 Title Page 
 Acknowledgement 
 Table of Contents 
 List of Tables 
 List of Figures 
 Introduction 
 The partitioning technique and... 
 Lower bounds to energy states 
 Lower bounds to the reaction... 
 Lower bounds to the 3s states of... 
 Appendix 
 References 
 Biographical sketch 

Full Citation 
Material Information 

Title: 
Lower bounds to eigenvalues of the Schrödinger equation 

Physical Description: 
v, 56 leaves. : illus. ; 28 cm. 

Language: 
English 

Creator: 
Wilson, Timothy Michael, 1938 

Publication Date: 
1966 

Copyright Date: 
1966 
Subjects 

Subject: 
Quantum theory ( lcsh ) Boundary value problems ( lcsh ) Eigenvalues of the Schrödinger equation ( lcsh ) Chemistry thesis Ph. D Dissertations, Academic  Chemistry  UF 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Thesis: 
Thesis  University of Florida. 

Bibliography: 
Bibliography: leaves 5455. 

Additional Physical Form: 
Also available on World Wide Web 

General Note: 
Manuscript copy. 

General Note: 
Vita. 
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Bibliographic ID: 
UF00097889 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

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All rights reserved by the source institution and holding location. 

Resource Identifier: 
alephbibnum  000577298 oclc  13957473 notis  ADA4993 

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Table of Contents 
Title Page
Page i
Page ia
Acknowledgement
Page ii
Table of Contents
Page iii
List of Tables
Page iv
List of Figures
Page v
Introduction
Page 1
Page 2
The partitioning technique and the bracketing theorem
Page 3
Page 4
Page 5
Page 6
Page 7
Page 8
Page 9
Page 10
Page 11
Page 12
Page 13
Page 14
Page 15
Lower bounds to energy states
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Page 22
Page 23
Lower bounds to the reaction operator
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Lower bounds to the 3s states of He and Li
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Appendix
Page 50
Page 51
Page 52
Page 53
References
Page 54
Page 55
Biographical sketch
Page 56
Page 57
Page 58

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 PerOlov 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.
TABLE OF CONTENTS
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 timeindependent 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
$i = E*, where the Hamiltonian R is assumed to be hermitian, i.e.,
a = ( and bounded below. Let 2 = { i be a complete ortho
normal set of functions. As an immediate consequence there is a
resolution of the identity,
(2.1)
I = < ^><9. .
where the sum over the index i implies summation over the discrete
indices and integration over the continuous ones.
Let us define a selfadjoint projection operator O such that
it projects out of the space spanned by 9 a subspace Ta of order g:
(2.2)
From this definition it follows that satisfies the relations
J (2.3)
Let P = 1 I be a projection operator associated with the subspace
b', the orthogonal complement to the subspace T :
f^$^
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)
= (En 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= [Co9+ 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+(EiP)P]. (PRGO) .
Operating on (2.10) from the left by P gives
p?= fP f[. 4 PE)PjTPF ] 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.+ PCEs)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 4c 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,
Oi&. 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
CA 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 meanvalue 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
+ AL (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 gvalued function 1 as a function of the vari
able E can be made by imposing the noncrossing 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
jo=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 (AB)1 there are the
following identities:
(A8)'' = "'+ ,?"8( B)"/,
(3.7)
= "+ ()FY' ",
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 + vrv) = T, ,
where 4 V + V 7 V  V + V 7 4te (39)
is the reaction operator.2 The last relation corresponds to the
LippmannSchwinger integral equation in scattering theory.
Provided VI exists, we find from (3.9) that (VT )te = 1;
hence,
I
S= (V'T) (310)
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
multivalued 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 selfadjoint 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 selfadjoint
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 selfadjoint operator A
are greater than or equal to the eigenvalues of a selfadjoint 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 noncrossing 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 noncrossing 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 (EA0 (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 nonvanishing 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
tlkl
0
0
S+tk+l,kl
tgk
g,k1
0 ck+
tkl, 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)
tkl,
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
AI
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 procedurediscussed in Chapter IV is used to calculate
3 +
lower bounds to some of the lowlying S states of He and Li For the
twoelectron 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)>
VL>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
E4
CLI
r4 H
i. C/)
E
0c
0
Ci
Z
3
H
Er
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 r4 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
14
+1
.r
u
C:
,4
01
0
LW
'I
4
C
4
0
0
0
0)
14
co *l 0 C
L) (L) L) 0)
(B 0 '
r4
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
2S 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 lowlying 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
44
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
re
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' SStL 7Yb < S 2i(&t b b2At Spta
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
(SWtz)
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...
e7
(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
(St') tQ et
(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
(gb)e' f' ,
,H ,.2
L t+M +^
(4 1,)
.b)2^ /
(a)b, /)= b)
L ( )'
r 4ch+;)Q +/)
( 4b .
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
Ipfrm~!
;P!
Cet~) (a~~ 'IC
(4t~)(P+I) ii
I' clr~
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).
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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 Postgraduate Teaching Assis
tantship award in Chemistry for the academic year 196566.
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
Dean, Graduate School
Supervisory Committee:
Chairman
/ J ^

