General Sp;n Orbitals for Three-Electron Systems
by
Nelson H.F. Beehe
A DiSSERTATiO;J' P2ESENTEP TO THE GRADUATE COUiNCIL OF
THE' UNIVERSITY OF FLORIDA I'i PARTIAL FULFiLL!' ENT OF
THE REO'UIJ EMU TS FOR THF DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1972
ACKNOWLEDGEMENTS
I would like to thank all the members of the
Quantum Theory Project for helping to provide such a
stimulating environment for research, and especially
Per-Olov L8wdin, for having the wonderful idea of an
international group working in quantum science.
I am very grateful to Sten Lunell, who first
suggested the problem examined here and with whom I
have closely worked, he in Uppsala, and I in
Gainesville.
Yngve Ohrn and Charles Reid have often helped
when problems arose that I could not stumble through
myself.
Support of the Computing Center and the
Chemistry Department of the University of Florida, and
of the Air Force Office of Scientific Research and the
National Science Foundation through grants AFOSR-
71.1714B and NSF-GP-16666 is gratefully acknowledged.
Finally, I wish to dedicate this thesis to my
wife, Thesa, for her sacrifice over the last four years
has greatly exceeded mine.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . .
LIST OF TABLES . . . . . . ..
ABSTRACT . . . . . . . . .
INTRODUCTION . . . . . . . .
CHAPTER 1 HARTREE-FOCK AND BEYOND ..
1.1 The Hartree-Fock Method . . .
1.2 Extensions to Hartree-Fock Theory
Relaxation of Restrictions ..
The Symmetry Dilemma .. . . .
On Spanning the Angular Momentum
Spaces . . . . . . .
1.3 Other Methods . . . . . .
The Configuration Interaction
Method . . . . . . .
Interelectronic Coordinate Methods
Bethe-Goldstone Perturbation
Theory . . . . . . .
CHAPTER 2 REDUCED DENSITY MATRICES . . .
2.1 Introduction to the Density Matrix
Literature . . . . . . .
2.2 Construction of the Reduced Density
Matrix . . . . . . . .
Definition of the Reduced Density
Matrix . . . . . . .
The Reduced Density Matrix for a C
Wavefunction . . .. ..
The Reduced Density Matrix for a
Non-CI Wavefunction . . .
2.3 Properties of Density Matrices . .
CI Expansion Convergence . .
Bounds on Occupation Numbers . .
Page
* ii
* vi
. vii
S 1
. 10
. 10
. 23
. 23
. 25
. 30
. 39
. 39
. 42
. 44
. 46
. 46
. 47
The Carlson-Keller-Schmidt Theorem . 71
Symmetry Properties . . . .. 73
Density Hatrices of Some Special
Functions . . . . . ... 78
2.4 The N-Representability Problem ..... 85
CHAPTER 3 ATOMIC PROPERTIES . . . . . 88
3.1 Introduction . . . . . ... 88
3.2 Energies . . . . . . ... .89
3.3 Specific Mass Effect . . . . .. 90
3.4 Relativistic Mass Increase . . .. 92
3.5 Transition Probabilities and Oscillator
Strengths . . . . . . . 94
3.6 Fine and Hyperfine Structure . . .. 95
CHAPTER 4 THE PROJECTED GENERAL SPIN ORBITAL
CALCULATIONS . . . . . . 104
4.1 Introduction . . . . . ... 104
4.2 Matrix Fornulation of the PGSO Method 105
4.3 Choice of Bases and Initial Orbitals . 108
4.4 Evaluation of the 1-Matrix . . . 112
4.5 Evaluation of the 2-latrix . . . 118
4.6 The Hyperfine Analysis . . . ... 119
4.7 Comparison with Other Methods . .. 126
4.8 Hyperfine Structure Results by Other
Methods . .. . . . . . . 128
APPENDIX 1 VALUES OF SOME PHYSICAL CONSTANTS . 130
APPENDIX 2 THE COMPUTER PROGRAMS . . . .. 135
APPENDIX 3 CONVENTIONS FOR SPHERICAL HARMONICS
AND SPHERICAL TENSORS . . ... .139
APPEND IX 4 0N' THE CUSP CONDITIONS . . . . 142
APPENDIX 5 SOLUTION OF THE MATRIX SCHRUDIDINGER
EQUATION, HC = SCE . . .. . 148
APPENDIX 6 LIST OF ABBREVIATIONS . . . .. 152
APPENDIX 7 SOME THEOREMS ON DIRECT PRODUCT
MATRICES . . . . . ... 155
BIBLIOGRAPHY . . . . . . . ... 194
BIOGRAPHICAL SKETCH . . . . . . .. 238
LIST OF TABLES
Table Page
1. Energy conversion factors . . . ... 159
2. Comparison of the convergence of some
properties with convergence of the
energy for various basis sets . . .. 161
3. Sample bases, properties, and natural
analyses for Li 2 2P and 2 2S ....... 171
4. Contributions to one-electron properties
by NSO . . . . . . . . . 177
5. Comparison of energies and Fermi contact
terms from different methods . . . . 180
6. Angular parameters for the evaluation of
hfs matrix elements . . . . . . 184
7. Experimental hyperfine structure
parameters . . . . . . . . 185
8. A and B hfs parameters for several states
of atomic lithium . . . . . . 186
9. Summary of hfs parameters for Li
calculated by different methods . . .. 188
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
GENERAL SPIN ORBITALS IN THREE-ELECTRON SYSTEMS
By
Nelson H.F. Beebe
June, 1972
Chairman: P.O. Lw.d din
Cochairman: NI.Y. thrn
Major Department: Chemistry
The use of general spin-orbitals in a spin-
projected Slater determinant for a number of states of
several three-electron atomic systems is studied. The
spin-optimized self-consistent-field method, which is
an alternative way of treating the spin degeneracy
problem, is shown to be a special case of the projected
general spin-orbital approach, and comparisons are made
between the two methods.
Computation of the hyperfine structure
parameters has been carried out for all the systems
studied, and the results should be a useful guide to
future experimental work.
vii
The method of construction of a general p-th
order reduced density matrix from a configuration
interaction wavefunction built from non-orthogonal
orbitals is derived as part of the work.
The general conclusion of this work is that
while the projected general spin-orbital method is a
conceptually satisfying approach, particularly for
treating the spin degeneracy problem, it is not
competitive with configuration interaction calculations
of comparable accuracy.
viii
INTRODUCTION
The time-independent Schridinger equation,
H/ = E (1)
in principle describes most of the physics and
chemistry of matter. Unfortunately, neither the
Hamiltonian -operator, H, nor the wavefunction, Z are
ever known exactly, and consequently, neither is the
energy E. For the non-relativistic case of a single
particle, such as an electron, moving in a central
field of force, such as that provided by a nucleus, H
is known fairly accurately, and (1) can be solved
exactly. The series of hydrogen-like atoms, H, He+,
Li++, etc. fall in this category. The relativistic
counterpart of (1), the Dirac equation, cannot be
solved exactly, even in this simple case, unless the
nuclear mass is assumed infinite; a covariant many-
particle Dirac Hamiltonian is not even known.
As soon as another particle is introduced,
the equation, like the classical Kepler three-body
problem, is insoluble, and approximation methods must
- 1
- 2 -
be resorted to.
The non-relativistic Hamiltonian is known
sufficiently accurately to describe chemical
properties, so that determination of the wavefunction
is the principal problem. An approximate relativistic
Hamiltonian may be written for a many-particle system;
in the limit of low electron velocity, it can be made
to reduce to the usual non-relativistic Hamiltonian.
The physicist may be interested in studying the
construction of the proper Hamiltonian, while the
chemist is more concerned with the results, that is,
the wavefunction and the properties that can be deduced
from it.
Although the calculations'described in this
thesis use a non-relativistic Hamiltonian, many ideas
and corrections are borrowed from the relativistic
case. These will be introduced as we go along.
The non-relativistic atomic Hamiltonian that
we shall use may be written
7 e2 + e + (2)
H iri
where the summations go over electrons, and
- 3 -
-e = electronic charge,
h = Planck's constant,
J> = h/2n,
ri = distance of the i-th electron from the
center of mass,
r.. = distance between the i-th and j-th
IJ
electrons
/ O= (1/m + 1/M)
= reduced mass,
m = electronic rest mass,
M = nuclear rest mass,
and Z = atomic number.
Since it is inconvenient to introduce
experimental values of h, e, and p, we shall convert (2)
to atomic units. To distinguish between ordinary units
and atomic units, we will temporarily put primes on
quantities expressed in ordinary units. We begin by
'2
introducing some atomic unit of length, a. Then, since V
is homogeneous of degree -2, we replace it by -2 72 and
r' by ar. Multiplying by a2/A 2, we obtain
1-1'- ji ,2
227
-2: (ez #z
217
ly
(3)
and E = E'( A2/ 2). (4)
We can remove the experimental factors now by requiring
which gives
which gives
a = )2/(, 2),
(5)
Setting
E = 2/ (/2) = e2/a = ,e4//2, (6)
we see that the atomic units of energy and length, E
and a, both depend upon u, and hence upon the nuclear mass
M. We shall call the atomic unit of energy, E, the
Hartree (H.) and the unit of length, ., the Bohr (B.).
If we let M become infinite, the reduced mass becomes
equivalent to the electronic rest mass, and we find
E 2/me4, (7)
or E = (1 + m/M)E0, (8)
and a = /me2 (9)
or a = (1 + m/M)aO. (10)
It is also convenient to introduce the fine structure
constant,
= e2/Ac = 1/137.0888
(11)
We then find that the Hartree is also given by
- 5 -
e4/2 = dc2 (12)
That is, the Hartree corresponds to the rest energy of
the reduced mass times the square of the fine structure
constant. We observe that the Hartree is therefore on
this scale a rather small unit, although Appendix I,
which tabulates the most recent values of the
fundamental constants and conversion factors, shows
that for chemical energies, it is a large unit.
According to (6), the energy levels of two
isotopes of a given atom will differ by a slight amount
given by
_____ %I (-I
(13)
(M'- N) +
= --+ m ( - / *
The spectral lines of the heavier isotope will
therefore appear at higher energies (higher
frequencies, shorter wavelengths). This effect is most
pronounced for the three isotopes of hydrogen and is
called the isotope shift, or normal mass effect. Even
then, it is small, since /XD//ZH = 1.000817. Its existence
- 6 -
led to the discovery of deuterium in spectra of
residues of liquid hydrogen in 1931. Its importance in
spectroscopy lies in its very simplicity; since the
shift can be exactly predicted, it can be used as a
tool for identifying spectral lines. In the case of
the isotopes of hydrogen, for example, it may be used
to separate the specific mass effect, or mass
polarization, which will be discussed later.
Our assumption of a point nucleus, which was
implicit in writing (2), is not entirely correct.
Nuclei of different mass will have different finite
volumes and shapes, and this will influence the energy
level separations. This effect is small since nuclear
dimensions are of the order of 0.00001a0, and will be
discussed in'more detail later.
In (2), we used the reduced mass rather than
the electronic mass. This is an approximation that
comes from the transformation to the center of mass
(CM) coordinate system. It is usually ignored in
textbooks, although a few, such as Shore and Ilenzel /1/
and Bethe and Salpeter /2/ do treat it. The
implications for molecular calculations have recently
been discussed by Fr8man /3/. le will sketch the
results for an atomic system here.
-7-
If P is the momentum of the nucleus and -i is
the momentum of
the i-th electron,
then, by
conservation of momentum, in the CM system, we have
P t
pt
/
=e k 2 pe T, is-
The kinetic energy, T, is
~2pi2
Making the usual
transformations from classical to
quantum mechanics,
p -i1f
we obtain
T7
PE
- i ) .-z
AiD
* .
+ Z, l.
(14)
a~vr1_
S, i '-~ D 2
- 8 -
T -= .2 (15)
The second tern is generally omitted since it is of
order p/i/, which is small, and also, it is a two-electron
operator, which is more difficult to calculate than the
first term. The correction due to the second term is
usually called the specific mass effect, or sometimes,
the mass polarization effect, in contrast to (11),
which is usually called the isotope shift. In the
relativistic case, it is not possible to perform a
rigorous separation of internal and translational
motion by going to a CM system. This sort of
difficulty is quite general, and consequently, one has
to be extremely careful about attaching physical
significance to correction terms derived from non-
relativistic theory. This difficulty is compounded by
the fact that experimental data is frequently
interpreted on the basis of these approximate
corrections. A simple example of this is the spin-
orbit effect, which is really a two-electron effect,
and yet experimental data is usually reported in terms
of a separation parameter based upon the one-electron
effect found in the one-particle Pauli extension to the
Schrodinger equation. In view of our non-relativistic
approach in this work, it may be better to view
- 9 -
calculated small terms as empirical parameters; they
contribute to level shifts and splitting, but they in
general represent only part of the physics.
Nevertheless, such quantities are often useful in
intercomparison of wavefunctions and can help to point
out deficiencies in the description of various parts of
a given wavefunction.
The reader who is interested -in the
relativistic CM problems is referred to Bethe and
Salpeter (Ref. 2, section 42) and elsewhere /4,5/.
Finally, we remark that for one-electron systems,
Garcia and Mack /6/ have given theoretically calculated
tables of energy levels containing all known
corrections to the relativistic Dirac Hamiltonian.
CHAPTER 1
HARTREE-FOCK AND BEYOND
1.1 The Hartree-Fock Method
Let us assume that the total wavefunction for
an N particle system is built up from a set of N
orthonormal one-particle spin orbitals, q5 (x), where x
= (, O ) is a combined space and spin coordinate. The
Pauli exclusion principle requires that the
wavefunction be antisymmetric, so we write it as an
antisymnetrized product of spin orbitals, i.e. a Slater
determinant:
_L L L, ) X... ?j (1)
where a (- P (2)
is the antisymmetrizer, P is a permutation of
electrons, and p is the parity of the permutation.
Application of the variation principle to the
Schridinger equation then leads to the well-known
- 10 -
- 11 -
Hartree-Fock equations (see for example, Ref. 7,
Chapter 16) which may be written
S,. - ,.) (3)
Jy
where it has been assumed that
-0. 64 ^[(4)
The -*.are Lagrange multipliers, and by a unitary
transformation on (1), which leaves the wavefunction
invariant, one can diagonalize d and obtain
A. (5)
where i= -
and the #i in (5) are now linear combinations of those
in (3). The one-particle effective Hamiltonian, h, is
given by
; --Z -/ 2 Z
r> (6)
)j Sd 5^( (2) f-,)D V^U)^
- 12 -
where P12 permutes the coordinates of electrons 1 and
2, and g12 = /r12. The Hartree-Fock (HF) equations have
received extensive study in the literature, and we will
refer the reader to the extensive bibliographies in
Slater's books /7,8,9/.
To simplify writing, we will let
f-- (7)
and
(.1, -,) (8)
so that
S(9)
and
-i < ilRili> (10)
j=/
The total energy is given by
- 13 -
(11)
21
(#
i.C212Ij>
?Note that = 0, so that in (11),
w
Substituting (9) into (11), we
find
-x..
j Z*
*
/1212 IJ>
+2
<7
11212/j>
- 2, < /If.i>
> J
so that the total
energy is not given by the sum of the
orbital energies.
Let us see what happens if we remove an
electron from orbital m without changing any of the
remaining orbitals. We find
(Al-I)
(13)
6 <2mI 12/2II M>
I.
2:t
-z
(12)
~ = c 4' iHl 4'> ~ 4J 1 41~
-! 57
- 2
iy
- 14 -
- Z / (Z> / /<
i-j u Z Cm
-Z
(A) < -l- ,z, 2 <)inm ,Ivm >
tf770m
- Z < YYj/ /-/,mj/>
(X4 j
= 61
(14)
Equation (14), known as Koopnans' theorem /10/, says
that the ionization energy is approximated by the
negative of the one-electron energy of the removed
electron. If we allowed the orbitals to relax, then
this would no longer be true. Equation (13), which is
not generally remarked upon, implies that the orbital
energies are decreased by the term
< ,'J2,2 / I'M >
which is always a positive quantity.
follows simply from the the fact that
(i(Pj)1 .- ( & / 1)
Positiveness
= i )
I
- Z -
Z' _L f-P,.
- 15 -
so (1 P12)/2 is a projection operator, and is therefore
positive, and since g12 is also positive, .,is positive,
and
< -0Z im -- ILM ?
> 0
If we assume that the orbital energies are ordered
6, 62 63 -- !S-N
and then we successively ionize electrons N, N-1, N-2,
...,1, without allowing orbital relaxation, we obtain
S= E. -1 6
IA/- E N-Z -
-< +A < 1 2 A-1I >
- EZ-2
(v-a)
c/,-/
hr-j
11-.2
- 16 -
N-2
A/-2
S
* < N-2, A/ I2,, /A/-2, n/>
(A/-k)
A= /
(15)
() -
A/-kC
=k-
'Z= < J 1A N ?,0
and finally,
'N- k
-E_ -E
- 17 -
- 7 6 (A )
(k)
~2
2>*
J
This is of
ionization
energy.
course expected since the sum
energies must be the same
of all the
as the total
Similarly, we can study the effect of adding
an electron to the system in spin orbital k = N + 1.
Assuming again that the N occupied orbitals do not
change in the process, except that an orbital N + 1 is
added,
(A//)
a'
-' < 1 i,/z-> i
(17)
(4)
+
kt
Z Ik
kZi
(16)
AI'/
AN J
- 18 -
2--
1
r c^
61
I,/
- ,.,<.> 2 y'
y J J1^
A/tI
1/
-z: <)tv 'j//AiLf
(18)
However, in order to determine this last quantity,
which is the electron affinity, we need to solve the
additional equation
(19)
/4/ 41t/
where h is determined by the original l orbitals only.
The third quantity of interest in addition to
the ionization potential and the electron affinity, is
the excitation energy of an electron in orbital rn going
to an unoccupied orbital, say U + 1. A similar
" A.
<0/
- 19 -
analysis as before gives
e I/ p > i l
er' "- ,,
(21)
J- t < 1vp/ 1 2 z V/ A l
and again, we must solve (19).
We have emphasized these three quantities --
the ionization potential, the electron affinity, and
the excitation energy, because they are often cited as
the major use of HF theory. Ideally, these should all
be computed by making accurate total energy
calculations on the two states involved and then taking
the difference of the two total energies. The
assumption of no orbital relaxation in the final state
enables one to approximate these quantities by a single
calculation (for the ionization energy), or a single
total energy calculation and one single-particle
calculation, (19), which involves less computational
effort.
The orbitals, however, do relax
processes, and the effect may be significant.
the difference of total energies approach has
practical drawback of accuracy loss due
subtraction of large numbers to obtain
in these
Because
the very
to the
a small
(20)
- 20 -
difference, schemes have been introduced to enable the
energy difference to be expressed in terms of an
orbital energy and a correction term (see for example,
Ref. 13 and references cited therein). Mention should
also be made here of Slater's statistical exchange
approximation /11/ to the HF equations because simple
equations can be derived for excitation energies,
ionization potentials, and electron affinities, which
do not require total energy differences. Instead, one
makes a single calculation in the so-called transition
state /12/ which is a state halfway between the initial
and final states. The method also has the advantage
that it is computationally much easier than HF, and
work is in progress in extending the calculations to
molecular systems.
The HF approximation and related empirical
and semi-empirical one-electron theories have carried
chemistry and solid-state physics a long way. In
particular, the mechanistic approach which has been so
useful in synthetic organic chemistry is based
essentially upon the one-electron picture, and
certainly chemical concepts of molecular geometry and
bonding are modelled upon the localization of electrons
into regions of space called orbitals. For atoms and
molecules, we can go beyond these ideas, but for the
- 21 -
solid state, they are still very much a computational
necessity.
However, from a purely theoretical
standpoint, the HF approximation is unsatisfactory in
many ways. Consequently, there have been suggested
many different schemes for improving it. Before we go
into any of these and show how this leads into the
subject of this thesis, we will give the four major
restrictions that are made upon the orbitals in what is
(naturally) called the restricted Hartree-Fock (RHF)
method.
a) The space and spin dependence is
separated.
b) The radial and angular dependence is
separated.
(see Appendix III for definitions of
the spherical harmonics, Ylm).
c) R(r) is taken to be independent of ml;
this is not a restriction for states
with L = 0, if L-S coupling holds.
- 22 -
d) R(r) is taken to be independent of ms;
this is not a restriction for states
with S = 0, if L-S coupling holds.
All of these are made for computational and
conceptual simplicity. The first and fourth permit the
determinant, for certain special cases, to be an
eigenfunction of S2, and the second and third, of L2.
The assumption in going from (3) to (5) was that a
unitary transformation could be applied to the
determinant (1). Because of this, a) introduces a
further restriction -- double occupancy of the
orbitals; that is, each spatial function occurs twice
in the determinant, once with c( spin, and once with /spin.
This restricts the method to closed-shell systems.
Since the latter are of less interest chemically than
open-shell systems, various methods have been worked
out to deal with these. The most important are
discussed in the next section.
- 23 -
1.2 Extensions to Hartree-Fock Theory
Relaxation of Restrictions
In the general open-shell case, with the
restrictions noted in the preceding section, the off-
diagonal Lagrange multipliers cannot be eliminated, and
the total energy expression becomes considerably more
complicated. We will refer the interested reader to a
review of open-shell methods by Berthier /14/.
If we reexamine (9) and carry out the spin
integration, we find that the effective Hamiltonian,
h, is given by
Oz (- R2 Z
.z fJ = (^.( + 27 C )f(1ij*(2)
d J
~ O(D ( 1)
(22)
Because of the delta-function in the second term, which
is the exchange tern, only those orbitals having the
same spin as orbital i will cone into the operator. If
there are differing numbers of d and 3 spins, i.e. S
f 0, then there will be different effective operators
for electrons in the same (n,l)-shell if they have
different spins. lWe have already noted the double-
- 24 -
occupancy restriction, and it was just this observation
that led to the introduction of what is called spin-
polarized HF (SPHF), or different orbitals for
different spins (DODS), in which orbitals from the same
(n,l)-shell are allowed to have different radial parts
if they have different spins. This is a relaxation of
restriction d) in the last section. Similarly, for
states with L 1 0, one finds different effective
Hamiltonians for orbitals of different m1 values. Relaxing
restriction c) then gives orbital-polarized HF (OPHF).
In principle, there is no reason to require
that orbitals have a fixed value of 1. A partial
relaxation of b) and complete removal of c) and d)
leads to the unrestricted HF (UHF) method. A UHF
orbital will have the form
U 4 Y i 4, (23)
The term unrestricted is not particularly good, because
the orbitals are still restricted to a single ms and a
single ml value, so that they are cigenfunctions of sz
and 1,, and to definite parity (which is why only odd
or only even values of 1 occur in the expansion (23)).
The SPHF/DODS and OPHF methods are relatively easy to
implement starting from an existing RHF program, but
- 25 -
the UHF method is much more difficult. As is usually
the case, the more we generalize, the harder the work
becomes.
The Symmetry Dilemma
We are still far from removing the HF
inadequacies. One of the major difficulties is the
famous "symmetry dilemma" /15/. This has to do with
the fact that the eigenfunctions of the Hamiltonian
operator should also be eigenfunctions of the normal
constants of motion, that is, 'the normal operators
which commute with the Hamiltonian. This is required
because the wavefunctions and energy levels of a system
are classified by their symmetry properties spin and
angular momentum values, parity, rotational or point-
group symmetry, translational symmetry, time-reversal
invariance, and so on. The four restrictions noted
previously are all chosen to make determination of
these symmetry properties simple. For example,
restrictions c) and d) guarantee that the wavefunction
is an eigenfunction of Sz and Lz. In fact, since most
N-particle symmetry operators can be written as a
simple sum or product of single-particle symmetry
- 26 -
operators, the temptation has always been strong to
restrict the one-particle functions to be
eigenfunctions of the single-particle symmetry
operators. This has had unfortunate consequences,
particularly in chemistry, where one often hears
discussions of the importance of d or f orbitals in
molecular bonding or of the importance of hybridization
in the determination of molecular geometry /16/.
It was thought for a long time that if a
variation were carried out on a wavefunction that the
end result would be a function corresponding to the
lowest energy, and therefore would automatically have
the correct symmetry properties. This is not
necessarily so, and it has been .shown by L8wdin /17/
that if a wavefunction has mixed symmetry, then it has
at least one symmetry component which will have an
energy at least as low as the mixed symmetry function.
By the introduction of symmetry production operators,
L8wdin /18/ showed that one could select a specific
symmetry, component of the wavefunction. The
wavefunction for a particular symmetry k is then
written in the form
1' =(9k D (24)
- 27 -
where D is the Slater determinant and the projection
operators Ok fulfill the usual relations:
(25)
(25)
0 Ok
0-^ C9 /-
That is, they are idempotent, Hermitian, bounded by the
zero and unit operators, and form a resolution of the
identity. The energy then takes the form
E /- < /IH/> //O/Z>
S< O)lHl/O >/< < /OIOD>
/
(26)
This may be regarded either as a modified expectation
value of the operator OH = OHO = HO, or as the
expectation value of a wavefunction that is a sum of
- 28 -
single determinants, since this is the form that OkD has
in (24). L8wdin /17/ then carried out a variation of
the individual orbitals in the determinant and showed
that one could still obtain HF-like equations, which he
called extended Hartree-Fock (EHF); these differed from
the HF equations only in having a more complicated
effective one-electron operator. These equations are
very much more difficult to solve, and it is only
recently /21,22/ that attempts have been made to put
them into a computationally feasible form. Instead, a
number of simpler approaches have been tried.
The first was simply to take a RHF or UHF
determinant, project it, and obtain the total energy.
This is not difficult to carry out, and the energy
improvements found are quite small for the state of
lowest energy. This is to be expected since the
unprojected variation does approach a minimum or a
saddlepoint, although it need not be the absolute
minimum. The name applied to this method starting from
the UHF determinant is projected unrestricted HF
(PUHF). This approach may be criticized because the
variation is carried out before the projection. It was
emphasized /23/ that the variation should be done after
projection, and this has recently led to the spin
projection of a Slater determinant followed by orbital
variation, the spin-extended HF (SEHF) method. Kaldor
- 29 -
/24-27/ and Sando and Harriman /28/ perform direct
variations of the total energy, abandoning the use of
the one-electron effective Hamiltonian for
determination of the orbitals. Goddard and Ladner /29-
33/ solve the one-electron equations and determine the
SEHF orbitals. The two approaches are equivalent for
the total energy calculation since the final orbitals
in each case are related by a linear transformation,
which leaves the determinant invariant (except possibly
for a constant factor which vanishes in the
normalization). Both are difficult because a large
number of non-linear parameters must be determined.
Solution of the one-electron equations is particularly
difficult because of the very complicated effective
operator, so their neglect in favor of a total energy
minimization is understandable. If we abandon the one-
electron equations in the computations, we lose the
simple pictures of ionization and excitation energies
and electron affinities, concepts which can be
generalized from the simple HF results given earlier to
the case of these more complicated one-electron
equations. In one sense, this is regrettable, but in
another, the fundamental limitations of the independent
particle model (IP I) -- that N orbitals describe the
motion of N electrons -- presently do not justify the
- 30 -
cost and effort of obtaining the results. There are
better methods available.
On Spanning the Angular Momentum Snaces
Aside from the inadequacies of the IPM, one
of the major problems remaining in the projected
Hartree-Fock methods is that the projected determinant,
even though it is a pure symmetry component, generally
does not span the complete spin or angular momentum
space. This has recently been emphasized in a note by
the author /34/, but since it i.s directly concerned
with the subject of this dissertation, we will repeat
some of it here.
For- N electrons and total spin S, the number
of linearly independent spin functions is given by
/35,36/
f = (2S + 1) N! < 2N (27)
(N/2 + S + 1)! (N/2 S)!
As N increases, f becomes very large. For example,
with N = 10 and S = 1, f = 90. The previously
discussed projected Hartree-Fock methods obtain only
one of these. The choice of which spin function to use
- 31 -
is not immaterial, even if the Hamiltonian is spin-
free. For example, for lithium 2 2S, E(RHF) = -7.432725
H, while E(SEHF) = -7.432813 H, an improvement of only
0.000087 H. There are two independent spin functions
for this system, one of which SEHF uses. If the other
one is used, one obtains E = -7.447560, an improvement
of 0.014835 H, nearly one hundred seventy times the
SEHF improvement over RHF! The reason for this
difference is clear. The SEHF (Goddard's GF)
wavefunction for doublet lithium is
A ( )s)l s (c a^ -p (28)
and consequently, the first spin factor forces the
inner shell to have unpaired spins which is
energetically unfavorable. Instead, it would be
preferable to have paired spins in the closed shells.
This scheme is known as maximally-paired HF (MPHF,
Goddard's Gl). For doublet lithium, the MPHF
wavefunction is
4 (s) (S/))(2 )s (~gca ,'J)/vi (29)
Although the energy of this function is better, it does
not correctly describe the hyperfine structure. The
MPHF function gives zero contribution to the Fermi
- 32 -
contact term for closed shells, thus predicting a zero
contact term for closed shell systems such as He, Be,
and Ne, in contradiction to experiment.
A way of taking into account all the spin
functions, but still remaining within the IPM, was
first suggested by Kotani /37,38/ in 1951 and has
recently been applied by Lunell /72/, Ladner and
Goddard /33/ and Kaldor and Harris /39/. This is to
use as a wavefunction a product of N spatial orbitals
and a linear combination of all the f independent spin
functions 0k for the value of S desired:
3-
4S 0 (1) 0 (N) (30)
Ladner and Goddard call this method spin-optimized GI
(SOGI); Kaldor and Harris name it spin-optimized self-
consistent field (SOSCF). The results that have been
obtained with this method will be discussed later.
There is an alternative way to span the spin
space, however. We recall restriction a), that the
spin orbital be represented by a product of a spatial
orbital and a spin function d or /3. This obviously can
generate only one spin product in the determinant for
a projection operator to act upon. However, we have a
choice of one of two spin functions for each of the N
spin orbitals, so we could generate 2 different spin
- 33 -
products, which completely span the N-electron spin
space, since there are no other possible products. If
we carried out a spin projection on this set, we would
obtain all f linearly independent functions. We then
recall that the solution of the relativistic Dirac
equation for a single particle is a single-particle
wavefunction, or orbital, with four components. Two of
these correspond to electrons, and two to positrons.
Each pair has one component with a-spin, and one with
/-spin. Since we are dealing with non-relativistic theory,
we can ignore the positron components, and write a
general spin orbital (GSO) for an electron as
40 (") = # (): (c F;)/ (31)
where 0 and are independent spatial functions. This
possibility seems to have been first noticed by L8wdin
/40/.
A projected determinant of these orbitals is
a more general function than the SOSCF one (30),
primarily because there is more spatial flexibility.
Since there has been some disagreement about this in
the literature /34,11/, we shall sketch the proof.
If we restrict (31) to the case
- 34 -
OK k X, 5 (32)
where Ak is a constant, the projected GSO (PGSO) function
takes the form of (30). Since there are N A's to be
determined, the SOSCF function is a special case of the
PGSO function, provided f does not exceed N. The first
case where this happens is for N = 6, S = 1, when f =
9. For f > N, the condition (32) is too restrictive.
The PGSO function consists of a sum of orbital
component products multiplied by spin function
products, which may be written in the form
Y ^ /(1u ... (33)
where ;K will in general be a sum of orbital component
products, and @k is the spin function used in (30). At
least g different orbital components must appear in the
PGSO function, where N g 2N. Now, suppose that for
computational purposes, we expand the orbitals in (30)
and (32) in the same M-function basis. The SOSCF
orbitals will then be functions of M variables each,
and the GSO, functions of 2M variables. Since we can
perform a linear transformation on the orbitals in a
single determinant without changing its value, except
for a constant multiplicative factor which vanishes in
- 35 -
the normalization, we can without loss of generality
orthogonalize the GSO orbitals, but not the SOSCF
orbitals. Note however, that this does not imply that
the sets
are individually orthonornal. The number
independent variables for the SOSCF function is
= NM + f 1
and for the PGSO function,
nPGSO = gM N
since the 'orthonormalization has removed one
independent variable from each orbital in the PGSO
function. We can always satisfy npGSO nSOSCF provided
we choose H sufficiently large that
nPGSO nSOSCF
= (g N)H N f + 1 0
M (f + I 1)/(g N)
ILI
nSOSCF
or,
- 36 -
The case with g = N corresponds to that state of
highest multiplicity, for which f = 1, independent of
N; in this case, the PGSO and SOSCF functions are
equivalent. In this case also, since f = 1, the SOSCF,
EHF, and PGSO methods are equivalent. Except for one-
and two-electron systems, these states of highest
multiplicity lie high in the continuum and only
recently have become of interest. For most chemical
applications, they are of no concern.
The PGSO and SOSCF methods are substantially
more difficult to carry out than the other methods that
we have discussed. PGSO is essentially an EHF method
with general spin orbitals rather than pure spin
orbitals. In both cases, one has to deal with non-
orthogonal orbitals, and this makes the integral
computation considerably more time-consuming.
Computations using the PGSO method have been carried
out on helium and its isoelectronic sequence by
Lefebvre and Smeyers /42/ and Lunell /43/. Lefebvre
and Smeyers however, after expanding and projecting the
GSO determinant, orthonormalize the components, which
does not leave the original determinant invariant.
Consequently, their wavefunction is not a true PGSO
wavefunction. Lunell is therefore the first to
actually carry out the solution of the EHF one-electron
equations for GSO. For all but the smallest bases, he
- 37 -
encountered convergence difficulties in their solution
and consequently abandoned them in favor of a total
energy variation for his larger basis sets. He also
made preliminary calculations on lithium using the
restriction (32) which makes the PGSO function
equivalent to the SOSCF function for this system. At
his suggestion, the author began work on the general
case for three-electron systems work that forms the
main results reported in this dissertation. These will
be discussed in Chapters 3 and 4.
Thus far, we have emphasized the spin
degeneracy problem in the projected Hartree-Fock
methods. There is also, however, an orbital angular
momentum degeneracy problem which has not yet been
satisfactorily treated in the PHF approaches. This is
much more difficult, because unlike the spin problem,
where we had a finite space of f 2 2N spin functions,
we have an infinite space to deal with. In the spin
case, the result of applying a spin projection operator
to an arbitrary product of spin functions is well-known
/36,91/. The result is a sum of spin products with
fixed coefficients, called the Sanibel coefficients,
which can be obtained from closed expressions without
having to carry out the rather tedious operation of
applying the spin projector directly. For the case of
a product of an arbitrary number of spherical
- 38 -
harmonics, closed forms for the coefficients are not
known /45/. Of course, one can still work out the
result using the projection operator. One approach
that has been used to avoid this is to set up a matrix
2 2
eigenvalue problem for the operators L L, S and S
and solve for the eigenvectors and eigenvalues /46/.
!e have ignored this problem in this work for
two reasons. One, it introduces many additional
complications in a method that already is perhaps too
complicated for practical everyday use with our current
computer capabilities. Two, the problem can be handled
more readily in the configuration interaction method
discussed in the next section. The angular problem is
important, however. For the ground state of lithium,
the RHF energy is -7.432 H, the best PGSO energy, using
orbitals restricted to 1 = 0, is -7.448 H, and the
exact non-relativistic energy is -7.1478 H. The correct
treatment of the spin space has yielded only about 350
of the correlation energy, the difference between the
RHF and exact non-relativistic energies.
- 39 -
1.3 Other Methods
We shall ignore all the empirical and
semiempirical theories. These have of course a great
deal of use in systematization and preservation of
simple concepts, but their reliability for any
particular system is generally questionable. Rather,
we are more interested in the ab initio methods -
methods which could be trustworthy, if we could carry
them out to a sufficient degree of accuracy.
The Configuration Interaction Method
The first, and most important of these, is
the configuration interaction (CI) method. It was
shown by LB.wdin /18/ that any antisymmetric
w.avefunction may be written in the form of a linear
combination of determinants formed from a complete set
of one-electron functions; the CI idea itself goes back
to Hylleraas /17/ in 1928. The fact that all the
complete sets of one-particle functions of interest in
quantum chemistry are infinite means that the CI
wavefunction is an infinite expansion, since a basis of
MI functions for an (l-electron system can generate (1)
F
- 40 -
determinants. For a fixed basis, variation of the
total energy with respect to the configuration (i.e.
determinant or projected determinant) coefficients
leads to a secular problem of the same dimension as the
length of the CI expansion. The variation principle
and the separation theorem guarantee that the
eigenvalues of the secular equation will be in order
upper bounds to the exact eigenvalues of the
Hamiltonian used. This fact is extremely important,
for it means that we can treat excited states,
something that cannot be done easily with the HF
methods (unless the state is the lowest one of its
particular symmetry). In addition, the CI function can
be optimized for each state, while -the RHF function can
be optimized only for the ground state. The virtual
(i.e. unoccupied) orbitals that come out of the HF
problem can be substituted into the HF determinant to
obtain an approximation to an excited state; the
resulting determinant by Brillouin's theorem /48/ will
be orthogonal and non-interacting with respect to the
ground state determinant, and therefore its energy will
be an upper bound to some excited state energy, but not
necessarily the one desired.
With a sufficiently large basis for the CI,
one can span a much larger section of the spin and
angular momentum spaces, and the determinants can all
- 41 -
be projected. Consequently, these difficulties of the
PHF methods can be minimized. There are two general
problems in CI calculations however. The first is that
the number of configurations for even a moderate basis
is usually much more than can be handled, so that one
has to make a selection of configurations. This aspect
of the method is currently more of an art than a
science. The second problem is the selection of the
basis. If the functions are fixed, and if one can
include all possible configurations, then any linear
transformation on the basis only alters the
configuration coefficients; solution of the secular
problem yields the same energies. For a truncated set
of configurations, there exists an optimal linear
transformation of the basis which will give the lowest
energy for a particular state, although finding it is
non-trivial. This is discussed further in Chapter 2.
An alternate approach would be to use a sufficiently
small basis so that all configurations could be
handled, and then to vary the orbital basis. Lowdin's
EHF equations /18/ can then be used to obtain the
optimum orbitals. This has been implemented in a
somewhat restricted fashion, and is known as the
multiconfigurational self-consistent field method.
- 42 -
Interelectronic Coordinate Methods
A difficulty with the CI method is the slow
convergence of the expansion. This is partly due to
the difficulty in satisfying the cusp condition on r 12
that is, the proper behaviour of the wavefunction as r
12
--> 0. This is discussed further in Appendix IV. This
difficulty can be avoided if interelectronic
coordinates are introduced into the wavefunction
because the cusp conditions can be satisfied exactly.
The use of interelectronic coordinates was first
introduced and used by Hylleraas /49/ in 1929 for the
helium atom; wavefunctions containing interelectronic
coordinates are now generally called Hylleraas
functions. James and Coolidge /50-52/ in 1933 used a
thirteen-term Hylleraas function on the hydrogen
molecule. The helium calculations were carried further
by Kinoshita /53/ and then in a monumental piece of
work, the wavefunction was extended to up to 2300 terms
for the singlet and triplet S and P states of the
helium isoelectronic sequence by Pekeris and coworkers
/54-56,103,104/, with an accuracy of about 0.001 K,
10
about one part in 10 The method has been extended
to lithium by Burke /57/, Ohrn and rordling /106/,
Larsson /58/, and Perkins /59/, and to beryllium by
Szasz and Byrne /60/ and Gentner and Burke /61/. The
- 43 -
Ohrn and Nordling, and of Perkins, is notable in that
excited states have also been treated. These
calculations are currently the most accurate available
on two and three electron systems. The method has not
been carried out for atomic systems with more than four
electrons, or for molecules other than hydrogen,
primarily because of the difficulties in integral
evaluation. (See however, ref. 107 and references
therein.)
Recently, Sims and Hagstrom /62/ have shown
that a combined Hylleraas/CI wavefunction of the form
2J= Z cV k
K
where
.. G 0 g ,; ... ,l
is feasible for atomic calculations with any number of
electrons and have reported results for the beryllium
atom.
For atoms at least, the combined Hylleraas/CI
method seems to be the most promising, if it can
compete with ordinary CI in terms of computer time.
For small diatomic molecules, several Cl programs are
currently in use, written by Kouba and thrn, by Harris,
- 44 -
Michels and Schaefer, by Bender and Davidson, and by
Hagstrom and coworkers. The IBM research group in San
Jose, California is running a linear triatomic CI
program called ALCHEMY, and work is currently in
progress there on the production of a general
polyatomic CI program using the Gaussian orbital
integral package from their polyatomic RHF program,
IBMOL.
Calculations with such programs are currently
rather expensive; assuming that computational costs
continue to fall and computer facilities continue to
expand, these programs may be in general use in a
decade or two. In the meantime, we shall undoubtedly
continue to work on simpler methods to try to increase
our understanding of atoms, molecules, and solids.
Bethe-Goldstone Perturbation Theory
Finally, we should mention a method brought
over into quantum chemistry from nuclear theory,
principally by Brueckner, Nesbet, and Kelly /97/, which
they call Bethe-Goldstone (BC) perturbation theory. A
great deal of work has come out lately on atomic
calculations of excitation energies and hyperfine
- 45 -
structure. The method relies heavily on the pseudo-
physical (i.e. intuitive) pictures that one has in the
Feynman diagrams which represent the terms in a
perturbation series, based usually upon a Hartree-Fock
starting point. The method, while giving good results
by careful workers, does not seem to provide the simple
conceptual pictures of electron densities which are
desirable in chemistry. Further, the wavefunction, if
indeed one exists, is not exhibited, making comparison
with other methods difficult. In the author's opinion,
one of the main functions of quantum chemistry is to
provide chemists with a language that can be used to
systematize and if possible, quantify, chemical ideas.
We feel that the BG methods, however close their
results agree with experiment, do not succeed in this
respect.
With these closing remarks, we shall now pass
over to a discussion of density matrix theory.
CHAPTER 2
REDUCED DENSITY MATRICES
2.1 Introduction to the Density Matrix Literature
The concept of a density matrix goes back to
von Neumann and Dirac in the late 1920's, but the more
useful concept of the reduced density matrix was
introduced by Husimi /63/ in 1940 for use in the
Hartree-Fock problem. Husimi's work was significantly
extended to arbitrary wavefunctions by LBwdin
/18,19,20/ in 1955, who should be credited with
demonstrating the quantum mechanical utility and
significance of reduced density matrices. For reviews
of the subject, see IlMcWeeny /64/, Coleman /65/, Ando
/66/, Bingel and Kutzelnigg /141/, and the proceedings
of two recent conferences on density matrix theory
/67,68/. In her Ph.D. dissertation, Ruskai /69/ has
given an excellent survey of all the known theorems in
reduced density matrix theory, with a bibliography of
the principal theoretical papers.
- 46 -
- 47 -
2.2 Construction of the Reduced Density Matrix
Definition of the Reduced Density Matrix
We define the p-th order reduced transition
density matrix as
J' ^ 1. -..,A) w I :'.'. p' ,,;
(34)
The term density function would be preferable,
particularly since the term density matrix will also be
applied to a quantity with discrete indices introduced
later. However, the dual use of the word matrix is
well-established, so we will continue to use it. The
primed variables always come from the complex conjugate
of the wavefunction; also, when K = L, we shall drop
the subscripts K and L and the word "transition". It
is often convenient to economize the notation by
letting x = (1,2, ... ,p) and y = (p+l,p+2, ... ,N) so
that (34) becomes
[I' IT 0-' rJ -
The factor ( ) is chosen so that the trace operation,
p
r --7p)
gives the number of groups
I and J are orthogonal,
transition p-matrix is zero.
(34) is 18wdin's normalizat
use are Coleman's,
of p particles. If states
then the trace of the
The normalization used in
:ion. Two others in common
D p) (A) /-7(P)
and McWeeny's,
For our purposes, the L8wdin normalization is most
convenient. The wavefunction in (34) may be symmetric,
for bosons, or antisymmetric, for fermions. Many of
the theorems can be developed for both cases. However,
for most quantum chemical applications, we are
interested only in fermion systems, so in the
following, antisymmetric wavefunctions are to be
- 48 -
Z frI^wa/
(35)
- 49 -
understood.
conventions
We shall also use the usual notation
We obviously have
" .- p-)l .
i- (2P' 112'"~1
f7 -p -
Pi I
and the p-th order reduced density matrix, or simply,
p-natrix, is antisymmetric with respect to interchange
of any two indices on the same side of the vertical
bar:
(37)
where P is any two-particle permutation operator such
that P 0= -z. We also have
(38)
r I ') -
XT J
When K = L, the p-matrix is therefore Hermitian, and
in this case, it is also positive, since for arbitrary
functions f(x),
(36)
rOp) (xl P') =- F ( Ix)
FI IIT XI
(P)
- 50 -
f&(P) -/'~(x/x) j ) -
J rJ*' c/i-
(39)
The diagonal elements have the interpretation that
fl(x 1)cA',.. C/Y
is the probability of finding any p particles in the
volume element dx dx2...dx regardless of the positions
of the remaining particles.
Suppose that we have an arbitrary operator,
Q, which is symmetric in the particle indices, so that
we may write
ZQL 3 Z'Q3 /
'fXJ2,t J
(40)
o, I~e :L
The primes in the first set of summations indicate that
we omit terms with two or more equal indices. Then
I < > I/CI> <4 >_
< a >,j
QZ' :
=%
- 51 -
ffz -A 4ch J Y.2
(41)
- .
where we follow the convention that the operators act
upon the unprimed variables only, and after this the
primes are removed and the integration carried out.
The Reduced Density Matrix for a Cl ,lavefunction
Following LBwdin's original work /18,19,20/,
we introduce a CI expansion for the wavefunction of
state I:
, i
(42)
The index K represents an ordered configuration defined
by the one-particle function indices (kl, k2, ...,kN )
where k1 < k2 < ... < k ; that is,
SQ, r, (121r~~ dr, dr,
- fQ,'(llw, ,
613Z- !QJ)^ ( v 4 42
-- Z, c,<
v S1
- 52 -
-A^#b-- (A"
'1? a'e(2)
The one-particle
orthonormal. We i
D])(CVIL;T) J
basis is not assumed to be
introduce
I4 T -
V ST L
(43)
where
(011)VL)
Y
(44)
For notational convenience, we introduce "fat" symbols,
g= (k1, k2, .. ,k ) and = (k < k2 < ... < k ). We
- 2 p 1 2 p
also need
)( V, r; 1, ,; /p IL,; .. ,...P)- T;CKI; L/,,PJ; ) (45)
the cofactor of the determinant (43), which is formed
by deleting rows containing functions kl, k2, ...,k and
columns containing functions 11 ...,1 and evaluating
1 2 p
r
d- f S IT, KLI
c brlp~~~
- 53 -
the resulting determinant and applying an appropriate
parity factor. The parity factor is -1 if the sum of
the cardinal positions of the function indices in the
original determinants is odd, and +1 if the sum is
even. With these definitions, we obtain
JW (i\.I p ) f- 21 1 2 (LI CL ** LI))
K L
The notation 4 above a summation means that all
configurations containing the orbitals in set A are to
be included. We can simplify (46) by introducing a
discrete p-matrix,
E < (< ms i l l1 U-
[^ LI C^( M LI) ^ LJ,/J
(47)
The presence of a tilde under a matrix signifies that
it is indexed by the non-ordered sets k, f. We drop
the tilde wthen we mean the part indexed by ordered sets
of indices, b and We then have
- 54 -
(?) 7 X (.. (') ('T
,K ) ~(48)
The peculiar reversed form of the indices I and J Is
simply to make the matrix form of this equation
simpler. The discrete transition p-matrix satisfies
r ---- r (49)
Pr k(j/i ) (50
where P permutes any two indices in a set. That is, it
is antisymmetric with respect to interchange of any two
indices on the same side of the bar, and is Hermitian
if I = J. The summations in (48) are over all values
of the indices. With a little study, one can use (50)
to reduce (48) to
_L 3 (i... 51)
ft p) (51)
where the summations are now over ordered sets of
indices. This is extremely important for computational
- 55 -
use, since we will only need a small part of the full
p-matrix in (48). In fact, for M orbitals, the full p-
matrix in (47) has dimension M ; the part with ordered
indices has dimension ( ) which for large M goes as
p
rP /p!.
Thus far, we have essentially followed L&wdin
/18/, except that we have used the transition matrix
throughout and introduced Slater determinants in (51).
At this point, LBwdin specialized to orthonormal
orbitals (Ref. 18, eq. 59); we shall avoid this
restriction.
Now we would like to study the properties of
the discrete p-matrix. Using (44) and (48), for I = J,
.....,..
For orthonormal orbitals, S.. = .., and (52) reduces
Ij IJ
to
^ r(P- i(V Ps Fej(P)ij (53)
'S < '
- 56 -
.z EF )(k 1)
-LK
In the general non-orthogonal case,
re r i tten
(52) can
& rF"
where
f S = s ...
is the direct product of p overlap matrices. The
product matrix in (54) is not Hermitian, which is
rather unpleasant to deal with computationally. We
note that since the trace is invariant under
permutations of the product matrices that
Lt = C(fPIRYM ']- tb(li
(55)
so we define
(54)
PiI tr ~B~
- 57 -
f7T')
Y C^'ff"
P")
fR2 'TY2
and then in the general case, we will have
ir 61(/
-/ N
From now on, we shall refer to the primed Hermitian
matrix in (56) as the p-matrix, because it has the nice
properties that hold for the case of an orthonormal
basis.
To proceed, we need some theorems on direct
product matrices, which we state here, and prove in
Appendix VII.
Theorem:
Theorem:
Theorem:
(A x B)(C x D) = (AC x BD)
If AB = CD and EF = GH,
then (A x E)(P x F) = (C x G)(D x H)
(58)
(59)
If r is a rational number, then
(AI x A x x A )A r (Ar x Ar .. x ANr
1 2 ,iI 1 X .. X )
(60)
(56)
//rA'
(57)
- 58 -
provided that all the powers on the right exist.
t t t t
Theorem: (A1 x A2 x ... x A) = (A1 x A2 x ... x AN) (61)
From these easily follows the results
fl(LUX LX
SLJL U
and
_ [ S S ... J (63)
z. ( xLi.... )r )XX )
Since S
therefore
for the
so we can
> 0, all rational powers r exist, and we can
compute (56) from the eigenproblem solution
basis overlap matrix. Now (56) is Hermitian,
diagonalize it:
(64)
(c17u) A lfYJJ)
... < J) = ( x ...x -U)(AlXA -- xA) (62)
T'P''
= },uI
r"' /m-~2-Y2 6'"~`"
- 59
For I = J, (48) can be written
-S X .
- 0 / x j ... x '-
S# r.... x
,a-^Os-Yj['yf--'-16 ]
(65)
where
(66)
and
r 1
(67)
There is-an infinite number of ways of orthonormalizing
a basis. It is interesting that the particular one
(66) comes into this quite naturally. It is the well-
known symnetric orthonormal ization /70,71/ which has
the particular property that if the basis set is
symmetry adapted, the orthonormalized set is symmetry
adapted also to the same symmetry operations.
Fr7 (x/x) =
[ (f)X (U .. (,] [.(ti Y (2)... 'p
0-= :y'
t(r/'c = S ^^/^ S1">
- 60 -
Substituting (63) into (65), we can bring the expansion
of the p-matrix (48) to diagonal form:
r~iui=
(68)
~i (69a)
f I (69b)
The general form (69b) is again more convenient for
computation because we need deal only with ordered sets
of indices.
Practically speaking, the operators that one
is usually interested in contain at most two-particle
interactions, so according to (41), we need at most the
2-matrix. Let us therefore specialize (68) to the
cases p = 1 and p = 2. We obtain for p = 1,
) oL J~)il
--- \ ) < 4(n'\ (70)
?I [( 6',%- c a ] (5-6-...r)dI]
where
^= ^ L
#Vz LA
and i is obtained from
and for D = 2,
r1l) 21'2')
C de (6k, 6, (1L (11
J
*^^~ z IO
rz-Pr
.,n (J tz(,) A* (1
(72)
whe re
(,,(1,2) =
(73)
The orbitais defined by (71) are cal led the natural
spin orbitals (NSO); the functions defined by (73) are
called the natural spin geminals (NSG). The p-matrix
- 61 -
(71)
I -3
7- 2j Jez^) m.
,2.
- 62 -
eigenvalues are called occupation numbers, and the
eigenvectors, natural p-states.
Obviously, had we used the orthonormal basis
(66) to begin with, the primed p-matrix in (56) would
have been obtained directly from (47) for I = J. For
this case, for the diagonal elements we find
so that
O -a _r i o f ( ) 1 (74 )
The diagonal expansion of the p-matrix in
terms of the occupation numbers and natural p-states is
particularly convenient for the computation of
expectation values. We obtain from (70) and (72)
Z < ,I >, X > (75)
K
and
- 63 -
ce/>- L Zr
; 2 A,-1o <;/&^ ^,,Z> (76)
The NSO and NSG have the physical significance that
they are the set of functions for which expectation
values are strictly additive. The occupation number
factor in (75) and (76) is computationally important
because it means that the sum may usually be truncated
after the larger occupation numbers.
The discussion of the eigenproblem has thus
far been centered on the non-trans'ition p-matrix case.
To the author's knowledge, no work has been published
on the transition matrix eigenproblem. In fact, Bingel
and Kutzelnigg (See ref. 67) seem to be the only
authors who have carried out derivations in terms of
the more general transition p-matrix.
As we noted earlier, according to (49), the
transition matrix is not Hermitian. It is well-known
/73/ that an arbitrary matrix may be brought to triangular
form T by a similarity transformation with a non-singular
matrix :
W TW= T (77)
The eigenvalues of Fare the diagonal elements of A.
The eigenvectors are found as follows.
J r WT Vh'
t 1 (78)
This is a triangular set of equations for the vector
JV from which V may be determined according to
VV') (79)
Since we do not have a diagonal form of the matrix ,
the simple results (75) and (76) do not hold. There is
consequently no value in determining eigenvalues and
elgenvectors of the transition p-matrix because doing
so does not simplify the situation. Instead, one
- 64 -
- 65 -
simply computes transition values directly from (41)
and (51):
<^Ef ^-
^>^' Qo frI4>( rl}~4
< Z
lur 1i
J,
ii
(80)
Thp RPnduired Density Matrix for a lon-CI
Wavefunction
If the wavefunction does not have the Cl form
(42), one can still obtain a reduced density matrix
front (34). Obtaining a representation of it in a
discrete basis is not difficult. The p-matrix may be
considered the kernel of an operator such that
f F (x/x, ,)/'xdx' /
(81)
for arbitrary functions
eigenfunction of then
f(x).
If f(x) is an
1 _X ~__~_Y___ ______ _I _~_ _~_ _
(Jz, P 0
<,.O IQ, O 0 k,( f ^
F' f() =
- 66 -
r"F ? ANW
(82)
Larsson and Smith /74/ have recently used this relation
to derive NlSO's of the 1-matrix of Larsson's Hylleraas-
type wavefunction for lithium. They introduce an M-
function basis and expand the NSO's in this basis:
(83)
Using this in (82) gives
(8 i)
which leads to the secular problem
<^/ >C Cr< ^ ~
(86)
where
(85)
? = 4 C
a~-~~
IC sck
- 67 -
- f e)f (y ) k 4& (87)
J4. ,
-<, jI >-
f07 (x) C&
(88)
The secular problem (86) may be solved by the methods
discussed in Appendix V.
The occupation numbers calculated this way
will be lower bounds to the exact occupation numbers.
The proof is not difficult. Let the set of exact
normalized ISO be with eigenval.ues A :
9',% K- A
(89)
Now let us order the exact and approximate solutions
according to
14z 7
>- 0
(90)
and then construct the operators
- 68 -
9x, 21 (91)
P E 2 IZ<>1 / (92)
k.-M+/
which are projection operators satisfying the usual
relations. From the theory of outer projections /75/,
we know that for an arbitrary projection operator 0,
and any operator I bounded from below, the eigenvalues
of O0O are upper bounds in order to those of /.
Now oSO= 6 (93)
and 01 O i (Z' /)
so () n^ -z v : (94)
so that 9j has eigenfunctions X and eigenvalues and
we have immediately
,i /i (95)
The approximate occupation numbers are therefore lower
bounds as stated. The sum of the approximate
N
occupation numbers approaches ( ) from below and provides
a convenient measure of the adequacy of the chosen
basis.
- 69 -
2.3 Properties of Density latrices
Cl Expansion Convergence
In Chapter I, we mentioned the convergence
problem in the CI method. In his original paper on
density matrices, L8wdin /18/ showed that the natural
spin orbitals are actually the orbitals which give the
most raid convergence of the Cl expansion, the HSO of
highest occupation number being the most important. Of
course, one needs to know the wavefunction to begin
with in order to obtain the p-matrices and the natural
p-states. However, if a truncation is made of the CI,
one can obtain NSO for this truncated function, put
these back into a new CI, perform a new truncation
based on the size of the NSO occupation numbers or
other criteria, obtain new NSO, and so on. This
natural spin orbital iteration technique has recently
become quite a popular tool in Cl calculations, but the
convergence of the scheme does depend on the quality of
the initial truncation.
- 70 -
Bounds on Occupation Mumbers
Since the p-matrix is positive and of finite
trace, its eigenvalues obviously satisfy
o ~ ()
(96)
Coleman (See Ando, ref. 66) showed that
(97)
p = 1 and p = 2 takes the form
OLX
(2)
O"i1L
L Ni ( )
1 CMn> !)
A/L ( >3)
(,'>
it can be shown that the upper bounds are never
attained except for p = 1 and p = N 1. Sasaki /76/
obtained better bounds than these, the first few of
which are
which for
(98)
(99)
- 71 -
;11
21. '2)
;1
'3
(100)
_ 1+ 3.[i V-3)]
where [x] is the integral part of x.
He also proved that the bound for p = 2 is the best
possible.
The Carlson-Keller-Schmidt Theorem
Carlson and Keller /77/ showed that the non-
zero eigenvalues of the p-matrix are identical to those
of the (N-p)-matrix, and if the number of non-zero
eigenvalues is finite, then these two matrices are
unitarily equivalent. In addition, if
and
p (Ac~l~f(jt 21 c (i ~ 1G~ry) i
(101)
1,11~4
. 1 <'/P > --
,/<.?, > (102)
- 72 -
then
",^-A (ovfSY )^ (104)
and
7 y) (A/ (105)
If and were derived from an antisymmetric wavefunction,
then the resolution (103) of the wavefunction is
automatically antisymmetric already. The
eigenfunctions of the p-matrix are called natural p-
states, and those of the (N-p)-natrix, co-natural p-
states. Coleman /65/ later pointed out that this
theorem had already been discovered more than fifty
years earlier by Schmidt /7'/. Schmidt's results, in
the terminology of density matrices, show that the
expansion (63) gives optimal convergence in the least
square sense to the wavefunction; this, coupled with
the fact that the natural n-states can always be
expanded in terms of the ISO /65/, leads to the CI
convergence theorem independently obtained by L8wdin
which we referred to earlier.
- 73
The Carlson-Keller-Schmidt theorem is of
particular significance for N = 3, since the 1- and 2-
matrices then have identical non-zero eigenvalues, and
the NSO and NSG can be obtained from each other by
virtue of (104) and (105).
Symmetry Pronerties
We mentioned earlier that the wavefunctions
should be required to be eigenfunctions of the group of
the Hamiltonian, and the question of how the symmetry
properties of the wavefunction carry over to the p-
matrices and the natural p-states has been extensively
studied. We shall merely list some of these results
here which have significance for our own work.
Theorem 1: If 1 is an N-electron Hermitian operator
of the form
4"- z (106)
or a unitary operator of the form
.n j=4 (107)
-I
or an antiunitary operator of the form
- 74 -
i=/
(Al) IT fi(108)
where, in (107) and (108), R ;is unitary
and K denotes complex conjugation, and
if P is an eigenfunction of then
the natural p-states can be chosen as
eigenfunctions of .
Theorem 2: If 4 and J transform as the irreducible
representations 9 and d respectively of
some group, then /7 transforms as the
direct product representation ex X
The particular significance of these results
is best illustrated by a few examples. If the
wavefunction is an eigenfunction of L S or rarity,
z z
theorem 1 applies, and the p-matrix blocks by ML, M'S
or parity value, and the natural n-states are
eigenfunctions of L S or parity. If the wavefunction
9 2
is an eigenfunction of S2 or L2, the natural p-states
2 2
can generally not be chosen eigenfunctions of S or L
except when S = MS = 0 or L = ML = 0. Of course, for
special choices of approximate wavefunctions,
additional symmetries may be introduced. Garrod has
shown for example that if the wavefunction is taken as
- 75 -
an average of M components with identical space and spin
parts, then the NSG's can also be made eigenfunctions
of L2. Theorem 2 is probably more useful for molecules
and solids; for atoms it essentially duplicates theorem
1.
In order to better see the structure of the
1- and 2-matrices, it is sometimes useful to expand
them in terms of separated space and spin parts. If
the wavefunction is an eigenfunction of S it may be
shown that
l^')c o 9 i
,-~ ,-~
(109)
F 61 W'~~i:7bd
Scd / ./rr, cc
, dc/^' rj-dr,
dc I /r, dc
(110)
where
a-c d
C r -I{CLA^^CL)
(111)
dJ L (c-e
_V-2 Ig -sc
The presence of the cross terms cd* and dc* in (110)
shows that the 2-matrix is generally not an
F Cx, 'iY/2
- 76 -
2
eigenfunction of S2. Also, one sometimes introduces the
charge-density 1-matrix,
(112)
the spin-density 1-matrix,
?d (riy)
(113)
zfS'z6- I/'C')
and the chare-density 2-matrix
and the charge-density 2-matrix,
(114)
The eigenfunctions of the charge-density matrices are
called charge-density natural orbitals (CDNO) and
charge-density natural geminals (CDM'G), or simply,
natural orbitals (NO) and natural geminals (NG). The
eigenfunctions of the spin-density 1-matrix are called
spin-density natural orbitals (SDIO). To find bounds
on the eigenvalues of (112) and (113), we can use the
matrix representations
S(^ ')- = f I )d6;
a ^ a ^
fo; r6i C ) cla- r'
I~~vEi ~rL ;j
- 77 -
dr7 c
t P/I.
(115)
(116)
We then use the result that if matrices /4, and C have
eigenvalues a,, /, and respectively, arranged in non-
increasing order, and if
--= /f *(117)
then /73/
nnu- (L //4/ c// A,/A
(118)
From this result, we obtain the following bounds.
0 x40- x M7.2 "i>)
l\m~ ^:;jc. I )~~ ~ S d
-fc, /)Z 1
(120)
The interest in the CF!O and CDING is two-fold. First,
if the wavefunction is an eigenfunction of S2 and S
and if MS = 0, then nd are identical, and the NSO's
are NO's with l or spini; F and /'vanish, and the I!SG's
are NG's with one of the four spin functions (111).
,,,_ ~
75' --
- 78 -
Second, for an t-function basis, there are 2M NSO, only
ft of which can be spatially linearly independent.
Consequently, the Il linearly independent NO's have
sometimes been suggested for the CI iteration scheme
discussed earlier. In general, both the NSO's and N!O's
will have mixtures of either odd or even values of
angular momentum; that is, s orbitals will have s, d,
g, i, ... admixture, and p orbitals, f, h, j, ...
admixture, and so on. This mixing poses a
computational difficulty in that most programs are set
up to deal with orhitals of a single (l,ml) value rather
than of a single mi value; angular-momentum projections
become considerably more involved if 1-mixtures are
allowed.
Ihile we have not made explicit use of them,
we have generated the CDHO's and SDNO's for all the
systems studied in this work.
Density Matrices of Some Special Functions
For a single Slater determinant of N
orbitals, the 1-matrix has N occupation numbers equal
to 1, and the remainder equal to 0. If the orbitals
are orthonormal, the 1-matrix is diagonal directly from
- 79 -
(46), and the orbitals are the NSO. This is a
particularly important case and has been discussed
extensively by L8wdin /18,19,20/. In this connection,
it is worthwhile to introduce the extended Hartree-Fock
(EHF) equations which LBwdin derived for an orthonormal
basis set. We mentioned these briefly in the last
chapter, but deferred a derivation because the density
matrices provide a particularly convenient tool for
this. We begin with the expression (41) for K = L,
where Q is now the Hamiltonian operator.
= Ho
* J.h-/, ( )1 ,
9. rwd, ..
(121)
Varying the expression (51), we obtain
S z
yi'^ ..- ) s iPrWk) 7 / <-( j
, ,. (ii.%.. p &
. c ... c i 0
+ Complex coyj'j._ e. (122)
- 80 -
Using this result, we find
So t d et )
where we have introduced a Hermitian matrix of Lagrange
multipliers to maintain orbital normalization. By the
usual argument, the expression in brackets must vanish;
we then multiply by 8() and sum over Y obtaining the
EHtF equations:
EHF equations:
- 81 -
P, 1iir) < .j^ r [7 d'), > ...
+ f P ,,....p ^ ('< ...rl ,,z,... ^') d,.. dx
w= .(( (124)
where
(l/'J = /l ) < } (125)
Note that nowhere have we assumed an orthonormal basis
or a particular form of the orbitals; unless we start
with an orthonormal basis, there is no need even to
introduce the Lagrange multipliers, and the right-hand
side of (124) then vanishes. For a non-orthonormal
basis then, there is no need to determine Lagrange
multipliers, but we have a more difficult p-matrix to
compute. In general, it is not possible to
simultaneously diagonalize the 1-matrix and the
Lagrange multiple ier matrix, so we essentially lose the
concept of orbital energies.
It is often useful to introduce a quantity,
called the "fundamental invariant", defined by
- 82 -
p 0
which satisfies
='
where M is the number of orbitals in the basis.
invariance follows from the fact that a nonsi
linear transformation on the basis leaves
unchanged:
IrA> T V' < >Tk TI
-- /
(129)
For the case M = N, L8.wdin /19/ showed that o determines
all the p-matrices, and these are given explicitly by
(126)
(127)
(128)
The
ngu ar
(126)
< ^ '<^
! > '+14>' 4
- 83 -
S= J (/...(p,)) (130)
The fundamental invariant therefore contains all the
information contained in a single-determinant
wavefunction, regardless of the form of the basis
orbitals. This noint has lead to some confusion in the
literature. In an often-quoted paper, Bunge /44/
arrived at the result that for a PGSO wavefunction, the
EHF equations do not yield unique orbitals; i.e. the
fundamental invariant is not invariant. This result is
incorrect; the error in the paper is the omission of
the factor <>> in ,p ; this simplified form holds for an
orthonormal basis. Bunge then proceeded to vary the
orbitals, destroying the orthonormality. The EHF
equations are perfectly well-defined, even for GSO.
That p determines all the p-matrices for a single
determinant is true, even for a Drojected determinant,
OD. The occupation numbers are 1 and 0 if OHO is
considered the modified Hamiltonian and D the
wavefunction; however, if OD is considered the
wavefunction, the occupation numbers are in general not
0 and 1 because the projection introduces new orbitals.
In this case, the fundamental invariant must be
constructed from the complete set of orbitals,
including all the ones introduced by the projection.
- 84 -
The p-matrices must still be determined by the
fundamental invariant, but the form of the natural p-
states and occupation numbers is not obvious. For the
case of a spin-projected determinant of pure spin
orbitals, Harrinan /79/ has derived explicit formulae
for the 1-matrix, NSO, and occupation numbers,
Hardisson and Harriman /80/ derived a formula for the
2-matrix, and this has recently been extended to point-
group and axial-rotation symmetry projection by Simons
and Harriman /81/ to obtain formulae for the 1- and 2-
matrices. The first two /79,80/ are derived for a
projected DODS determinant; in the last /81/, the
orbitals are only assumed to be orthonormal. The form
of the occupation numbers and the natural p-states for
a PGSO wavefunction is not known in analytic form,
although we have calculated the 1- and 2-matrices
directly from the projected determinant treated as a CI
expansion over non-orthonormal orbitals. The formulae
for the DODS case are already very complicated; in view
of the great increase in complexity in going to GSO, we
feel that an attempt at obtaining an analytic formula
for the p-matrices of a PGSO wavefunction would not be
worthwhile.
- 85 -
2.4 The N-Representability Problem
The Schr8dinger equation, (1), has never been
solved exactly for a system with more than one
electron. As the number of electrons increases, the
approximate wavefunctions become increasingly complex.
The Hylleraas coordinate functions discussed in the
last chanter have not been extended beyond four-
electron systems; the CI programs mentioned are limited
to less than forty electrons. Systems of chemical
interest frequently have hundreds or thousands of
electrons which we have so far been unable to treat
accurately. One can imagine Colenan's excitement in
1951 when he first observed the significance of the
equations (41) and (121); since the usual Hamiltonian
employed contains at most two-particle terms, the
energy, and all one- and two-electron properties defend
at most upon the 2-natrix, from which the 1-matrix can
be derived according to (36). The 2-matrix is a
function of only four particles. Thus, by varying a
certain four-particle function, one should be able to
obtain practically every result of chemical interest
for any system, no matter how large. Rather than
launch a calculation on DNA, Coleman contented himself
at that time with a calculation on lithium, a three-
electron system. The calculation gave an energy 30%
- 86 -
below the experimental value, in seeming violation of
the Rayleigh-Ritz variational principle. The
difficulty was that the four-particle function had been
varied over too wide a class of functions. This
problem has since become known as the "N-
representability" problem -- the problem of finding the
conditions under which a 2p-particle function, such as
a p-matrix, can be shown to be derivable from an N-
particle antisymmetric (or symmetric) wavefunction
without actually exhibiting that N-particle function.
This problem has received a great deal of study in the
last two decades. The indications so far
pessimistically are that either the general solution
does not exist, and therefore cannot be found, or that
if the solution exists, and is found, implementing it
will be at least as difficult as carrying out a
calculation with the N-particle wavefunction. This
thought is rather depressing, considering that a
feasible solution has the strong possibility of
revolutionizing a good part of chemistry, physics, and
biology. More optimistically, one might hooe for an
approximate solution so that variation of a reduced
density matrix could be implemented in such a way as to
provide a useful alternative to ab initio, semi-
empirical, or even empirical theories. Some progress
87 -
has been made along these lines by a number of authors
/82 90/.
In the meantime, reduced density matrices
provide a convenient tool for interpretation of
wavefunctions and nronerties.
CHAPTFR 3
ATOMIC PROPERTIES
3.1 Introduction
Reading the current quantum chemical
literature gives one the feeling that a total energy is
the only property atomic and molecular systems possess.
Since the total energy, like the thermodynamic enthalny
and free energy, is meaningless except when compared
with another total energy, one miFght even begin to
question the motivation of the calculations. In fact,
of course, there are a good many oronerties of interest
which we can in orinciole compute. A recent hook by
Mfalli and Fraga /92/, although somewhat concise and
uncritical, does at least give an idea of some of the
orooerties of interest. A review article by Doyle
/105/ discusses relativistic and non-relativistic
corrections to atomic energy levels and a number of
numerical tables with these corrections is given. We
will content ourselves in this chapter only with givinr
a short indication o4 some of these properties with
- 88 -
references to work where greater detail may be found.
3.2 Enprries
The calculation of the energy determines the
wavefunction. Fxcept for one-electron systems, which
can be solves exactly, and Pekeris' work cited earlier
on two-electron systems, calculations of energy levels
cannot corrnte with exoerimnnt in accuracy.
Consenuently, xr-ent for determination of the
wavefunction and com)nrison with other theoretical
results, for atoms, calculation -of energies is of
little interest because the exnerimental data is so
much better. Cor nolocules, even small di-tomics, this
is not the case, and one can often cet better
characterization of nntential curves by theoretical
comnutat ions than current exner mental n thds can
five. It is perhaps one of the sad facts of quantum
mechanics that determination of the energy is the only
route to the wavefunct ion, ann that even if an
aonroximate wavefunction gives a good enerTy, other
properties calculated from it may be rather Door.
In this work, in addition to the energy
determination, we have also evaluated the scale factor
- 89 -
- 90 -
and scaled energy given for atoms by /98/
(131)
.2
E V< > __ (132)
2 4 =c-'/
An atomic wavofunction may always be scaled to satisfy
the virial theorem; if the unsealed wavefunction
satisfies it already, then the scale factor is
necessarily unity. We have found this useful in that
a scale factor differing from unity by more than about
0.001 indicates that the basis is poorly chosen.
3,3 Snecific Mass Fffect
In thp introduction, we derived the snpcific
mass effect, or mass nolarization, correction to the
kinetic energy, en. (15). FrmHan /99/ has estimated
-6
the effect from Pxnprimental snectra to be about 10
!1 (0.2 K) for Li 2 S and 10-7 H for Li 4 2S. He also
states that tho effect should be approximately
independent of Z, so that the same estimates apply to
the rest of the isoelectronic sequence. However,
- 91 -
Prasad and Stewart /100/ have recently evaluated the
effect from Weiss' 45-term CI wavefunctions for the 2
S and 2 2P states of the sequence from Z = 3 to 8; for
the 2 2S state, their data gives the shift proportional
to 1.29; or the 2 2P states, the shifts decrease with
increasing Z, becoming negative for Z > 4. The shift
9 7 6
for the 2 'S state of Li is 2.587 K, and for Li6, 3.017
K, a difference of 0.430 K. By contrast, the normal
isotope shift, (13), causes the same level of the two
isotopes to differ by 21.353 K. The specific mass
effect is therefore small for light atoms, but
important for accurate determination of energy level
separations. For heavier elements, the specific mass
effect can be many times larger than the normal isotope
shift. Dalgarno and Parkinson /102/ have estimated the
specific mass effect in lithium by perturbation theory
applied to the results of Pekeris and coworkers /54-
56,103-104/ on two-electron systems, and obtain results
in agreement with Prasad and Stewart.
According to Kuhn /101/, theoretical
determination of the specific mass effect would be a
valuable contribution, and we therefore intend to
compute the effect with our wavefunctions at a later
date.
92 -
3.4 Relativistic !ass Increase
Relativistically,
the electron
mass varies
with velocity according to
eWo
where
/V_
C
and m is
0
energy is
energy is
the electron rest mass.
The relativistic kinetic
-T= -t' M 0o C
3 i
C2
- . .
and the relativistic four-momentum is
p= (-YM, imc)
but its magnitud- is constant:
pp D- = -III C
We therefore take t-e non-relativistic momentum,
(133)
(134)
(135)
(136)
*
* |