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||Lower bounds to the eigenvalues of Hamiltonians by intermediate problems
||v, 72 leaves : illus. ; 28 cm.
||Gay, Jackson Gilbert, 1932-
||Quantum theory ( lcsh )
Nuclear physics ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
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||Thesis - University of Florida.
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notis - ACY7451
LOWER BOUNDS TO THE EIGENVALUES OF
HAMILTONIANS BY INTERMEDIATE
JACKSON GILBERT CAY
.A DISSERTATION PRE'ENTEP TO ITH I.R.PDUtTE COUNCIL OF
THE LINr\ERSIT' OF F.LOPIJT1
iN PARTrA1. FULFILLMENT OF THE FEQ UIREfIEN'I FOR THE
DFCPEr OF DOCTOP OF PHILOO-FH'V
UNIVERSITY OF FLORIDA
I t s a pl(.,
bE~th dirL nO i ndirect, provide by my Supervsory Corim tteu, ,an
by nwiernus other members of thn Physics Department Faculty and the
s taff oF the Quantum Theory Project.
I a rtclrindebtedf t the Chairmanm of my committee Pro-
fe!ssor Per-010v Ldw)i1n, for his, invariable willingness, in spite o h
many deirnuns cn hi7 *Iimes to give me- the opportunity t icusm:e
_--nrch with hin. Thi( )dvMce and encouragmrent thus obtained weren
valuable, I df-,i also- indebted to Dr. David W 1. Fo f r a very helpful
65,cussion. Thanks jr due Mrs. Carole Puller for her careful typing
f ta nuscript.
Iwuuld like to ackn(owlIedqe thL financial support provided by
the United States Gmvernmert In thf. form of a National Defense Ac
Fellowship, iid thn donation iof tcom-piuter time by the University o
Firjrld') Computin center.
I om profoindly grateful to riy wife, Pairicia, for supporting my
decision to ,nter Griduate School and trr h,r cootinue support and
enouagmetir circumsttances often miore try ing (). h.--r than on we
It Is to hter Ohat thi, dissertation is dedjcatcd.
TABLE OF CONTENTS
ACKNOWLEDGEKENTS .. . . . . . . . . . .. I
LIST OF TABLES.. . . . . . ......... .. . .... IV
LIST OF FIGURES.. . . . . . . . . .
1. INTRODUCTION. . . ........... ... . ...
11. INTERMEDIATE HAMILTONIANS. .......... ... .... .... 3
111, THE METHOD OF SPECIAL CHOICE _ ....... ..... 10
IV. THE METHOD OF TRUNCATIIOIS ............. .. . .... 13
V. A SECOND METHOD OF TRUNJCATIONS.......... .. .... ... 16
VI, ELEMENTS CONTAJNING THE INVERSE OF THE RKE50LVENT.. 18
Vil. DESIRABLE FORMS FOR NITERMEGIATE [(GENFUNCTIONS.-... 25
Vill. A TWO-PARTICLE HARMONIC( WELL _....... .. ._... 21
IX. LOWER BOUNDS TO THE GROUND STATE OF HELIUMl ......... 37
. . . . . . . .... . . . . . . . .- 5 8
. . . . . .... .. . .. . . 1 . .. 0 4
LITERATURE REFERENCES _, ..... ... .. .. . .. . ... .. 7
LIST OF TABLES
Tab Ic p ,,
-1.7 7 7
1. The matrices (V and V .... . . . ........
2. Upper and lower bounds to thc singlet S tat es c," the
the harmonic wel I ...................... ..... . .
3. Normalized eigenfunctions rbta;ned from the sc'.enth
order upper and lower bound calcul ari.ns... ........... .
4. Comparison of upper and lower bound energies and
wave functions tor helium............................ .
5. Comparison of the errors in lower and upper bounds
to helium....................... ...... .. ............ 4
6. Lower bounds to the ground str.:c c hel ium b, the mt rlhod
of truncations i......................... ..... ... 2
LIST OF FIGURES
1. Roots of a secular equation of order two............. 43
2. Lower bounds to the ground state of helium as a
function of a scaling parameter..................... 45
I I NTRODUCT ION
The method of intermediate problems fur obtaining lower bounds
to eigenvalues was introduced by Weinstein [I] I sa means of esti-
matinq from below the eigenfrequencies of a vibrating clamped plate.
His procedure involves constructing a solvable base problem to the
gven problem by relaxing boundary conditions so as to removecn
straints from the system. The eigenvalues of this base prob lem
thus give lower bounds to the eigenvalues of the given problem
Solvable problems intermediate between the base problem and the
9ivQen probicem are constructed by systematically restoring there
moved constralnits. The elgenvalues of these problems give lower
bounds im~proving toward the elgenvalues Of the given problem.
The method has subsequently been formulated in terms of the theory
of operators in Hilbert spaces [2,31.
A diffe-rent procedure! for constructing intermediate problems
has been givcrn by Aronszajr; [4). In this procedure, a base oper-
ator, whose eigcnIvalue! equation can be solved, Is constructed
From a given operator by sujbtractinq a positive definite part.
The e,;genvaiue!s Lo the bnse operator are lower bounds to those of
the ciiveG operator. By meins of a projection technique operators
INumbers in bracket5 refer to the literature references located
after the appendices.
smaller than the subtracted operator prr added to the bas- operator
to i?;v sol'..ible intermediate operators whos- e-Ignn.3lues appro-ch
those of the gi\en operator rrom below.
Recently, Eizle, und FoA (5. 6, 67 hi.e us.-d the projection
te..hnique of Arc.ns-a.in to construct intermediate operators whichh
may be sol.ed or approximated from belcw in terms of finite matrix
el en.alue problems. Their methods are applicable to many of the
Hamiltonians of atonc and molecular systems.
In the First part of this dissertation we develop the procedure
of Bales and F.-r. for constructing intern ediae Hamiltcnians along
wvith three of their methods of solution. lie also introduce another
method of solution which, while possessing undesirable features,
appears to offer sorie ad..'ntages in deternmning lowI r bounds to the
ground states of atomic and molecular systems.
In the last part ~e apple' these procedures to the helium atom
ind to a quasi-atomic system consisting of two electrons confined
within an isotropic potential well.
...... ...... ..' 1 1111
...... ...... .....1111~ ~ 1111 iiiii;;; ;~ l
II INTEP'lEDIATE HAM ILTOI' I AIS2
We consider a time independent Schrtdinger equtic.n,
-- = E (2.11
for which the Hamilconian N can b. deccmpc.ed into. two p;ars,
OT = + -.?2,
such that V is positi;.e definite, and hC h s kncwn oigenaluc-s
and eigenfunctions, is bounded from below, and possesses a sp;ccrrum
consisting cf at least an init;ally discrete set .j f le.'ls.
Since ~C r' is the positi'.'e dcrin;te .perator tV we have
for every function e in the domain cf
If we order the eigenvalues of and I' lying below their
respective first limit points, so as to Fcrm increasing sequences
E E, ( 2 4)
The major part of this Section and Secticns III, IV, and V are
enindensat Ions of the original development given in (6] and 
(2.3) is Suf ficient to guarantee (8 Ithe inequalities
Wenwconstruct a sequence of intermediate Haniftonin
s t s ying the inequal tie
< Y O> e '> :!E ,-t ( '') !5- (T) (2-7)
-o that their ordered eigenvalues saif
and ivelowr bounds rImproving toward the- eigenvalues of T
:-onstrijct the 67Y we introduce a projection operator de' cI ned
inavctor space S characterized by the metric operntur .The
inner product in S is therefr give, by However,
elmnts of 5 are taken to be normalized in the sense
In Appendix I we show that the projection p rtr for the, sub-
space spanned by the ekements A S C p-- fastA
linearly independent in Si'_
whe-rc is Lhe nvrs ot (h, P h order riatrix ;with eleents
tiatr ices ire indicted by, bco!d Tybos Te sybo ra iI pp IIed
a S S U p Cr serC Pt to an o er 3 VrGr M3 trx I Y, otS i ts HermVi iti I inlt
(0C!) = 0" ,
'0"t = O'tV = VO"
so that the cpc-rator V is self-adio;nt.
Further, from (2.3) and Bcssel's inequality in ,
0 : (1/0-) ( a'"'')^ C) (2.12)
The relation (2.12) forms the basis for construct in of the X
for, if we define by
I 2. 3 )O",
the inequalities (2.7) are insured by (2.12).
Thus, provided the eigenvalue equations of the operators 12.13)
can be solved, we have a means of obtaining a sequence of lower bounds
which increases toward the eigenvalues of W .
A formal solution has been given by Bazley and Fox  which we
repeat in part here. The eigenvalue equation for the ,
may be written with the aid of (2.10) as
(U, }^ =- -0 dI.
ii . 2.1 5)
C, = E >. (2.16)
If E";is not an eigenvalue to &7"V we may multiply (2.15) by
the resolvent operator ) to obtain an expression for the
Unfortunately, the resolvEnt operator is seldom known in closed
form being generally only expressable as
C _- E ")-'= 2?/ : >
o le Z (2.12")
t/t c Pt
where the rA. represent the discrete eigenfunctions of P and
the 9 ('E)continuum eigenFunctions normalized on the energy scale.
'E denotes the first limit point of 0O, which is, incidentally,
also the first limit point of the I 
5 and C (nay have common eigenvalues in which case the
solutions take a different form  but the difficulties associated
with the resolvent remain.
In the following sections we discuss procedures which remove
these difficulties and lead to practical methods for computing lower
bounds. Prior to this, however, we wish to comment on the convergence
of the elgenvalues of the ;Y~ to those of 2(
It has been shown that  provided and K have compdletal
continuous Inverses, Y is bounded relative to and the setdi
is complete in S the eigenvalues and eigenfuncticns of the U
converge to those of . Of equal interest in applications is
the rapidity of convergence. In this connection, we demonstrate
that the error in a lower bound E is of second order in the error
in the eigenfunction 4 and in the error in the operator f.
To this end we introduce the orthogonal complement p to Omde-
/P"= I/- VO", (2.191
having the properties
(p7m) p ,
0 'VP = P"'V = (2)
omtK pfn 0 C
We may now write ; as
= " t (O" tP")
= C j l/PO ,
and one of Its elgenfunctions as the sun
,The energy corresponding to 4S is then4
4We point out that D 4 does not in general vanish.
E = (J/ I/>)
If we denote )0 e by 6 and retain cnly terms of second order
we find for the error in the lower bound
0o A-= E-E"
observing that the first twc terms arise from the error in e while
the last arises fromri the error in .m
We give for comparison the well-kncwn equivalent relation for the
error in a upper bound obtained by the variational principle. le
take !l to be a normalized trial function crthogonal to all eigen-
functions of d with energies less than C We write 3 as
< (i i = 1
If A/ is the upper bound obtained with i ,i.e,,
= E (4 ) + (/lJl >, /
we find for the error in the upper bound
0 A W- E = < I / > -< >).
~II' II IIl;;;;;ii* iii sii
4' and ,I. rot differ greatly it is reasonable t.-. supp..se
that these terms will be comparable in magnitude. If th;is i the
case we see th-it the error in the lower bound I ll be greatE'r by
We may therefore c..pect that procedure; based (n the ;ntermed;ate
Hami 1 tonians gi ver here w; Il net demns, trate convergence proper ties
comparable to upper bound procedures based .n the .ari2tic n prir,-
ciple. We find confirmation in our applicat;cin to the helium atc(m
In Section IX.
III THE METHOD OF SPECIAL CHOICE
Bazley  proposed the special choice of the f,
A = ~, (^ :/,z,.. (3.1)
where the 9 are again elgenfunctions belonging to the discrete
spectrum of r 0. With this choice (2.17) becomes
4 = -( g-ME') C +,
Upon substitution of (3.2) into the elgenvalue equation for r ,
(.7 -Eo L it c -: = ; ."
0 M V E e E E-
1-- i" i2 -Ec-
By forming the inner product of (3.3) successively with ; ***'";
we obtain the in equations
"= -_ _;-. C_-
which constitute a matrix eigenvalue equation of order r This
equation can be put in more conventional form if we successi.ely
multiply on the left by the diagonal matrix with elements (F; -&'9j
and the inverse of the matrix aith elements
= 56p'/ '^^
which we have previously designated b / .A The resulting eigen-
L [(E; -E)W + A,.] = o, .(6)
differs from a Raylelqh-Pitz calculation based on the trial function
only in that the matrix r is replaced by /
(3.6) determines 7a of the eigenvalues of r and the cor-
responding elgenfunctions are of the form
The remaining elgenfunctions of V are all eigenfunctions of o
different from ;,, * These have corresponding
7ii y =- E; ~> (f > ) ) (3.9)
Il ei these persistent eigenvalues are ordered with those determined
front (3.6) in a non-decreasing sequence
we have the lower bounds
S" E ( = /42,..) (3.lll
This procedure is sometimes simpler to apply from a computational
iewpcinr than the other procedures covered in this paper and can be
expected to give relatively good estimates for excited states much
more easily. We note, however, that for problems where a portion of
the spectrum of At is continuous, the ft, given by (3.1) do not
constitute a complete set, and the procedure cannot converge.
For an application to a problem involving continuous spectra
we refer to Bazley's calculation on helium [5 ]. For an application
to a problem where Oa has only bound states see Section VIII.
This procedure has been generalized [6,7] to include .c" given
t 04s "'. (3.12)
The eigenvalue equation equivalent to (3.6) Is slightly more in-
volved but may lead to simpler calculations in some cases [7 ]
T:: mid1~ 1111111 1: AH.'i
S c ,,,....... . .
IV. THE METHOD OF TR.UNCATIOIIS
This method [6,7] involves replacing C by the smaller
,'" = L E01:/ < )< /
< r / -0
to obtain the truncated intermediate Hamiltonians
W+ =V/ (4.2)
These satisfy the inequalities
< We -g') 2 )- () < 4.3)
so that their eigenvalues give lower bounds
.. (j t..(4.4)
T..i|he advantage of this procedure Is that the resolvent
l b& "i) L ,s known in closed form,
S ..... .............
/ *. / (4) <
ThS are taken to be ordered according to (2.4).
The relation equivalent to (2.17) is
..-* .-** E < ~ E
+ v- -- --- -- -- --r^ ---- L- tZ VIa
which, when substituted into the cigenvalue equation for ",
00t V V
v, <.., ,c. .=a
By taking the inner product of (4.7) with the elements j -
,f, we obtain an n order matrix eigenvalue equation
l-tti ~ l
(jV i V,> ,./ X: /I/ ) (
E =.e E ; ,
whose solutions ge al e u ions and eigeal of the
whose solutions give all elgenfunctions and elgenvalues of the&'
1 u :4Bi~ltlll k'.. ........1'
different from those of ;Wo If there are less than / t such
solutions, it remains to be determined whether any of the eigenvalues
a, ," E persist in the spectrum of Procedures
for finding such eigenvalues and their eigenfunctions are gisen [6,7]
but we do not cover them here as they are not needed in our application
of this method. Finally, it is shown (6,7] that all other eigen-
functions ofa ? belong to the infinitely degenerate persistent
eigenvalue ';Oi .
Equation (2.25) for the error in a lower bound requires modifi-
cation for this procedure. We have, instead of (2.21)
so that L becomes,
0o_ = E -E 0-
= ( /~ S"/f) EL" 5/f) + 6'/VI'f) (4.'0i
The additional term cannot, in general, become arbitrarily small unless
:io has bound states only, showing that the procedure is not con-
vergent for problems involving continuous spectra.
Nevertheless we note that this procedure can always give results
superior to those obtained by the method of special choice for it
contains the method as a special case. This Is seen by observing
that when the at are chosen by (3.1), (4.8) reduces to (3.4).
Bazley and Fox have applied this procedure to helium. We give
Some additional calculations in Section IX.
V. A SECOND METHOD OF TRUNCATIOrIS
Bazley and Fox have given another method of truncation [6,7]
which can be convergent for problems involving continuous spectra.
Tco illustrate this method we write 0 as
S= z /0 ( o) (5.1)
where the vt are given by (4.1). Since the operator
is positive the operators
R =g( -A(^ (5.2)
are intermediate between 9I and 0 is given by (2.10
with V replaced by N .en .
The solutions to the eigenvalue equations fcr the rl' are
developed exactly as in Section IV. The equations corresponding to
(4.6) and (4.8) are
L- E } 4:
To distinguish the two truncation procedures we will hereafter
denote the present procedure by truncations ii and the previous pro-
cedure by truncations 1.
"i~ii !riii iiiii iiiiii, miii~i, ii% i;;iiii
A-, E: if ,"
S(,(. o /(_x ,. /), / >
Procedures for finding persistent eigenalues parallel those
for the method of truncations i. Here FI, is also an eigen-
value of infinite multiplicity.
Sufficient conditicns frr convergence f.r this procedure are
just those stated for the R "in Section II with V replaced
by'(-i /, and -f replaced by Co. We point cut that,
should r have a spectrum extending to infinity, then ( z- /)
is not bounded relative to t l fcr Z finite.
This procedure has been applied to the helium atom. Our
results are given in Section IX. We Find the lower bounds gi.en
by this procedure to be inferior to those of the other three
procedures covered In this paper. For a possible explanation
we refer to Section VII.
VI. ELEMENTS CONTAINING THE INVEPSE OF THE PESOLVEINT
We present here a new procedure for soling the eigenvalue
equation of the intermediate Hamiltonians (2.13). Our procedure
avoids the appearance of the resol.ent operator (2.18) by selecting
the according to7
P0 = /"(c -E" -- -.. ,(.1)
where the i are essentially an arbitrary set of elements in the
domain of c .
In the development of this procedure it is convenient to introduce
a family of j; more general than those of (6.1) defined by
6E -- V,' /-a), ./,2...), (6.2)
where E is regarded initially as an arbitrary scalar parameter.
We require of the 4" that they be linearly Independent and such
that the (El)are square-integrable in S I.e., such that
that the operator y/ considered as a linear transfor-
mation from the set I f to the set i$i be invariably non-
singular for ail values of E
7The author is indebted to his advisor, Professor PO, Ltfr.din
for suggesting this choice.
From (2.10) we obtain for the operators 0() projecting on
the subspace spanned b, ,f (' k ( ) " (
Using these we construct, as in (2.13), a family of intermediate
Hamiltonians parameterized b,/ 6
;? '(C) = (-f (6.4)
According to (2.17) their eigenvalue equations,
SC c) )= E Q) (6.5J
are satisfied, for elgenvalues different from the L by eigen-
functions of the form
eC) =7 (6-E ) C (), 6.6)
with the C g given by
&c = A IA/ Ce)<>,l()/IVlsa)P. (6.7)
Using (6.2) in (6.6) gives an expression for the eigenfunctions in
terms of the
4 = (= o g )~, (z.eQ)f/. (.8)
We inquire now as to whether there exist specific operators
If 1/of the family (6.4) having the eigenvalue . for, in this
event, the eigenfunction (6.8) reduces to the finite linear combi-
nation of the S
If such operators exist the, will satisfy eigenvalue equations of
(Ol- E) C C /'-= CV.E-) C & f
4=S .C =
= E"vE Ct V
By forming the inner product of (6.10) successively with ([ ,
, (E ),- ., i l")we obtain the matrix eigenvalue equation
which has solutions provided
del [c<(C-E")( / V" I(-' -E')
+^\ = o (se
for real -. The existence of real roots to the secular equation
(6.12) is thus the criterion for the existence of operators i)
We observe that, since the resolvent has been eliminated from
in going from (6.8) to (6.9), (6.9) Is not restricted to
eigenvalues different from the .
...iiiiiiiiiii.iiiiiiiiiiiiiiiiiii : ii ii
having the eigenvalue E .
The left side of (6.12) Is a pol,ynomial of decree 2/1ln C and has
an even number/ of real roots (0 n,1 ) Each such root 2,
is an eigenvalue of an intermediate Hamiltoni3n e ( .i if
there are f distinct roots (-m,.), we find from (6.12) eigen-
values to the 4 operators) Se7), "
SShould a root be multiple it is a degenerate
eigenvalue of f CE The associated eigenfunctions e)f
S( *"),... are determined from (6.11).
The fact that the procedure supplies only one eigenvalue for
a given operator is a serious disadvantage for without the remaining
eigenvalues we cannot, in general, locate the known eigenvalue /E
in the spectrum of k Fj .and are consequently unable to deter-
mine to which eigenvalue of ra it is a lower bound. However,
from the relation
Iwhch serveLi|| to prove the following Theorem.
TII HCRE: If the known eigenvalue E orf CF E sat isvf ies
IIIIII ThIII subscr on A4 in (6.14) enumerates the ordered elgen-
and if upper bounds to oc e;genvalues of Yt lie below E ,
then ;is a lower bound to oc For the ground state the theorem
reduces simply to
c' ; .(6.16)
Fortunately. (6.16) is sufficient to identify lower bounds to the
ground states of many atomic and molecular systems.
In connection with the con.vergence oF this procedure we first
consider the entire family of operators (6.4) without regard to
whether they have solutions determined by (6.11). Since these are
just special cases of the intermediate Hamiltonians of Section II,
the conditions stated for convergence there are sufficient for
convergence of any member of (6.4) to O However, the com-
pleteness of the elements (6.2) requires, in addition to the com-
pleteness of the 4 that the transformation defined by (6.2)
be non-singular. In Appendix I we show that the necessary and
sufficient condition for the elements 1b, 't),. (6) * *, ift
i: :ii. .i::
to be linearly dependent is that some eigenfunction 4 f ofi
be included in the span of /,i /, and that
Therefore, if we separate the family (6.4) into those operators
for which C PO 4 =/,A,- -* and those cr which
-7 = ,,. ,the former will converge to AH
the 4 become complete wh lie the laI tter will rot .. rce (
will agree with on eI
< (6; "O 'E; / 613)
We return now to the members of (6.4) which have an eigenvalue
and eigenfunction determined by (6.11). We emphasize two aspects
cf the solutions to (6.11) in which they differ from the solutions
for the other procedures covered In this paper First, it is
entirely possible that a given equation rill have no solutions at
all corresponding to a situation in which (6.12. has no real roots
Second, we have no guarantee that the subspace of
will contain that of 0 )E'7since, in general, E E
We therefore have no assurance that the sequence E "
is monotonic increasing. Because of these complicating features
we discuss only a special case which, hocrever, is the one of in-
terest in applications. We suppose that, by a judicious choice
of the i we have succeeded in obtaining a set of equations
(6.11) of orders q t/ q-2 etc. whose secular equations
have at least one pair of real roots. Further we suppose that
each member of the sequence Z " formed of the
4, formed of the
't roots of the secular equations satisfies the theorem pre-
viously proved for some value of cc We then have
E2 E7 -ECC (ei=qjr .). (6.19)
We now make use of the convergence properties established above
to state that as hrh increases without limit and the A/
become complete one or the other of the qualities in (6.19) will
hold so that sequence will (on the average) either decrease to-
ward t c or increase toward Eoc. We are, of course, in-
terested only in lower bounds belonging to sequences exhibiting
the latter behavior.
As with the method of truncations i, this procedure contains
the method of special choice as a special case. When the ', are
chosen as eigenfunctions of o, (6.11) may be seen to reduce
We mention a feature of this procedure which is, to our knowl-
edge, unique among procedures based on intermediate problems. It
is that the procedure, while requiring a knowledge of the eigen-
%alues of ; does not use its eigenrunctions. This can be a
definite advantage in applications to systems such as diatomic
molecules where the f may be accurately determined but where
the 9 are not known in closed form.0
We are of the opinion that the success of this procedure In
our application to helium in Seqlction.iIX lies in the ease with which
may be chosen which lead to good intermediate eigenfunctions.
This feature is shared in large measure by the method of truncations
i. We close the theoretical part of this paper with some comments
in this connection.
For diatomic molecules the E d are constructed from
the solutions to the hyHrogen-mo lecule i|on Hamiltonlan whose
Functions are not Ra:w ijlln closed form 9IJIll i
VII. DESIRABLE FORMS FOP. INTEPHEDIATE EICENtFUNICT IONS
We consider the errur ;n a Iower bound due to the crror in
the eigenfunccion which according to (2.25) is (<6/ 5 I'- ./J
in general we expect a reduction in to ir.prcove the lower bound
so we wish to have the quantity I/ large. This is
most easily accomplished by selecting the $,. so that their con-
tribution to the eigenfunction is a linear combination of functions
known to serve as a good trial function for a variaticnal upper
bound calculation. If, e.g.,
#^ =A: E7.1)
Is such a good trial function we would choose P, given b,
= V" *; = Y,.-, -1.4) 7. ,
for the method of truncations i and by
1;-i = / '( '- '" ) 4 ('- : /,2, .. ) o (73)
tfor the method of Section VI. The respective eigenfunctions are
then of the desired form,
This is invariably true for the ground state.
However, if ue attempt a similar choice for the method of
truncations ii. i.e.,
I r -(7-
4?) = A (7.5)
However, ir tie attempt a similar choice for the method of
truncations ii i.e.,
^ = C( '0)"^ ,foyf .,m (7.6)
we find that we are required, in evaluating the elements of (5.11),
to compute the inner products,
This is generally not practical, being a task of the same
nature and difficulty as is found in applying the resolvent operator
in Section II. We are of the opinion that our poor results for
helium given in Section IX are a consequence of our not being able
to select our ij according to (7.6).
VIII. A TWO-PAFTICLE'HAF.MOHIC WELL
We determine lower bounds to the eigenvalues cf .j s',steri
consisting of two electrons confined b, an isoiropic harmncnic
well having the Hanri I tc.nian
I- 1 2 1
r a f j .t "VtJ 'I)
f g -j tw r t t(3.21
we write N as
S= " ( .4)
and see that it is of a suitable form since 0 and j are posii.iv-
definite, and the eigenfunctions and eiqenvalues of C may be de-
termined from the known eigenfunctions and eigenvalues rf the one-
= -2 a /. (C.5)
1Atomic units are used throughout Sections VIII and IX.
We use the method of chicice of elements detailed in Section III.
We note that the Harmiltonian (2.1) is well suited to this method since
its spectrum consists entire/ of bound le.tels.
le cutl;ne hre the procedure for constructing the eigonfunctions
and eigenalues of iC from those of c d, Since y marr, be
written as the sum
,(l) + Hf2), (8.6)
the eigenfuncticns of are linear coiTbinations of products of
the eigenfunctio'ns of f(Il) and Nt,(f), and its eigenvalues are
sums of their e;genvalues. It is shown [10, p. 16631 that the
eigenfunctions ofi are
and that the corresponding eigenvalues form a discrete spectrum
extending to infinity,
E,, = (,m )w ( = ,, )/ (8.8)
with all levels above the lowest degenerate. In (8.7) the W are
spherical harmonics and the are Laguerre polynomials
[10, p. 784] P and/lt are the usual total and magnetic orbital
angular momentum quantum numbers and on is a radial quantum number.
In cur calculations w; consider only singlet S states of 0t
so that we need only singlet S states of X The space parts
of these are linear combinations of products of the form
,m, ) ~m', '(2J ) w' ) *S)1
syr.netric in the cuorJinates. For convenience we designate therrm
with the usual spectroscopic notatic.n, (Is ), (?p 4p), etc.
Hyllcraas  has sh.-sn that these spac, pur ts imay be wri tten as
c, ,,, ) = i [ S,,,,.) S,,,I,,] Yo /,,...0.
or as I
(/M In S- S,
when m'1 t Si is the normal ed radial part of (1.7) and
19 is the angle between und / The symbol Q here represents
one of s, p, d, etc..
In this notation the eigenvalue equation for 6 is
:0VY M I n ) = (M *,",* l),l(, p (8.1,1
We list for reference the seven lowest singlet S e.iqenfunctions
of Nt and the corresponding energies:
State Designation Energy
S(1Is2) 3 CU
i, (1s3s) "5 2
Stare D.s i-in t ion En.2r.,
19 "i5s)5 7.w2
6i' Zpp4p //
96; (3d-) //
Having determined the lower part or the spectrum cf g e O. e
def; ne the o 6 according to (3.1), i.e.
A^ = ^ ( /1 2 ,. -.,7), (3.';)
and compute the elements
V,. = <(#: h/,p =< l ^ .> i4
The procedure used to evaluate the inteqrals is covered in Appendix II.
For an nh order calculation ( : 7)we select a subrnatrix of
order /' from the matrix Formed b' the elements (8.12) and invert
it to obtain the marrix A Having determined the matrix A
we form the eigenvalue equation (3.6)
Z [(E; -E^) / i A^ C2 =o, (0i,"
which may be solved for the eigenvalues /" and the eigenvectors
C by conventional methods. When the eigenvalues of (8.15) are
ordered with the eigenvalues which persist from owe have lower
bounds to the eigenvalues of N
In order to get an idea of the accuracy of the lower bounds we
have computed upper bounds by the well-known Rayleigh-Ritz procedure
based on trial functions of the form
The upper bounds are given as the solutions of a matrip equation
identical to (8.13) except that A is replaced by V wth
elements l i.e.,
EKE; -mJA + 0. 6.17)
For our calculations we chose the values 0.64 for This
choice gives to the ratio
S= <4/xl . ,S
a value of about 0.3 which is comparable to chat for the heli.jn atom.
Our results are given in Tables 1, 2, and 3. Table I displays
the matrices (V' ) its inverse A a, and the matrix V used in
the upper bound calculations. It is interesting to note the close
correspondence between / and V Our upper and lower bounds
are ilnpared in Table 2. The first order calcula tons were based
oiin ; and the third order on 4, 4'; and T ;. The
elements added for each of the fourth order calculations are given
in the table. We compare the normalized eigenfunctions obtained
fIrom the seventh order upper and lower bound calculations in
Ilile 3. We find that eigenfunctions representing the same states
areI substantially the same. Outstanding is the nearly complete
,Ireerient of the eigenfunctions for E 4 -
We see that the rate of convergence of the upper and lower
:- o .--
' C s' N .
Ci i i-
I?. Ir .
Ci Z L. o r
r-, -3 uj O ~ (Nj
oI P 1 -J 0 S C1
0 O LA C'
co 'o co
- a Q
c', N C;, J -, '
-T N L. -
SJ ul -3 -
. C 'iJ 0 LN
So I -
in -- I' -
"" O 03 L"i, .
o 0 ,*
CC C C
.I .01 -
Q. -4 NI LA
(NJ CO LAt 0
Table 2. Upper and
Icwer bcunds to the sinqlet S states of0 the hjrronic
E E E.
E E E. E
I i 7
5.0100 5.0673 5.1156
0 -- N r.
I I i I
0. nO "-CO C
: r r- r,
ar 0 0 0
S -- ( N O
M C.- 0u .
I .I. I -
m L CO o
. 0 C
fi LL LL
U II) 1 CL I-
.3 m3 32
C N re% -D n r..
-7 -^ -7 LP '-
-i^r: r t it'Li) r-II
u r. r" (
ST 0 .r' u'C, C1
i 0 r-i 0 II I
0 u'. CT .- p
0V, N- GC, Cj r -
c- ( 'r(N -rZ 0)
1t-- .7 0 '.- -a'. c
0 4 0 C, 0
fiC NNI NC) jt r
m N 3* a 0
0 r C -L C, r.
r 0- o r r
-r L.o oo o- -
LZ 0 .3 C'7 - ?
S O I". 1ML a 1
I-5 ^,2~i Dd ?^ ^
r. LL LL L.L 3j
bounds are generally comparable. However s since we do not know the
exact eigenVae weS Wc annot make an accurate judgeme nt. We find
nothing in these results which contradicts our assertion in Section
11 that the lower bournds give poorer estimates than the upper bounds.
We would not judge that either the upper or lower bounds exhibit
outstanding convergence. We feel that this may be attributed to
the functional form of the .1; I t is known that sets of functions
whose angular dependence is given by the spherical harmonics do not
exhibit rapid convergence for operators containing the
operator  I t is a shortcoming of the inethod of choice of
eeetthat such functions could not be avoided in this calculatikon
IX. LOWER BOUNDS TO THE GROUND STATE OF HELIUri
The helium atom serves as an ideal system For the testing of
lower bound procedures. Its non-relatikistic Hamiltonian possesses
a typical atomic spectrum consisting of an infinit,' of bound states
lying below its first limit point at -2  and a continuum in which
are imbedded other bound states above this point. At the same time
the Hamiltonian is of a relatively simple form so that the procedures
do not lead to unmanageable calculations. Further, as a result of
highly accurate calculations, the ground state is known to lie be-
tween the limits.1
-2.903726615 (au) E, -2.903724375 (au) (9.1)
so that the error in a gi'.en lower bound may be precisely determined.
*!iiie!!!!!!!jii jBazley  and Bazley and Fox  have previously applied the
Imthod of intermediate problems to the helium atom. The best estimate
given for the ground state is -3.0008 (au)  obtained as the
firit eigenvalue of a truncated intermediate Hamiltonian ~LC
in this section we give lower bounds to the ground state using the
proiedurIes of SectiornsIV, V, and VI.
eI1hsse bounds are those given by PCkeris [14, 15] The lower
hound iwas determined from Temple's formula [16 ]using the 1078 term
lritlllinlllll wave function which gave the upper bound. A better upper
und having an estimated error of about one-half that of Pekeris has
ivee by Sthtwartz [ 17] From his work we estimate that the error
|owir b|jd of (9.o ) Is about 1,000 times greater than that in
Wc consider the non-relati.istic helium Hamiltcnian,
= a 2 2
which may be put into. the form (2.1) by taLing as
S 2 2
and V as
~ may be wr it ten as the sum of two independent h/drog.n- I i k
whose solutions are well known  is therefore sclvable for
its eige-nvalues are just summed pairs of the eigenvalues of / and
its e;genfunctiorns are products of the corresponding elgenfunctions
properly synrietrized to account for the Pauli Principle.
Since we estimate only the ground state which is singlet S in
character, we require only the lowest singlet S states of R .
These consist of its ground state and a sequence of excited states
with cigenvalues converging toward the First limit point at -2,
c;= f[/+ 4 1 ( /= 2 ,...),
E,'- -f, s = -z ,
V ^, f = ^5-^ =. -?,, *2
The corresponding eigqnfunctions ha.e as pace parts the ,,mmrietric
#" = ,,. (', R .o rr ,
where the oare the norrralized radial wa.e funcr ions oi h.aing
zero angular momentum.
Having constructed the necessary solutions tc. Lwe turn tc.
a consideration of a choice of the elements w"hih are iser.ntilly
unrestricted in the procedures iwe employ here. Our aim is [c select
the ,bj so as to lead to qc.:,d 'nt.rmediatL wa..e functions js dis-
cussed in Section VII. At the same time, we wish to a,'oid elements
which lead to unduly complicated calculations. Elements which meet
these requirements very ve ll (for the procedures of S.cticns IV and
VI) are those selected so as to give eigenfunctions expressed as a
finite number of terms of the Hylleraas series [ 19 The scrics
is defined in terms of the coordinates
S -- + = u = 1 19.sl
73 ii' r0
^vV2 /o (
er is an arbitrary scaling parameter. The series has the
i[b!here, is an arbitrary scaling parameter. The series has the
ad- antige that fiat'.- functions composed of its terrs ar' automrat;ralI,
of singlet S character
H; leraas 1 19) fund thas onl! si. terms of (9.9) werc required
tu gq; an excll- I nt upper bound. It is these s\ w which tLi 'emplC,'
in .ur caIcuIat i ns We l i t them. h re normalized to 16in f.r
, ( 75
3 fr (q S
We present first lorerr
introduced in Scction VI.
bound d e cede
bounds determined b;' the procedure
Elements Containinn the Inverse of the Resolvcnt
We note from (9.1) and (3.6) that
so that we may identify lower bounds to the ground state by (6.15).
Here, in order to obtain intermediate wave functions of the form
(9.9) we have only to identify the AI of Section V1 with those of
(9.10). Our lower bounds obtained as cigenvalues of ) of
orders '.*ne to six require the e-aluartion of the elemernts,
M1, = <(Y-E'")//V'NI~'-E')X+//I/'-E7'[> (,I2
each oF which is a quadratic in L-
S= .(71 ) +/f ,E" E ML ,(E 2. 1.,)
We do not list all rwenti-one of th-es but rather gi\ two typical
examples. le find for M22 thi components,
/ /sZ= i '/ o./9 (5s- )[30o/. Fz -
/sfce.soss0/)-J -/ 309o -289(1/5-? ]
M2. j fiazo 6eo80( -11) -192
' [, /927
and for Z # the components,
MR4 ts3.-99,Ti Z f [og5 [ (z5 ( 15)
120o (/-)] -/808 ('-) t -i zz5 -360(i-) ,
M =253. 99/21 ,OO 20/56(1.25-- -25
In the abo~e, I is the ratio 2/ The procedures for computing
these elements are given in Appendix II.
We construct from these quantities eigen.alue equations of the
form (6.11) of orders one through six and determine their eiqen-
'alues (if any) by finding the real roots of the associated secular
equations (6.12). The technique we employ for finding the eigen-
values and eigenfunctions is an iterative one due to Ldw.din  .
We gi'e a brief description in Appendix II.
We have noted In Section VI that the real roots of (6.12) occur
in pairs, with an equation of order -, having from zero up to ^M
pairs. In our calculations we never find more than one pair of real
roots. This is despite the fact that, e.g., our secular equation of
order six is a polynomial of degree twelve in the energy. In Figure
I we give, as typical, the behavior of the roots of a secular equation
of order two based on the elements C and We see that, in the
region centered around I= 2 two roots are found which merge near
7-= /75and 7-= 2.2. Outside this range no roots are found. Both
roots are lower bounds by (9.11), but we retain as our lower bounds
only those given by 2
It is of some Interest to note that as the order of the call
~~111111111111111111IliiCi;;llr 111111 ; rllr1111lrr
1 .4 2.0 2, 2.452
1he cjrvel- IbeIed Eqtn E gvc, 'n"s VAlue co th grsound statu
of diFferont cpEralors H' 2Ead/E
culations increases the curve flattens out rapidly, so that at
S= 6 the lower curve is quite close to a straight line, E = -7 .
This supports our discussion in Section VI regarding convergence to
the E, The exact value -4 is attained at ,= 2, for an, order,
since at this point is identicall with 4'
In Figure 2 we give the curves -;obtained from our calculations
of orders one through six. It is gratifying to observe the monotonic
increase of the lower bounds with order since we have not proved that
this must necessarily occur In this procedure. At points where two
curves tcuch as, e.g., the point near P-s2.O2for if and ,
we find that the higher order eigenfunction reduces to the lower
order eigenfunction. Thus at such points
and f' (1E ) have a common eigenfunction as well as a common
In Table 4 we compare energies and wave functions of our optimized
lower bounds of orders one, two, three and six with those of well- III
known upper bounds of the some orders. We see that, though individual
coefficients and the scaling parameter differ somewhat, corresponding
wave functions have the same general form.
For some time two lower bound procedures, not connected with
Intermediate problems, have been known and used rather ext E lsivly
in obtaining lower bounds to helium 2 22, I Thee are due to
Temple  and Stevenson and Crawf 231 ihE proiedure of
Stevenson and Crawford is more diicult to Apply than Templ
A ca 1 cu lat i mofiii;, order I s based on the functin
taken in thei.order ti ha tiiiithev anDi riiiiii
oL 0 :,
i r. C, -a I U I
+ + 0,
- I, ,' ,' I
a) ( C. C> Lfl MTi
C L- O r_ o N O
%C r0 0 0 0 0
C "- = :- L
C 3 M n
u + + C+ + + L
.0 C o_ al o. *n a
a, o Ln a a a a a a a
Ec IC I - IC O I
0 di '1 t0 Ldi di O 01 E
0 f Lf C, r LA V' C
o I IN 0 r- C 1
>r - 0 0 0
L 0 L
-i Lf I' I 0 rI 3
a .0 0 O- 7 0- 04 30 0 -
0 0 1
La Aa 0 0 0 0 0 0 0
ID w m > i L
Li -I 0 3 L 3 0
EiEiEEEEEEEEii~~iEiEm i EiEinEiii iiiini l:::EEEEEEEEEEEEEEEEEEEEEEE: EEEEEE E :::. E E: .........
is somewhat more efficient.15 Actually it contains Temple's formula
as a special case  Their lower bound is given b,
E, oc ( oc<(/2/# >-cc'],(9. 16'
where the quantity (fC OlHi ?Cc ( /la f ) is min;imied ti,
variational means and where oc must satisfy
oc 6 E, + E, (9.17)
Optimum lower bounds are attained when the equality, in 19.171 holds.
In their original paper [2.] Stc.'enson and Crawford computed
lower bounds to helium using trial functions cons;stinq of our
functions ; f, and 1 ; through /1 ; and a nine
term function in which three additional terms of the series (9 91
were appended to the six term function. We giae their results
in Table 5. We also compare our lower bounds with theirs and with
the upper bounds of Table 4 by listing the ratios of the corres-
ponding errors. We see that our lower bounds, while quit.: inferior
to the upper bounds, are superior to those of Ste.enson and Crawford.
(Actually our lower bound of order six, -2.9094, is ter, close to
theirs of order nine.)
Since we judge the procedure given here to be roughly comparable
in difficulty with that of Stevenson and Crawford, we consider the
data of Table 5 to be suggestive of the possibility that this pro-
1 Kinoshita [21, 22] using a 39 term trial function reduced the
Ieirror In his lower bound by a factor of two In going from Temple's
Formula to the procedure of Stevenson and Crawford.
Table 5. Comparison of the errors in lower and upper bounds to helium.
Order Lower bounds of
Ratio of the error
in the lower bounds
of Stevenson and
Crawfr.rd to the
errors in ours
Patio of the error
in our lower bounds
to the errors in the
upper bounds of
1 -3.5410 3.9 2.9
3 -2.9481 2.6 13
6 -2.9215 3.1 12
9 -2.9089 --
Taken from [23) The calculations of orders one, three and six
could be improved slightly by using a less conservative value of a
[see (9.17)]. The ninth order calculation used a value of a near
cedure may provide a practical means of obtaining more accurate lower
bounds than those obtained to date.
The Hethod of Truncations I
For this procedure ue construct the elements ?, from the
functions (9.10) according to (7.2), i.e.,
p; 1 = f ( = ,, . ) 9.,1 )
so that our intermediate wa.e functions are expressed in terms of
the #a and the /A as in (7.4).
We consider first truncated intermediate Ham;l tonians
m = /,2 6 The eigenvalue equation 14.8) reduces in this
+'^ ./,//> +
+ ( _- <4/-lc" (9.1?)
-25 J' _
Evaluation of the inner products appearing in (9.19) is eleren-
tary. The formulas we use are given In Appendix II. As an example
we give the numerical results for the equation (9.19) of order two.
Cenoting the elements in brackets by /A we find,
N,= 8.75 7 + 65./7'_ + 7 25/r
-9 r ", -2.5 E'
//, = 9.79795927 + 2286.1905/ '
+ 3.r52/726 ,' A.2,8./1905/l 3 (9.2
/ = /3,/25. 1f 4 'o9/7f666r/ S 94-Z#fl _/_
f- -C'" -Z.5 E"
where 0 is the quant; t j f 2)/2
By multiplying each element / of equation (9.9) by the
comrion denominator (4 t E"')(?5 "') we convert the elements
into quadratics in the energy so thac the technique used to
solve f6.11) may be applied here. Our solutions give, as the
lowest eigenvalues of the operators F /,r=/2,. f
the lower bounds in Table 6. The values of ? given are those
which optimize the lower bounds. The eigenfunction corresponding
to = is,
16This operation introduces a spurious root ,"- into the
corresponding secular equation but does not affect the already existi||ng|
,"' = 777 7 -7 367, -.0C0^ fC
i 9. 2, 1)
+./6951 /0185f +0753 T
The functions 9,, and the r are normalized to one in (3.211.
Our results in Table 6 ar. not particularly, qood. In fact. our
best lower bound, E f = 3 004-5 is inferior to the lower bound,
E, 3. 0008 obtained b, Bazle, and Fox [ 7 ] using this
procedure. It is clear upon ccrparinq these two valuess and noting
the slow improvement with / that in order tc iiTpro.e thcse bounds
significantly we would have to increase the index < J in cur OY
However, the numerical calculations increase in complexity ..ry
rapidly with so before proceeding to the morel in.ol.ed cal-
culations, we determined an upper bound to the impro.-erent which can
be obtained by increasing e
The upper bound depends on the fact that the operator
UL" = E t/'( /I Eq[I- /e >(#( /J
= /iX / z [/ /u+t +i]
is larger than any ,
Therefore the eigenalue = (9.2of
Therefore the eigenvalues ,I,, of
&L"'= &tf L. /
Table 6. Lower bounds to the ground
of truncations i.
E n (au)
state of helium by the method
We quote this .alue from B-zlev and Fox 
....... .. ...... ...I.ii.... ::...h
are upper bounds to the eigcnvalues of all i.e.,
EPi A: (/(,q..>c/2, 9.25)
The A different from -4 and -2 are c.btaaned from the solutions
of a set of equations equivalent to (9.19) except that Z-, is re-
placed by Eo
We find that the first eigen.alue of the 6 corresponding to
Sis optimized for 7= 2.30 and has the value
S 2.96 56 (9.261
Therefore we have
-3.049 5 5 E -" -Z 9656 (9.27)
(= ,2,,-) .-)
On the basis of (9.27) we have concluded tiat it would not be
profitable to extend our calculations to larger .alues of P
Incidentally, the rather large difference between the upper
bound AI and the true ground state f, is indicative of the well-
known fact that the continuum of O'o has a significant projection
on the ground state wave function J, of helium 14 .
The Method of Truncations II
As discussed In Section VII it is not practical with this pro-
cedure to attempt to construct lA,` which give intermediate eigen-
functions expressed as linear combinations of the 0: and the /,
*f (9.10). Our calculations with this procedure are confined to
operators 7 and are intended to illustrate the results ob-
trained when the a, ore chosen so as to led to straightforward
For the operators [ the eigenalue equation 15.4) reduces
t 1"' (9.2S)
For our first calculation e used the element,
r u= nd_ as2 i e ( 9.2.9
and rc.und as the First root of (9.2S) at = /.80 the optimum
jalue -3.8055 frcr / #
We next tried
2 r2 -75
c( _) Cu e (9.30)
and found for r the poorer value -3.8981 at ?. .
Finally we tried the element,
Wli iiiiiriiii1 i rii;ii~ii,
Here we found a lower bound of -3.3448 8 t 9-= 9 We consider
this value close to the best that can be obtained using a single
element based on simple functions such as those used here. Further,
considering the poor results obtained with the elements an, and ~ ,
we do not feel that significant imprrovemnent would result in gcing to
calculations in.holving the operators etc. Of
course, it may be possible to construct elements which do not lead
to unmanageable calculations and which at the sane time gqi.e grod
lower bounds, however, we have found no ja, .f constructing such
elements for helium.
We give, as an example, the eigenfunction 4f, obtained from
the calculation based on the element c ,
= #: [.752 +.59 -2 s- J .'3
We see that t appears as the last term in brackets. The wave
functions obtained from the other two calculations are more in-
volved than (9.33) and contain numerous terms in addition to 3, and
the appropriate function / .
We find this procedure the most difficult to apply of the
three used in this section. The expressions for the elements of
the eigenvalue equation (9,28) are even more complicated than
those for the procedure of Section VI. To illustrate this we
171n this case, at = 2 the method becomes identical to that
of truncations i. This results from the fact that atl =S? Z, fo
reduces to I, and we have
Using 9o in a '"'calculation Bazley and Fox  obtained the
better lower bound -3.29.
write out (9.28) for the calculation Involving .
I -35 7530
25 6/5'," 'F <
7 9 t- + 720(,5 ) 35 -/-
#120d2 -60(l./5- ) +S.. f7s '- s +
367s50 V4 67 o
2 5 6//P 0
In (9.34) is given byl2/ and Y by ( 2/ 17f The
techniques needed to evaluate the various inner products in (9.28)
are those used in the previous two procedures and are given in
Projection Operators In the Vector Space S18
We consider the vector space S introduced in Section II character-
ized bj the positi'.e definite metric operator V and the inner product
(4/I/ y> but having its elements normalized according to
(4/i4 > = / i (ALl)
We Introduce in a linearly independent set of elements [f.l
and consider the linear manifold mE consisting of the totality of
elements of 5 cf the form
-=2 ^ (Ai.z)
We take an arbitrary element 49 of S and write it as
for the purpose of showing that if the CA are chosen so that
< /V/fJ> = A = / i iii4)
18Ths development, with changes in notation to agree with the
main text, Is taken from a seminar given by P. 0. LOwdin at thii
University of Florida in June of 1962.
then the operator "defined b/
o0 = (Al .5)
is a projection operator in S projecting on the linear manifold M
From (A1.3) and (Al.4) we have
<(i'/ l7) = <(rIV/I >) (a.6)
(d = /,, * *,m )
so that if A is the Inverse of the nth order matrix with elements
iiiI iiiicri/v/4, = ',
iFr" (Al.7) we may define 0 as
The properies (2.11) are easily verified directly from (Al.8),
c n), are suf ficient to establish that tois a pro-
..(Al.) .. ..
Th!petiii e ( t w 2.1 ) are easily verified direct ly from (AI. 8),
jection operator in S.
The inequalities (2.12) follow from Bessel's inequality/ in 5
or directly' from the fact that OC is a projection operator ;n S
and that the subspace of l contains that of 0 [25, p. 68]
tie conclude this discussion by showing that when the CA are
giten by (Al.7' the quantity (6 ) V is a minimum. We first take
f to be
where the Q,, are not restricted to be of the form (A1.7). For 0
given by (Al.11)
<< vi > = (4 V/S)>
( I 12)
-2E7 i(4/ i/p) tC2JX jpyI/cst>,
which from (AI.A) can be rewritten as
<(/V ) <4d/itd> ( C,,ai cTaZ)
<-' ( P( / IV/ai p .
By adding and subtracting b, (c' ./V/IY C, we obtain
<(6/vl/ = (dlV/I? -Lk
(Al 1 ii
+ C2 < a ci / V/i ( -, iii
which, since the last term Is positive deffinite, is a minimum for
' i = i a "*"- i .t "
::EllEEEE::. 1 .............. EEEEE
Conditions Under Uh ch the Transformation
V"(VT- 1 ) is Sinqular
We consider the equation (6.2) defining the i, in terms of the
linearly independent set f.f
V7Y C"( c) ( =, ...,), (A.l,
and seek the conditions under which the transformation I/ (l oC- )
is singular. It is sufficient to examine the transformation ( .-)
since / being positive definite, is ne.er singular 
We suppose, for convenience, that 0O is positive definite
and define the functions by
The necessary and sufficient condition for the to be
linearly dependent is that there exists a set of coefficients L, ,
not all of which are zero, such that
0 ( I
iilii i:iiiiiiiiiiiii !
SE ) ,, = ,., (A.4~1
If XE! Is bounded from below we make It positive definite by
vinr the zero of energy,
whZr t th -* f c E---
Now neither side of (A.4) can vanish on account of the positive
definiteness of o and the fact that 6 Is a scalar constant so
that (A.4) Is just the eigenvalue equation for rC Therefore, we
state that the necessary, and sufficient conditions for V'-, (f c )
to be singular are that some eigenfunction 7'6" of do be contained
in the span of A, '- n and that simultaneously j .
S.................. .. i
Outline of the Procedures Used in the
Calculations of Section VII
Calculation of matrix elements
The elements of the matrices V and V 1of Section V(11 are
/1-1 1 14 /77 3 .2 /7 1, C? 2
Using the relation
(OS 2 >2(A2.3)
in equations (8,10) and (8.11) and insertint these in (A2.1) and (2
reduces the integrands to lI neAr combin,-t ions of fucions of
and 2 Us ing a method of C, lai s and Lthd in j261 For h nd nq s uch
Integrals permits us to) wri te them a I near cciribinatleons oi- the mul-
where we have let
S= C = z= (42..)
Upcn writing out the Laguerre polynomials and carrying out the Z
Integration these are further reduced to sums of the double in-
2 r X ,
(2Zr', Zd) =d x Ixe g
0 0 (A2.6)
which may be evaluated in closed Form,
(2-rn,,,) ( 3 (/x3.5 (2.cf 2.-,)-
qt r =it r- .7 (42.7)
The -formula (A2.7) is obtained from the formula [27 p. 176],
e x x 6 Y = /
e e d L, I (A2.8)
by successive differentiation with respect to 4 and A.
A program was written for use on an IBM 709 computer which
evaluated the integrals (A2.4) as linear combinations of (AZ.7).
|lThese integrals were then combined to give the matrix elements (42.1)
ine on of and solution of the eigenvalue equations (8.15) and
The A were obtained from the appropriate submatrices of V
and the eigenvalue equations (8.15) and (8.17) were solved by computer
ipr rans kindly supply ed by Mr. J. 0. Nordling of the Quantum Theory
PIIrojIel t f th.ini Fe University of Florida.
i i ii i i ; .. . . . . . . . . .
Outline of the Procedures Used in the Calculations of Section IX
Evaluation of matrix elements
The elements of equations (6.11), (9.19), and (9.28) require the
evaluation of one or more of the quantities ( -o
a nd where /1 is one of the terms of (9.9).
In [ 21] & eo is given in the coordinates ra, and )2
Using the relations (9.8), the express ion can be transformed to the co-
ordinates S, i and e,& w 1th the resu It
as" d t,- s d s ; t t
ds 25 +2 a.2 _Sz J A29
&( s -, -l dads e (S., t 2) udt
2 la ) 2 Id le -f a s
Therefore 0r operating on g aives a linear coribinatiC-T Of terms
of the form
(52-r") S (48(A2, 10)
w h ere ^e :r o r 0 0 ea l and -. being "
gives a single term of the form (A2,10) when operating on 7' There-
fare. 60 beng just the sum of d and also cives linear
comb Ina t ions of (A2 10). ope r,,t Ing on 7L g ives, a l;near comb ina tion
of /1 Ond 6 j (nor mal ized to) 16" 2 ) is exp Ii c It
= 37 8(A2,10
With this information It may be seen From an examination of (6.11)
(9.19), and (9.28) that all matrix elements have as integrands linear
combinations of terms which are at most quadratic in the quantities
(A2.10) and in (A2.11). The elements themselves may therefore be
shown to be (see, e.g., (10, p. 1737]) linear combinations of the
(n/t,&.n) = rn^f e s ds
S U (A2.12)
X u'dj^s-t' t'dt ,S
where a, r-/,0 ur / ,4 O ,0 / -/ and S is either
4? orf 7 These may all be evaluated in closed form although
several formulas result. Ue list them below.20
(0/4, Z 71) -2 69*2 1 ( t d o r 2) / (A2.13)
(2 .Zo + ,, + Y)!
F/r rnn i-Z^ 3
A__4__0o_____ L (A2. 15)
or convenience in our calculations we used functions normalized
CID To compensate we divided (A2.12) by 16 2 in our actual
t 2 --J C +, -2 rc)(2 -% -2/)
661(& L-/. \
( -//.1 1, -) = z rr ( ) --
,o (a -2,c -/)J
A large number of the Integrals (A2.12) were required for the
,arlous calculations, so a simple program for evaluating them was
written for the I1M1 709, embodying the formulas (12.13) through (A2.16).
Solution of the eigenvalue equations
The elements of the matri. eigenvalue equations (6.11), (9.19),
and (9.28) are, or can be made into, polynomials in the energy. Ue
therefore consider an n order equation
/ C = 0 (A2.17)
where the elements of M are
= Ay (E) r., = i,,. ;,- ). (A2.18)
LUwdln  has given an iterative procedure based on a partitioning
technique for solving such equations. The procedure was originally |
developed to handle equations whose elements are linear in the energy
but is readily adapted to equations where thr /n are more complicated
functions of [28) We sketch here the development of the pro-
cedure when the 11/ are given by (A2.18). We do not consider compli-
cations which arise In special cases as, e.g., when there are multiple
eigenvalues. These are covered in detail in 
We begin by partitioning M and C in (A2.17) to obtain
t= 0 ('42.19)
fiL /'bb Ch
where M is one-dimensicnal and I'bis( -/)-dimensiora l. Carrying
out the multiplications we obtain the separate relations,
.',, C, / n, Cb = 0
M/ ,C, -t1 C = O ,
which can be recombined to give
( ,, M,, ; M I)C, = o(2.^ )
If C, Is not zero we have the relation
F(E) = (1,, ', /6 i ,) = ~(A2.22)
which Is equivalent to (A2.17), and is suitable for solutic.n b/ iterat;.e
In  It is shown, by an application of the hewton-Raphson
formula to F(), that a convergent second-order iteration procedure
for can be obtained in the form
==.Ii lb bb .
/ tib MIU6 nis tbM f~b, in, n6 ( .
We note that rib66 is easily evaluated, having elements given by
The Iterative process is initiated by selecting a trial value a
for ,computing? FZ)and11 efZ)and determining a f irst Iterate
'C" fis then used in place of Zo to start a new cycle
which determines a second iterate /1.The process is continued
un ti I the careemn nt be tween success ive i tera tes i s sa t isf actory The
e i genvector bel ong ing to the e 'Jgenva Iue thus f ou nd i s, f rom the second
of (A2.20), given by
evaluated at the elgenvalue.
A program, employing this procedure and capable of handilinq
equations of orders up to six whose elements are polynomials cf degrees
up to six,2 was prepared for usc on the IBM 7()9. The program con-
'tains, as subaroutines, the actual ele-men~ts Ple'xpressed as functions
of l' .Exatiples rethe- clements display-A in) (9,14) and 95)
Input to the program i5 a tset of values of )for which solutions ire
,to be found and a ranye of enerqy which is to be scarnned for eigenvalucs.
21These vziluas are easily increased, being limited only by loss
,,r significant Figures and the capacity of the machine-.
I. Weinstein. A., Hem. Sci. Math. No. 88 (1937).
2. Aronszajn, N., Proc. Ilar. Ac. Sci. 34, 474 (1943).
3. Aronszajn, N., Proc. Nat. Ac. Scl. 24, 594 (1948).
4, Aronszajn, N., Proceedings of the Oklahoma Symposium on
Spectral Theory and Different;al Problems, Oklahoma A. and lM.
College (1951) (unpublished).
5. Bazley, N. W., Phys. Rev. 120, 144 (1960).
6. Bazley, N. W., and For, D. W., J. Research IlatI. Bur. Standards,
65B, No. 2, I05 (1961).
7. Bazley, N. W. and Fox, D. U., Phys. Pe.. 124, 483 (1361).
8. Weyl, H., Bull. Am. Math. Soc. 56, 115 (1950).
9. Bates, D. R., Ledsham, K., and Stewart, A. L., Phil. Trans. R ,y.
Soc. (London) A246, 215 (1953).
10. Morse, P. H., and Feshbach, H., "Methods of Theoretical Physics",
Vols. I and II, McGraw-Hill, New ork (1953).
11. Hylleroas, E. A., Z. Physik 4E, 469 (1928).
12. Shull, H. and Liwdin, P. 0., J. Chem. Phys. 10, 617 (1959).
13. Kato, T,, Trans. Am. Math. Soc. JO, 212 (1951).
14. Pekeris, C. L., Phys. Rev. 115, 1216 (1959).
15. Pekeris, C. L., Phys. Rev. 126, 1470 (1962).
16. Temple, G., Proc. Roy. Soc. (London) A119, 276 (1928).
i17. Schwartz, C., Phys, Rev. 128, 1146 (1962).
18. Schrldinger, E., Ann, Physik j7, 361 (1926..
IIIII Hylleraas, E. A., Z. Physlk 54, 347 (1929).
ii20. Ldwdln, P.O,, J. Mol. Spectroscopy 10, 12 (1963).
21. KInoshlta, T., Phys. Pe.. 10j. 1490 (1957).
22. Kinoshita, T., Phys. Re'. I15, 366 (1959).
23. Stevenson, A. F., and Crawford, M. F., Phys. Rev. 54, 375 (1938).
24. Pauling, L., and Wilson, E. B., "Introduction to Quantum Mechanics",
McGraw-Hill, flew York (1935).
25. Akhiezer, N. I., and Glazman, I. M., "Theory of Linear Operators
in Hilbert Space", Vol. I, Fredric Ungar, New York (1961).
26. Calais, J. L., and Ltwdin, P. 0., Preprint No. 6, University of
Florida Quantum Theory Project (1960) (unpublished).
27. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.,
"Tables of Integral Transforms", Vol. I, McGraw-Hill, New York
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Quantum Theory Project (1962) (unpublished).
" Eiiiiiiiiii EEEE:iiiiiiiii :
Jackson Cilbert Gay was born December 27, 1932 at Selma,
Alabama. He attended the public school system in that city and was
graduated from the Albert G. Parrish High School in June, 1950.
In June, 1955, he received the degree of Eachelcr of Science from
Auburn University. From 1955 until 1960 he .was empl/ed bt, Fratt
and Whitney Aircraft in Middletown, Connecticut. In 1960 he was
awarded a National Defense Act Fellowship to pursue work toward
the degree of Doctor of Philosophy at the Uni ersit.,' of Florida.
Since April, 1963, he has worked as a teaching assistant in the
Department of Physics.
Jackson Gilbert Gay is married to the former Patricia Gail
Varner and is the father of two children. He is a member of the
American Physical Society and Sigma Pi Sigma.
: ..........E : .... . ...........
This dissertation was prepared uinder the direction of the
charma ofthe candidate's superviso ry committe-e anid has been
approve by all membeers of that commrittee. I t was subr i td to
the Dean of the Cllege00 of Arts and Sc iences and to the Graduate
Coluncil, and was appro ved as partial fulfillmen of the require-
met o r the degree of Doctor of Phi losophy.
Deebr 2 ) 193
Dean, ColIlege of krts icc-ce,
Dean, CG- iduate cho
Supervisory C omittee:
Chairman n i71 -1-