical asymptotes for the values ^ = E., (i=l,2, ...)Â• It then follows that the bracketing function^ ^ has vertical asymptotes for the values Â£, = E., provided E. is not simultaneously an eigenvalue of J*\ . Let i|/. be a function satisfying the SchrBdinger equation \Pilr. = E.ijf.. From (2.5), we see that i|f. = (P^. + P\|f. . Therefore, if (J)\lf. = and P\|;. ?^ 0, it follows that }p. ^ P lf>. = Tp. , (2.33) Then, from (2.7a) and (2.7b) we find that (2-31t) ^^. = (PÂ«P)^i = Eji/^. That is, for those values Â£ = E. where E. is an eigenvalue of J^ associated with a state \|^. that is orthogonal to the reference manifold O , i|a. is also an eigenstate of J# having the same eigenvalue, i.e., E. = E. . 1 1 Let us make the separation P= P'+ P". (2.35)

PAGE 18

12 where P' = Ill^^K'^fJ for all the f^ = t^^ such that (tf^if^ = a a P" = 2lltB)<^^J ^Â°^ 3^^ the t such that Â©iL j' 0. The reaction operator Tg may then be separated: iS C* :i^ (2'.36) = 'e + U . It then follows from (2.22) that . u . ri\$LaiMiMiM_ (3.37) Since j^ ^Qj = E^^c^ and Â© t^^ = 0, we see that i(f^\)K\ ^^ = E <Â© I t /^ = foi^ all (p e. Cp . Hence the second term in (2.3?) drops out and we obtain the final result where Â•From (2.38) it is easily seen that for S = E^, where E^ is a solution of If t. = E\|f. and G)t= 0Â» the bracketing functionf^^ has no

PAGE 19

13 vertical asymptote. Further, we see that S , has vertical asymptotes only for those values of Â£ corresponding to eigenvalues of P ^ P not simultaneously eigenvalues of J^ , i.e., only for those cases where (P ^i ^ Â°' o It follows from the variation principle that the eigenvalues of ^ are upper bounds to the eigenvalues of ff in order, Â£^ >/ E J . (2.39) Hence the vertical asymptotes appear in the bracketing function Â£ for values of Â£ that are upper bounds to eigenvalues of /f corresponding to the states \|/. satisfying the criterion that ^ \|f. '^ 0. A plot of the g-valued function Â£ as a function of the variable Â£ can be made by imposing the non-crossing rule, which is valid for states having the same symmetry. The values of the g curves will satisfy the relation Â£fl >/ Ef , (i=l,2,...,g), (2.i+0) for all Â£ ^ E^ . If Â£^^ is taken to be the lowest value of the set of solutions j^-,.T , then for Â£ chosen equal to E^ we find that C--, = E provided that E^^, we see from (2.29) that c IT ^ E^ and the C-,-i curve will undergo a discontinuous change from 00 to + 00 as Â£ passes through E^ . In general, we find that the g curves have the behavior illustrated in Fig. 1, which is for g = 3* We see that each branch of the curves crosses the ti = z. line at an eigenvalue of ^ associated with a state \|f. not orthogonal to the reference manifold (Â© .

PAGE 20

Ik Â£.

PAGE 21

15 It is apparent from this discussion that for ^ ^. ^ and ^K S K ^Â•Â» O"^^ of ^^^ eigenvalues of (}v (6)> C ii,> i-s the lower bound to E., i.e., E. lies in the interval (f Â•,,,Â£) and for f = E., *" Ik 1

PAGE 22

CHAPTER III LOWER BOUNDS TO ENERGY STATES Assume that the Hamiltonian 1/ of a system may be separated into two parts; a zeroth order Hamiltonian or unperturbed part ^ and the interaction part or perturbation V. Let there be associated with the Hamiltonian i4 a complete orthonormal set of eigenf unctions i Q) .( and a known eigenvalue spectrum E., (i=l,2, ...)Â» such that Let it be further assumed that the perturbation V is positive definite: V > . (3-2) The space (p spanned by the set 1 CP Â• ( can be partitioned into two subspaces (|p and (^ , by defining the projection operators Â©=ii0<<3^i = t-o: , (3.3) LÂ«/ ^ tÂ»/ p= 1-0= ii(?f><9;' ^^i ' such that ' <' V d The operators and P satisfy the relations given in (2.3) and (2.U) From (3'l) we also see that 16

PAGE 23

17 and using (2.l6) we have the result Equation (2.I9) may be rewritten as Let us introduce a reduced resolvent T associated with the unperturbed o Hamiltonian X , where again the sum implies summation over the discrete states and integration over the continuous part of the spectrum. For any inverse operator of the type (A-B) there are the following identities: Lfl-Sr' = B'' + fl''BiB-B)'! = /?' 4-(^-3)"'6^ . provided the inverse operators involved actually exist. Using the first of the identities given in (3.7) we find that * (3.8)

PAGE 24

18 where :t'^ ^ V+VT^V Â£ V + V X" jfg (3-9) 2 is the reaction operator. The last relation corresponds to the Lippmann-Schwinger integral equation in scattering theory. Provided V exists, we find from (3.9) that (V~ -T )tg = 1; hence, A = iv-xy'<5-^Â°' Assuming the spectrum of ^f is ordered, E^ < E < E < ..., it is easily seen from (3.6) that for f 0, we see that V > and hence t^ > 0. Introducing (3'9) into equation (3Â«5) we obtain Hence, we see that the choice of ^ as given in (3-3) leads to a multi-valued bracketing function ^ ^ . From Chapter II, it foil that this bracketing function will have vertical asymptotes only for those states \|f, of jt that satisfy the relation C)i|^^^ f' 0. That is, only those states not orthogonal to the reference manifold (ff) , spanned by the set of g eigenvectors of V , will cause vertical Â• asymptotes to appear in ^ , The function ^^ has tne important property that the first vertical asymptote occurs for Â£ > E , ^ . Hence, if P Jf ^ has the ordered spectrum E^ < E < ..., then E^ > E ^. To prove this let us consider two self -adjoint operators A and B satisfying the relation /) > B. (3.12) Then, for any function y, < O / /) I ^> > ^' (3.11) Lows

PAGE 25

19 Let P be an arbitrary linear operator. Then replacing (p by P^ we find that >^(plP+BP|Â»> , i.e., P'^RP > P^BP. ^'-''^ Since V ;> we have the inequality i^ = if "^ ^ ^ H ' ^'^^^^ iC*" ~ W ar.d ^/> ~ = J? . Then from (3. 13) we find that for the self-adjoint projection operator P defined in (3-3) Hence, <0 |P^PJ0)><^^l Pi{<^|(p^ for any function ep. Let ^ = ijr^, where y, is the lowest state of P|;fP not orthogonal to (^ . Since the ground state of P^^P is E . , > it follows from the variation principie that It follows at once that the first vertical asymptote of cu appears -54 O for '^^ > E , 1 Â• In general, the eigenvalues of a self -adjoint operator A are greater than or equal to the eigenvalues of a self-adjoint operator 2 . B in_ crder if A > 3 . It is an immediate consequence of this that for Â£ < E , tg_ 2,9 Lsts and s^^isfies the inequality ^ Henc find tg^ ^''-"77 ^ ^"' > ^(3.16) e V ? tj > and Kq "^ ^ ^ ^o "^ ''^ ^ ^o' ^PP^^^^S (3-13) we

PAGE 26

20 and therefore. -^. > Â£f> Bl^ (i=i,2.....g). (3.17) This says that the eigenvalues of ^ ^ are bounded above by the average values of ^ with respect to the basis (SD and bounded below by the eigenvalues of vr ^il order for Â£ ^ E , , . g+1 For each value of c Â» C, will satisfy the equation Lt,-E,j) Â£; =0, ('-^'^ ^)(3.18) From this it follows that for any ^ the curves Ci-Â» (i'^ljS, . . Â• ,g)> have negative definite slopes. From (3.IO) and (3.II) we have (3.19) dT 2 VJe see from (2.26) that Â— Â«= T . Making this substitution in (3.19) we obtain Each of the curves has a slope at the point Â£ for which ^ ,^ and (J were evaluated given by dZ,[ I ^ ct Â«J?' (tf < , N ( ial>2, . . . ,g) .

PAGE 27

21 Hence, the curves have negative definite slopes and the bracketing theorem holds for each of the g eigenvalues of Â£ . . From (3.2I) and the inequality (3.I5) it follows that for every E^^ < ^ < E^j^Â» where E, is the k th lowest eigenvalue of the Hamiltonian ^ associated with a state \|;, not orthogonal to the subspace ^) , there is at least one solution to (3.18) that will provide a lower bound to E, . In fact, it is now easy to prove that under these conditions the k th lowest solution of (3.18) is the lower bound (or highest lower bound) to E, . From the discussion associated with Fig. 1, we observe that the curves of ^ , cross the Ci =S Hfie at the eigenvalues of >^ associated with states f. having the property that (5)\tf. ^ 0. From (3* 15) we found that the first vertical asymptote was situated to the right of E , T , i.e., Et > E , T . Let there be k eigenvalues of If having the g+i i g+i 'Â» ordering E^ < E < ... < E < E ^, associated with states satisfying the above property. Let the eigenvalues of ^ -, also be ordered, i.e., Â£^,, Â£',p < Â•Â•Â• 1 a solution ^ . such that C 1 ^ ~ E, for C ~ E^. Then, from the non-crossing rule and (3.2I), Â£ -.-^ must be < E, and hence crosses the S , = C line for some value of iJ < E^. This contradicts the assumption that E^ is the lowest eigenvalue of y^ associated with a state f satisfying the requirement that Q^ ^ 0. Similarly, Cip provides the lower bound (or highest lower bound) to E for E < Â£ 2 there exists a solution Â£ such that S i~ ^p ^Â°^ 2 ~ ^ , then by once again

PAGE 28

22 imposing the non-crossing rule and (3.2I) we find that tip ^^^st cross the Â£* , =2 line for a value E < ^ < E which leads to a contradiction. By induction, we see that, for any E. < Â£ < E ^^, Sn ^^ the lower bound to E., ( i=l,2, . Â• . ,k) . Figure 2 illustrates the typical behavior of the multivalued bracketing function (3.II) for g = i^ and k = 2.

PAGE 29

23 Fig. 2. Typical behavior of the multivalued bracketing function \p for g = U and k = 2.

PAGE 30

CHAPTER IV LOWER BOUNDS TO THE REACTION OPERATOR In principle, equation (3.II) can be solved exactly and provides lower bounds to those states of the Hamiltonian iP satisfying the criteria that ^i|/^ f and Â£ < EÂ° . In fact, there are few problems for which (3.II) can be solved exactly. From (3.10) we see that the reduced resolvent T must first be o obtained in order to calculate the reaction operator tg . From (3.6) we know that T < for Â£ < EÂ° . . If T cannot be evaluated exactly, o g+1 o one approach would be to estimate it by the operator T (p): o' E. (it.i) where it is assumed that the eigenvalues are arranged in the order E, < E < .... T (p) is an upper bound to T . Hence, from (3.IO) we see that and since it is assumed that V > 0, 2k

PAGE 31

25 -/ . / -J Since tp (py and tg are positive definite, the inverse operators tÂ£^(p) 2.9 and tc exist, are positive definite ^ and satisfy the inequality t^ >y :&^Cf) > . {h.k) Better lower bounds to the reaction operator maybe gotten by introducing the idea of inner projections. Let f = (fT,f^,...,f ) be a set of n linearly independent ^ 1 2 n' J r vectors spanning a subspace in Hilbert space having the metric A = A-'<|)-i.ii>h:Uij. (...) Let P = 1 Q be the projection operator associated with the orthogonal complement to f. The projection operator Q is positive definite since Hence, P = 1 Q ^ since P also is a projection operator. Therefore, Q satisfies the inequalities b^' q>^i (h.i)

PAGE 32

26 The reaction operator t^ is positive definite for P
PAGE 33

27 and the reference function . ^^+t^ > Z ("-^^^ and therefore From this we may conclude that the eigenvalues of ^, are bounded above by the eigenvalues of ^, and bounded below by the eigenvalues of W in order; From the bracketing theorem, g^. is a lower bound to E., where E. is the i th lowest eigenvalue of |f satisfying the criteria E , j^> Â£ >E. and Â£P y. f' 0. Therefore, ^' is also a lower bound to the same eigenvalue. From (3.17) we observe that the eigenvalues of *| ^ are bounded above by the eigenvalues of ^ with respect to the basis Â® in order. It follows at once that the latter also provide upper limits to the eigenvalues of ^, in order.

PAGE 34

28 More favorable upper bounds to the eigenvalues of ^^ may be obtained by introducing the inequality (3.I6) into the expression for tg which gives the result Hence, we have that where ^im t^ = V and V is simply a Bazley projection of V onto the space spanned by the manifold vV. It follows at once that V > V^ > . / (U.18) From {k.lT) we see that hence. The eigenvalues of ^ ^ are bounded above by the eigenvalues of ff + V and bounded below by the eigenvalues of Jf : hi ^ Â£,i 9Bl , (i=i,2,....g), (iv.2o) where h. is the i th lowest eigenvalue of V\ with respect to the basis (S> . The inequalities given in (i4-.20) hold for all ^ < E and tell us at once that ^ ' has no vertical asymptotes in this region for

PAGE 35

29 any Bazley projection. The equality Â£^. = E. occurs whenever^^ = 0: when the Bazley space is chosen orthogonal to the reference manifold The inner projected bracketing function given in (U.I3) can be computed exactly in only a few cases involving particular choices of the Bazley space; ' ~^' otherwise, T is truncated according to (li-.l), which results in an operator tg (p) that, for ^ < E ^, satisfies the inequalities ^E ^ ^s" >:^g^c^> >o . (1^.21) To remove the problem of dealing with the reduced resolvent T , o Lbwdin has suggested the substitution /j = ( C Jf )>f be used in making the inner projection. Substituting this into (14-. 12) we find that (1^.22) where since, from (2.I6). ( Â£ SfÂ„)T^ = ( Â£ ffo)PT^ = P( Â£ " i{^)'r^ = P. We shall call tg defined by (U.22) the LBwdin projection and the associated space A = ( jj^, jp, . . ., j^), the Lbwdin space. This projection has the interesting property that the reduced resolvent T has been o

PAGE 36

30 eliminated from the calculation. It also leads to some very interesting behavior not observed wnen using the Bazley or Aronszajn projections. This behavior can be explained by the fact that the linear manifold onto which the projection of tg. is made is itself a function of the variable ^ and varies with changes in ^ . An immediate consequence of the C dependence of the Lbwdin space is that the slopes of the eigenvalue curves of ^, are no longer negative definite. In fact, minima occur in these curves for values of t = E., (i=l,2, . . . ,g) . That is, one of the eigenvalues of**, for C E, , E, < E , ^ and ^i,^<|P > ist,^ = E, and E, will be a minimum value of the function C io* This is easily seen when we write down the expression for the iJl th element of^, using the LBwdin projection (i4-.22). The only non-vanishing term in the k th row or k th column of ^ , when I; = EÂ° is the kk element, (Cl^kW ~ ^k' ^^ ^^'^ write the expression for (U.I5) evaluated at the point C ~ E, ,

PAGE 37

31 Vii Â•Â•Â• Â•lk-1 t' Â• Â• Â• wÂ« i[ i-< k-1,1 .1 'k+1, 1 ^_,+t,' , Â„ , 1 k-l,k-l -k+l,k-l *^k-l,k+l Â•Â•Â• ^-1,1 Q Ef,,+t; k+1 "k+l,fcfl I Q 'k+l.g 'Â•8,1 -g,k-l 'g,k+l EÂ°+t' g gg, (,k.2k) Obviously, E, is an eigenvalue of ^' for ^= E". That this is also a ^kminimum in some eigenvalue curve, e.g., p\p Â°^ Â£ -i Is seen by looking at the first and second derivatives of ^ -, with respect to ^ evaluated at C= EÂ°. If Â£1^ = EÂ° =Â£ , then o Â£--^ > o are sufficient criteria for the point Â£ = E to be a minimum in the & In curve. From (3.21) we know that dÂ£.: = c/ e^' de 4Â£ &. 6.e| Ct.a?)

PAGE 38

32 where (J Â» is thei normalized eigenvector of (T ^ associated with the eigenvalue E, . Since k\ is the eigenvector for Â£ ip "^ '^i'^ ^ ' ^^ follows from (14-.25) that (0 ... 01, ... o) Â£*Et Â£.6Â° dt, dz a Â£Â•Â£ (It. 26) .-/ From (U.23) and the identity (i A S 4"'il A'' . we obtain the expression \dÂ€ = [2iÂ€-^<A-'<0:>(U.27) from which it follows that c?2 6^e; = 6>. (1^.28) Â£^Bl

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33 The expression for the second derivative is given by ST /rt Of S, 8=E de"Â£*E ^ = [^ < ^'"<^l 9P ~ "^ (^-^P <5^li^> OvA '*"'^<^â€¢>- ^{A-'#/S-'i<^i?J)] Â£r=e"^ ;i<(p;/^>A"'<^lfC> >o. (i^.29) Hence, from (l|-.28) and (k.2^) it is obvious that the g curves of^^ using the Lliwdin projection are no longer monotonically descending. In general, the curve Â£| Â•, Â« can have minima and cross the Â£ !. =Â£ line at several of the points Â£) = E.. It may also cross the Â£. =Â£ line at values of ^ not eigenvalues of ^ . From (l<-.20) we have the relation h. ^ Â£ . >y E. for any Â£ < EÂ° . It follows that Â£, . can have minima at any or all the points Â£ = E., where i ^ X and E. 4h^. These features of the LBwdin projection are developed further in the application of thi-^ theory to He and Li . In this section, we have developed a procedure for calculating lower bounds to the eigenvalues of a Hamiltonian t\ = CP + V, where

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3i^ V > 0, and \y has a set of known eigenfunctions . The subspace (^ is chosen such that ^ if . "^ 0, where \j/. is the eigenf unction of ^ associated with the i th lox^/est eigenvalue E. having a particular symmetry and such that it contains all the eigenfunctions associated with eigenvalues of vP less than E. having the same symmetry as f'.. The L'dwdin space -i must be chosen such that the overlap integrals or F & \(p-|j,^ with the functions of the reference manifold^? are re nonzero. It is then a rather straightforward matter to calculate the matrix elements given in (i|-,23). If 1'i^((^) = gj then by choosing the parameter p > E. and
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CHAPTER V LOWER BOUNDS TO THE "'S STATES OF He AND Li The procedure discussed in Chapter IV is used to calculate o + lower bounds to some of the low-lying S states of He and Li . For the two-electron series, the nonrelativistic Hamiltonian in atomic units is given by " Â• / (5.1) where z is the nuclear charge, r, and r are the position vectors of the electrons, r . = |r . | , i = 1, 2 and r^ = j r^-r j . We will take advantage of the symmetry properties of the o Hamiltonian (5.I) and restrict ourselves to states having S sjanmetry. For these states, the eigenf unctions of (5'l) depend only on r,, r and r, and are antisymmetric in the spatial coordinates of the two electrons, that is RzWcr. ,^) = '^Cf. ,n) , where P^ is the permutation operator acting on the coordinates r^ and r . The Hamiltonian (5.I) may be separated into an unperturbed part ^*r and a positive definite perturbation V, where i^o"^ '''^'^^" ^"^^ ""it ""i^ ' (5.3) 1/ = -fc > . 35

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36 The eigenf unctions of I," are well known and for the S 6\ o symmetry are given by where the R Â„ are normalized hydrogen radial wave functions, P,, is the Â£ th normalized Lengendre polynomial and Â©^ is the angle between the radial vectors r^ and r^. The C Â„ are chosen in such a way that 1 2 '^I'^p-^ o 2 the /K> /, are normalized to it , -Â» 1^2 ,/ /Â©\> = TT^. (5.5) The eigenvalue spectrum for the S levels of J^_^ is given by I: [ < "^ l^nJ^' Ri-^) /?..,^) J , (5.7) ) . By including in the reference manifold {?y only those functions ^^. having "^S symmetry, we obtain from (ij-.30) lower bounds only to those states of \v' having this same symmetry since ^^\]r = for any state \|/ of Cj.'not having S symmetry. Hence, the solutions of (^-30) will be in order lower bounds to the 2'^S, 3 S, l4--^S, ..., j S states of JT , i.e..

PAGE 43

37 forÂ£E.3g, With the solutions of jr available for the construction of the reference manifold^ , we must next select the functions for the LBwdin space J^ . It has already been pointed out in Chapter IV that the choice of functions for the A nianifold should be made subject to the condition that the overlap integrals ^(D.jjj^ be non-zero. Funct: satisfying the criteria that :ions P. 'Â•xii^'i^ Â• (5.8) where L is the angular momentum operator, can satisfy these conditions. The functions were selected on the basis of those introduced by Is Hylleraas and are given by ^. = c,^,.^^ e''*s*^ i-^'*' u^'. (5.9) where y\ is an arbitrary scaling parameter and S-Y,-
PAGE 44

38 The minimum size of the reference manifold is determined by the number of states of iP having the same symmetry as and eigenvalues less than those of the state of Cr to which a lower bound is sought. For the S states of He the ordering in atomic units (a.u.) is given by EÂ° = -2.500 ... ^EÂ° = -2.22 ... < E _ = -2.175229 '^^ < i ^ 2-^S EÂ° = -2.1250 ... < EÂ° = -2.080 ... ^ oÂ» ^),* Erom f:^i3>^9;9 h(5.12) we see that, to obtain lower bounds to the 2 S, 3 S and k S levels of Li , it is necessary to have g = 1, 3 ^^^ 4 respectively. Of course, as many additional functions as desired may be included but these values of g give the minimum sizes of the reference manifolds necessary in order to calculate lower bounds by this procedure to the levels indicated. The size of the Lbwdin space, which shall be denoted by NINT, can vary from one to all the functions of a complete set. If the set is compli^ce, then the exact eigenvalues of ^rx are found to be those

PAGE 45

39 values of q at which the curves ^ ^ . cross the c i ~ Â£ line. In general, however, this set is finite and the curves cross the C i ~ C line at the eigenvalues of VP associated with functions in the reference manifold. Crossing may also occur for values of fc jf' E., i are included

PAGE 46

ko w pa < H O + a z < X b^ O w < H CO on o c C O 03 XI U U C O Â•H 4J a to 01 to 3 D-u a. 5 cu -l c 3 ^-N O Â• XI 3 l-i CO (LP o f^ U) d 3 o .^> X3 3 O) a, 3 M 00 to 01 o CO g u >, V3 W OJ

PAGE 47

Ul in(^^Â°, little improvement is found when more reference functions are added. The magnitudes of the errors in the bounds to the higher states are seen to be much greater than that of the 2 S state in both the He and Li calculations. This is, however, not an unexpected result. It has been pointed out by LBwdin that a truncated set which works well enough for the ground state may give a surprisingly poor result for even the first excited state of the same symmetry. 17 Quite good lower bounds have been calculated by Pekeris to the o 20 2 S state of He using Temple's method. By solving a determinant of order 252, he obtained alower bound of -2.175239 a.u. The lower bound to this state, for g ' k, which is given in Table 1, is very close to this value. In order to obtain a lower bound to an energy level of ^ using Temple's formula, it is necessary to calculate matrix elements of 2 v# and to have a knowledge of the next higher level of the system. It is apparent from the work of Pekeris that in order to obtain a good lower bound using Temple's formula, the size of the secular equation must be very large. 21 Goscinski has shown in his dissertation that Temple's formula can be derived from (2.I9) for g = 1. 22 3 Miller has obtained lower bounds to the low-lying S states 2' of He using the method of intermediate Hamiltonians . "' Solving a secular equation of order 9> he calculated lower bounds to the 2 S and 3^S levels of -2.l802 a.u. and -2.070iÂ»a.u. respectively. His result for the 3s state is somewhat better than that given in Table 1. 21 Goscinski has shown that Miller's equation can be derived from an

PAGE 48

h2 expression for the bracketing function ^ using a truncated resolvent T (p) as is given in (l+.l) and that the points Â£=Â£)t, correspond to its solutions. By making this truncation, however, one reduces the contribution from the continuum, a problem that is eliminated in using the Lbwdin projection. A study was made to determine the behavior of the lower bound curves when the scaling parameter 77 i-S fixed and Â£Â» is permitted to vary over a wide range of values. For both He and Li , g was taken to be 4 and NINT to be hO. In Chapter IV we observed that the LBwdin projection greatly changes the character of the lower bound curves from that shown in Fig. 2 for example. We found that, using this projection, a minimum appears in one of the lower bound curves at each of the points where ^ equals an eigenvalue of ^ associated with an eigenf unction in the reference manifold. This behavior is shown in Figs. 3 a"^"^ ^Â« These figures demonstrate the effect the Lbwdin projection has on the eigenvalue curves of Â£^ . It is interesting to examine the effect that changing the number of internal projection functions in the LBwdin space has on these curves. Of course, using fewer functions in constructing the projection t^ will result in a poorer approximation to the reaction matrix t^ . Keeping ?7 = 1.2, the results of using NINT = 20 and NINT = 10 are shown in Figs. 5 and 6 respectively. We observe in Fig. 5 that the lowest bracketing curve Cii <^oss not recross the S i ~ Â£ line after the minimum at Â£ = E , as it does in Fig. 3 and the minimum at Â£) = EÂ° occurs in the Â£)-,o curve and not

PAGE 49

43 O OJ I m a

PAGE 50

^ o o 00 1

PAGE 51

^5 O oj I ir\ OJ I K? OJ U3 CO OJ I oj I OJ u T3 a H Z M Si II 00 O u 9 O Â•o c: 3 o ua (U I OJ I

PAGE 52

k6 ITS O OJ I LTN OJ I Â» (A OJ I OJ II px> a (0 *

PAGE 53

hi in the ^ ^ curve as it does in Fig. 3I^i Fig. o, we find that using only 10 functions to construct the LBwdin projection results in the minir.un at ^ = "X being shifted from the t i -i to the ^ ,_ curve. It is also apparent that the ^ , C to ^nd C^ , , curves vary only slightly from the unperturbed eigenvalues E^, E and E, respectively, over the region under consideration. Only the C -i -i curve is appreciably affected by t^ . For NINT = 10, we find, by permitting W to vary, that the minimum for 2, = E is shifted back to the ^-,1 curve for Y) = 1.35' For ^J = 1.35 and Â£ = -2.06, the lower bound to the 2 S state of He is -2.216 a.u. These results are shov.'n in Fig. 'J. We conclude this study of the lower bound curves by noting that, if the LBwdin projection provides an adequate approximation to the reaction operator tÂ£ , the i th lower bound curve will have minimums at each point ^= E., where E. .
PAGE 54

^ o I ir\ CM I I CO oJ I OJ I II px> c CO H 2 II 00 S-l o U-l QJ 14-1 o en > 3 O Â•^3 C D O ^ U d) o h4 60 Â•r-l CM I

PAGE 55

1^9 A similar analysis can be made of the Li curves shown in Fig. k and the He curves shown in Figs. 5Â» 6> ^^id J, Although the choice of functions of the type (5 '9) used to construct the Lbwdin projection is perhaps not the best possible selection, tho bracketing curves obtained using this choice illustrate the general behavior we may expect when using this projection and the material discussed in Chapter IV.

PAGE 56

APPENDIX This is an outline of the procedures used in the calculations of Chapter V. The calculation of the LHwdin projection (U.22) and the matrix ^ ^ (i4-.13) require the evaluation of the quantities ^t j., i=l,2, ..., I4O, and the ^ matrix whose matrix elements are given by <5>:/^;> . where g = h. The Hamiltonian S^ and the inverse of the perturbation V , ' o defined by (5.3)> Â™ay be expressed in terms of the corrdinates S, t and u: V' = 2<. Â» (A.2) 50

PAGE 57

51 From (a. 3) and (5Â«9) we see that V operating on the functions j. gives a linear combination of terms having the form where n = -1, 0, k ^ 0, ^g ^ and m >^ -1. From this we can see that the matrix elements of (A.l) involving only functions of the LBwdin space are simply linear combinations of integrals of the type Uu^)^ i'w'je"''stisjtrdujcs'-fr t"j)t . (A.5) where n = -1, 0, 1, k Â» 0, ^ * and m ^ 0. These integrals may all be evaluated in closed form: -^ ^' (A.7) _J I (^-IM^^ )-27r "jTzp^sTT;^/ ^ Urr^f^^ Z^ I v (a.s)

PAGE 58

52 The integrals involving functions of the reference manifold and the LBwdin space ^(p. [j, / can be expressed as a linear combination of integrals of the type "* ^ '^ (A.9) where i ^ 0, k -> and m > 0. The solution to this type of integral is also known in closed form and is given by r~ D y^ ~ *" Jle^ f.' J + ^2-^.' J (A. 10)

PAGE 59

53 The number of integrals of this type and of the type given by (a. 6), (a. 7) and (A. 8) was quite large, so a program was written for evaluating them on the IBM 7O9 computor. Once the integrals were solved, the matrix elements ^i^n were calculated and the ^ matrix constructed. The inverse of this matrix, -1 26 /^ , was obtained using the method of successive partitioning. It was then a rather easy task to determine the matrix elements of ^-i > which are given by the formula Try **t (A. 11) The secular equation (4.30) was solved using the subroutine JACFUL, that generates the eigenvalues of a real symmetric matrix by 27 the Jacobi method. '

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REFERENCES 1. P. 0. Lliwdin, J. Mol. Spectry. 10, 12 (I963). 2. P. 0. LBwdin, Phys . Rev. 1.39 . A357 (I965). 3. P. 0. Lbwdin, J. Chem. Phys. k^, SI75 (I965). k. P. 0. Lbwdin, Proceedings of the Madison Symposium on Perturbation Theory, I965 (J. Wiley and Sons, Inc., New York, to be published ). 5. J. H. Choi, "Lower Bounds to Energy Eigenvalues by the Partitioning Technique," Ph.D. Dissertation, Department of Chemistry, University of Florida (I965). 6. P. 0. Lbwdin, J. Math. Phys. ^, 969 (I962). 7. P. 0. LHwdin, Advances in Chemical Physics (Interscience Publishers, Inc., New York, I959), Vol. II, P. 2O7. 3. E. A. Hylleraas and B. Undheim, Z. Physik 6^, 759 (I93O). J. K. L. MacDonald, Phys. Rev. k^, 83O (1933). 9. P. R. Halmos, Introduction to Hilbert Space (Chelsea, New York, 1957). 10. N. Aronszajn, Proceedings of the Oklahoma Symposium on Spectral Theory and Differential Problems, 1959 (unpublished). 11. P. R. Halmos, Finite Dimensional Vector Spaces (D. Van Nostrand, Princeton, I958). 12. C. E. Reid, J. Chem. Phys. h^, SI86 (I965). 13. J. H. Choi and D. W. Smith, J. Chem. Phys.,.^, SI86 (I965). 111-. C. F. Bunge and A. Bunge, J. Chem. Phys., h^, SI89 (1965)15. E. A. Hylleraas, Z. Physik ^Jf, 3^7 (1929). 16. C. L. Pekeris, Phys. Rev. U^, I2l6 (1959). 17. C. L. Pekeris, Phys. Rev. 126 , 114-70 (I962). 18. C. E. Moore, Atomic Energy Levels , National Bureau of Standards Circular No. k6l (U. S. Government Printing Office, Washington, D. C, 1959), Vol. I. 3k

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55 19C. L. Pekeris, Phys. Rev. 126 . ll*.3 (I962). 20. G. Temple, Proc. Roy. Soc. (London) A119 Â» 276 (I928). 21. S. 0. Goscinski, "Upper and Lower Bounds to Eigenvalues by the Partitioning Technique,',' Ph. D. Dissertation, Department of Chemistry, University of Florida (I966). 22. W. H. Miller, J. Chem. Phys. kg, 14-305 (I965). 23. N. W. Bazley and D. W. Fox, Phys. Rev. 120 . Ikk (I960). 2k. L. Wilets and I. J. Cherry, Phys. Rev. 103 . 112 (I956). 25. J' G. Gay, "Lower Bounds to the Eigenvalues of Hamiltonians by Intermediate Problems," Ph. D. Dissertation, Department of Physics, University of Florida (I963). 26. R. A. Frazer, W. J. Duncan and A. R. Collar, Elementary Matrices (Cambridge, London, I938). 27. F. Prosser, "Matrix Diagonalization by Jacobi's Method," QCPE k (1963).

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BIOGRAPHY Timothy Michael Wilson was born August 3> ^938 ifi Columbus, Ohio, He moved to Saint Petersburg, Florida, in I95O. He was graduated from Saint Paul's High School in June, 195^^"^ June, I96I, he received the degree of Bachelor of Science in Chemistry from the University of Florida. Since that time he has been pursuing work toward the degree of Doctor of Philosophy at the University of Florida. He was awarded the Kopper's Summer Fellowship in July, 1963, and in September, 1965Â» he received the DuPont Post-graduate Teaching Assistantship award in Chemistry for the academic year I965-66. Timothy Michael Wilson is married to the former Iris V. Barron and is the father of one child. He is a member of the American Physical Society and Lambda Chi Alpha. 56

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy August 13, 1966 Dean, Colle Â• Supervisory Committee: C.f, Chairman 9. 9. / CW-r^Ld^r^<-

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f J t '] , "^"^ 46 60 ^

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