Unitary group approach to reduced density matrices
M. D. Gould, J. Paldus, and G. S. Chandler
Citation: The Journal of Chemical Physics 93, 4142 (1990); doi: 10.1063/1.458747
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Unitary group approach to reduced density matrices
M.
D.
Gould
Department
of
Mathematics, University
of
Queensland, St. Lucia, Queensland, Australia 4067
J. Paldus
8
)
Department
of
Applied Mathematics, Department
of
Chemistry
and
Guelph- Waterloo Center
for
Graduate
Work in Chemistry, University
of
Waterloo, Waterloo, Ontario, Canada
N2L
3G1
G.
S.
Chandler
School
of
Chemistry, University
of
Western Australia, Nedlands, Western Australia, Australia 6009
(Received 28
February
1990; accepted
17
May
1990)
A fully spin-adapted approach
to
many-electron density matrices is developed in
the
context
of
the
unitary
group
approach
to
many-electron systems.
An
explicit expression for
the
single-
electron spin-density operator, as a polynomial
of
degree two in the orbital U
(n)
generators, is
derived for
the
case
of
spin-independent systems. Extensions
to
spin-dependent systems are
also considered, leading
to
the
appearance
of
total-spin transition densities, whose general
properties are investigated. A corresponding formalism for
the
two-electron density matrix,
which is capable
of
further generalization, is also developed.
The
results
of
this paper, together
with recent developments
on
the
matrix elements
of
the
U
(2n)
generators in
the
electronic
Gel'fand basis, afford a versatile method for
the
direct calculation
of
one-
and
two-body
density matrices in
the
unitary
group
approach framework.
I. INTRODUCTION
Reduced density matrices
and
their application have a
long history
and
were introduced, in the
Hartree-Fock
framework, by Dirac. I Subsequent work
of
LOwdin
2
and
McW
eeny
3,4
demonstrated
the
wide power
of
such tech-
niques, particularly for evaluating one-
and
two-electron ex-
pectation values
and
for
the
determination
of
effective spin
Hamiltonians as required in
ESR
and
NMR
spectroscopy.
Since
then
there have appeared
many
applications
and
ex-
tensions which are discussed in several books
and
review
articles.4--7 We only mention here
the
numerous applications
of
density matrices
to
the
definition
of
bond
order
matrices,
various
bond
indices,
and
overlap populations,4,5,8-12 as well
as their recent exploitation in theories
ofhigh-T
c
supercon-
ductivity.13
It
has been recognized for some time
that
molecular
charge densities, in conjunction with x-ray-diffraction ex-
periments, afford a powerful tool for
the
analysis
of
chemical
bonding. Unfortunately, however, the valence electrons
(in
the
bonding region) usually contribute only a small fraction
to
the
overall charge density, so
that
it is first necessary
to
subtract
out
the
dominant
contribution from the core (lead-
ing
to
difference density
maps).
The
latter is frequency mod-
eled by spherically averaged atomic densities, although oth-
er
possibilities, leading
to
different distributions, can occur,
and
hence great care needs
to
be exercised in their interpreta-
tion.
An
alternative approach, which enables one
to
experi-
mentally observe
the
valence electrons directly, is afforded
by
polarized neutron-diffraction experiments, in which po-
larized neutrons are scattered by unpaired spins in a crystal-
line lattice. These experiments yield magnetic
structure
fac-
tors which represent
Fourier
components
of
magnetization
density, in complete analogy to
Fourier
components
of
the
a)
Killam Research Fellow 1987-89.
electron density
that
are afforded by x-ray-diffraction struc-
ture
factors. Assuming
that
in
the
ground
state considered
the
orbital contribution
to
the
magnetization is negligible,
these magnetic
structure
factors provide a description
of
the
spin density
of
the
system.
In
this way valuable information
on
chemical bonding effect
may
be
inferred, particularly for
transition-metal compounds with unpaired d electrons. This
technique
has
been developed
and
extensively exploited
by
Figgis
and
co-workers.
I
4--16
The
experiments
and
the inter-
pretation
of
such neutron-diffraction
data
are, however, dif-
ficult
and
theoretical information
on
the
distribution
of
un-
paired spins, as afforded by
the
one-
and
two-electron spin
densities, can be
of
crucial importance
and
help. We quote
from
the
pioneering paper
by
Figgis, Reynolds,
and
Wil-
liams
I4
(b)
in which spin density
and
bonding in
the
CoCI
2
-
• 4
Ion were first explored:
"the
overlap spin-density component
in
the diffuse region is strongly correlated with possible spin-
polarization effects,
and
within
the
accuracy
of
the
experi-
ment
it is
not
possible
to
be clear about its origin
without
aid
from theoretical chemistry."
Indeed, theoretical calculations have already proved a
valuable adjunct for
the
interpretation
of
neutron-diffrac-
. d
17
hon
ata
but,
to
date, most calculations have been per-
formed
at
the
Hartree-Fock
level.
This
leaves unexplained
certain discrepancies with experiments, which are
not
re-
moved when limited forms
of
configuration-interaction
(CI)
or
multiconfiguration self-consistent-field
(MCSCF)
calcu-
lations
are
used:
18
This is particularly
the
case for transition-
metal compounds exhibiting a large degree
of
covalency.
16
We note
that
although the
CI
methodology was employed
to
study
the
energetics, particularly in connection with optical
spectra, in
the
transition-metal complexes, it was
not
em-
ployed in computation
of
charge
or
spin densities,
18
which is
rather
demanding.
The
formalism developed in this paper
can
certainly facilitate such an undertaking.
In
order
to
develop methods for efficiently obtaining
4142
J.
Chem. Phys. 93
(6),15
September 1990 0021-9606/90/184142-12$03.00
© 1990 American Institute of Physics
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2016 06:47:08
Gould, Paldus, and Chandler: Reduced density matrices
4143
molecular charge and spin densities in
CI
calculations, we
investigate this problem within the framework
of
the unitary
group approach
(UGA),
which currently affords a powerful
and versatile method for large-scale
CI
calculations on mole-
cules. The
UGA,
which arose from previous work
of
Mo-
shinsk
y
19
on the nuclear shell model, exploits the fact
that
the spin-independent (electronic) molecular Hamiltonian
is
expressible as a bilinear form in the orbital U
(n)
generators.
This enables the exploitation
of
U
(n)
representation theory
for the efficient spin adaptation and for calculation
of
Ham-
iltonian
matrix
elements
(MEs)
that
are required in
CI
cal-
culations.
The
initial development
of
UGA
due to Pal-
dus
20
,21
and Shavitt
22
was followed by numerous important
innovations and extensions, which are described in several
reviewarticles
23
-
27
and books.
28-30
The
UGA
formalism en-
abled numerous computational implementations, particu-
larly in conjunction with direct
CI,31
ranging from the inte-
gral-driven,32.33 100p-driven,34 shape,35
or
internal
interaction block-driven
36
to ME-driven
37
approaches
based on harmonic level excitation diagrams.
38
Thus, as indicated above, the
UGA
affords a convenient
method for the calculation
of
CI
wave functions and hence
provides an ideal framework in which to develop a spin- and
charge-density matrix formalism. This
is
particularly the
case since, as will be shortly seen, the general
k-electron den-
sity matrix may be expressed naturally in terms
of
the U
(2n)
generators.
In
the present and following papers we thus aim
to develop a fully spin-adapted
UGA
to k-electron density
matrices, with particular emphasis on the evaluation
of
one-
and two-electron charge and spin densities.
In
this paper we present a spin-adapted approach to one-
electron density matrices.
In
particular,
we
show that, for
spin-independent systems, the one-electron density matrix
is
determined completely by two density operators, namely the
charge and (normalized) spin-density operators. We give an
explicit expression for these operators in terms
of
the
U(n)
matrix A
of
Ref. 39 (which is given by a polynomial
of
degree
two in the orbital
U(n)
matrixE
= [E J]). We then consid-
er extensions to two-body density matrices which, for spin-
independent systems, are shown to be completely deter-
mined by six density operators, which in turn are expressible
in terms
of
the one-body charge and spin-density operators.
The
MEs
of
the entries
Aj
of
the
U(n)
matrix A in the elec-
tronic Gel'fand basis have been recently determined
40
and
afford an efficient
direct evaluation
of
one- and two-electron
charge and spin densities in the
UGA
framework.
We also consider extensions to spin-dependent systems
and demonstrate that the full one- and two-electron spin-
density matrices are completely determined by certain total
spin transition density operators. which are explicitly con-
structed in terms
of
the U
(2n)
generators. together with the
(scalar) densities discussed above.
From
the viewpoint
of
computer implementation. it
is
worth noting
that
the U
(2n)
ME
formalism developed in Refs. 39-41 affords a conven-
ient method for the
direct evaluation
of
these spin transition
densities in the electronic spin-orbital basis.
The paper
is
set up as follows. We begin in Sec.
II
with
the unitary group formulation
of
the one-electron density
matrix.
In
Sec.
III
we
investigate the reduction
of
this den-
sity matrix for spin-independent systems and obtain an ex-
plicit expression, in terms
of
the orbital U
(n)
generators, for
the normalized spin-density operator.
In
Secs. IV and V
we
consider extensions
of
the two-body density matrix and con-
clude with a brief discussion and suggestions for future re-
search in
Sec. VI. Throughout, repeated use
is
made
of
the
characteristic identity methods
of
Green
42
and
Gould,43-45
which are shown to
playa
natural role in discussing density
operators.
II. ONE-PARTICLE DENSITY MATRICES
We shall consider density matrices arising from wave
functions constructed from
2n spin orbitals
"'il'
(x).
1 <;i<;n,
J1
= ± 112, where x =
(r,t)
denotes the combined spatial
and spin coordinates. We assume
that
the spin orbitals
"'il'
(x)
factor into the orbital and spin parts,
"'il'
(x)
=
,pi
(r)xl'
(t),
where the molecular orbitals
,pi
(r)
are orthogonal and
XI'
denote the elementary spin column vectors
so that
XI'
(t)
=
Dl's'
In
the second quantized formulation.
we
introduce corresponding fermion creation and annihila-
tion operators.
xt
and
XiI"
respectively. satisfying the
fa-
miliar anticommutation relations
{xt,xj,,}
=
DijDl'l"
{xt.X]v}
=
{Xil"Xj,,}
=
O.
The space
of
N-particle states r
N'
to which the N-electron
wave functions belong. is then given by all
Nth-order
prod-
ucts
of
fermion creation operators X t acting on the physical
vacuum state
10).
Following the
UGA.
2
0-
23
the particle number conserv-
ing operators
E
'il_xt
X
jl'
-
iJ.l
jv
(1)
form the generators
of
the spin-orbital Lie group
U(2n).
The space r N
of
N-particle states then gives
riseto
an irre-
ducible representation
ofU(2n)
with highest weight
44
<iN.O)
= (1,1, ... ,1,0, ....
0).
with
l's
in the first N positions, called the antisymmetric
Nth
rank tensor representation.
The
spin-averaged operators
+
1/2
Ei
=
'"
Eiv
J
£.,
JV
(2)
l'=
-1/2
form the generators
of
the orbital subgroup U
(n)
of
U
(2n),
while the operators
11
E
l'
-
'"
Eil'
l'-
L ;v
i=
I
(3)
form the generators
of
the spin subgroup U
(2).
The genera-
tors
(2)
and
(3)
collectively constitute the generators
of
the
subgroup
U(n)
XU(2)
(outer direct product)
of
U(2n),
referred to herein as the spin-orbit (SO) subgroup.
}n
terms
of
the U
(2)
generators
(3)
Athe
number opera-
tor
N and the total-spin vector operator S may be expressed
as
21
J. Chern. Phys., Vol. 93,
No.6,
15 September 1990
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2016 06:47:08
4144
Gould, Paldus, and Chandler: Reduced density matrices
S + = E
1~2112'
S = E
1/2112
AS
I
(E
112 E
~
1/2) N
A
E
1/2
E
~
112
(4)
z
='2
112 -
~1/2'
= 112 +
~1I2'
~her.!:,
as usual, S ± =
Sx
±
iS
y
•
We
note that the operators
S ±
,Sz
constitute the generators
of
the subgroup
SU(2).
In
this formulation
of
the many-electron problem, the
general single-particle reduced density operator
is
given
by4
112
n
PI(X;X')
= I I
f/lf';.(x)tPjv(x')Ej",
(5)
'",v=
~
112
i,j=
I
which may be alternatively expressed as a 2 X 2 matrix with
entries
PI
(r;r')~
=
PI«r,/l);(Y',v»
n
= I
¢f(r)¢j(r')Ej".
(6)
iJ= I
Throughout
we
shall refer to the matrix (6) as the single-
electron density
matrix.
We emphasize that in this approach
the entries (6)
of
the density matrix are regarded as opera-
tors.
The spin-density matrixPI
(r)
arising from a given mo-
lecular wave function 'I'
is
then given by the corresponding
2 X 2 matrix
of
expectation values
(
6')
It
is
of
interest to investigate the behavior
of
the density
operator
(5)
under convolution product. We have
J
A (
')
A ( ,
")d'
PI
x;x
PI X
,x
X
n +
1/2
I I
tP~
(X)tPla
(x")
iJ,k,/=
I
,",v,p,a=
~
1/2
X f
tPjv(x')tPtp(x')dx'
EJ:E7:·
Using the orthonormality
of
spin orbitals
we
obtain
f
A (
')A
( ,
")d'
PI
x;x
PI
x;x
X
n +
1/2
= I I
tP~(x)tPla(x")(E2)t,
i,/=
I
,",a
=
~
112
where
E2
is the
(2n)
X
(2n)
matrix with entries
(
E2)i,"
=
Eil'
Eka
Jl'
ka
JV
(7)
(summation over repeated indices
is
assumed). However,
on the representation
ofU(2n)
with highest weight
(iN,O),
which
is
pertinent to the N-electron problem, the U
(2n)
matrix [E
j~~]
satisfies the quadratic identity
42
(cf. the Ap-
pendix)
E(E-2n-I+N)=O,
(8)
so that
(E2)j~
=
(2n
+ 1 -
N)Et~.
(8')
Substituting into Eq.
(7),
we
thus obtain
J PI (X;X')PI
(x';x"
)dx'
=
(2n
+ 1 -
N)p,
(x;x"),
which may be expressed in the matrix form
+112
f
I
PI(r;r')~PI(r';r")~
dr'
=
(2n
+
1-
N)jJI(r;r")~.
a=
~
112
(9)
It
follows, in particular, that the normalized density opera-
tor
1'1
(X;x')
==
(2n
+ 1 -
N)
~
I PI
(x;x')
is
idempotent, i.e.,
J
A (
')
A ( ,
")d'
A ( " )
YI
X;X
YI
X
,x
X =
YI
X;X
.
( 10)
We
emphasize that the idempotency
of
the density oper-
ator
1'1'
Eq. (10), should not be confused with the well-
known
2
•
3
idempotency
of
the molecular density matrix aris-
ing from a closed-shell Hartree-Fock wave function.
Equation (9)
is
an operator equation and
is
a direct conse-
quence
of
the reduced identities
of
Eq. (8). Such reduced
identities, first encountered by Green
42
(see also the Appen-
dix), afford a natural tool for investigating density opera-
tors, as previously noted, and will be applied extensively
throughout the paper.
Taking the trace
of
the density matrix (6)
we
obtain
3
,4
the familiar charge-density operator
n
= I
¢f(r)¢j
(r')Ej,
(11)
iJ= I
which
is
expressible solely in terms
ofthe
orbital
U(n)
gen-
erators
E1,
Eq.
(2),
and hence
is
obviously spin indepen-
dent.
In
terms
of
the usual Pauli spin matrices
-
i)
(1
0'
O'z = °
the one-electron density matrix may be rewritten as
PI(r;r')~~
=
!p~
(r;r')B:,
+
R'a;,
where
n
R(r;r')
= I
¢f(r)¢j(r')rj,
iJ=
I
and
(
rx),i=
_1(Ei,l12
+Ei,~1/2)
'2
j,
~
1/2
j,I12'
(12)
(l3a)
(
ry),i=
_li(Ei,l12
_Ei.~1/2)
(I3b)
'2
j,
~
112
j,I12'
(
r
)i
=1(Ei,l12
_Ei,~I12)
z, 2
,,112
,,~
112 .
Alternatively,
we
note that the one-body density matrix
can be expressed in a convenient tensor form: This gives rise
to a scalar density, which
is
given by the number density
operator
of
Eq. (11), together with a vector density which,
in terms
of
the SU (2) coupling coefficients,
is
defined by
where the summation over
/l
and v is implied. Note that
these components are explicitly given by
J,
Chern, Phys., Vol. 93,
No.6,
15 September 1990
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2016 06:47:08
Gould, Paid us, and Chandler: Reduced density matrices
4145
PI
(r,r')
I = PI
(r,r')
1~21/2'
PI(r,r')_1
=
-PI(r,r')';2
1/2
,
PI
(r,r')o
= - 2
112
R
z
•
We remark
that
this latter operator, defined by the z compo-
nent
ofEq.
(13),
determines
the
usual (single-electron) spin
density
of
the
molecule.
3
.4
This tensor form for reduced den-
sity matrices is useful, particularly for discussing two-
and
higher-electron density operators, which will be considered
in
Sec. IV
and
V.
However, for one-body densities,
it
isjust
as
convenient to
work
with
the
matrix formulation
ofEq.
(12),
which we adopt
throughout
most
of
the paper.
It
should be noted
that
the
entries
of
the density matrix
(6)
are given in terms
of
the
V(2n)
generators, Eq.
(1),
which
do
not
conserve the total spin
quantum
number
S.
Consequently,
in
actual applications, we need a resolution
of
the
density matrix into components which effect well-de-
fined shifts
on
the total spin S. This is particularly the case
when dealing with density matrices arising from wave func-
tions with well-defined total spin (i.e., describing a spin-in-
dependent system).
In
the
following section we present a
solution
to
this problem using
the
spin-shift formalism
of
Refs. 44
and
45.
III. SPIN DENSITIES AND TOTAL-SPIN TRANSITION
DENSITIES
In
this section we consider the partitioning
of
the
single-
electron density matrix into components according to their
shifts
on
the total-spin
quantum
number S. This leads to the
appearance
of
total-spin transition densities, which increase
or
decrease the total-spin
quantum
number S, as well as a
density-matrix component which commutes with the total
spin
S. Only this latter density matrix is required when deal-
ing with wave functions possessing a well-defined total spin
and
is expected to yield the dominant contribution
to
the
density matrix even for wave functions characterizing spin-
dependent systems.
In
this latter case, however, the total-
spin transition densities will contain useful information on
transition probabilities between states with different total
spins.
We
first discuss the problem
of
constructing states with
well-defined total spin
S, which has been extensively studied
in the past (cf., e.g., Refs.
46-49).
In
the
VGA
this problem
is automatically taken into account by considering a basis for
the
space r N
of
N electron states which is symmetry adapt-
ed
to
the SO subgroup
V(n)
X
V(2).
The
resulting SO basis
states may be written as
(14)
where
[P]
designates
an
ABC
(Refs. 20
and
21)
or
Gel'fand_Paldus22-30.41.44.45
(G
P)
tableau, labeling the orbi-
tal
part
of
the state,
and
Ms
is the azimuthal spin
quantum
number.
The
states
ISMs)
constitute the usual basis for the
irreducible representations (irreps)
of
SV(2)
and
are
also
A
eigenstates
of
the
number
operator N with eigenvalue N
[and
hence are also V
(2)
states].
The
total spin
of
the
states
(14)
is given by S = b
/2,
where
p=
[abc],
N=2a+b,
a+b+c=n,
are
the
Paldus labels
of
the
orbital group V
(n)
(correspond-
ing to the top row
of
the
pattern
[P])
that
determine the
irrep
of
V
(n)
concerned.
20.2
I
,44
The
VGA
was originally designed
20
.
21
for spin-indepen-
dent problems, in which case the electronic Hamiltonian is
expressible solely in terms
of
the orbital
V(n)
generators.
For
such problems it suffices
to
obtain the
MEs
ofthe
V
(n)
generators [Eq.
(2)]
(and
of
their products) in the orbital
part
of
the
SO basis, Eq.
(14),
with a fixed value
of
Sand
Ms.
However, even when dealing with spin-dependent Hamilto-
nians, the
SO basis, Eq.
(14),
represents
an
excellent starting
point, since spin-dependent effects
are
usually small com-
pared to the spin-independent Coulomb interactions. Conse-
quently, in such a case,
the
MEs
of
the V
(2n)
generators are
required in the
SO basis, Eq.
(14).
This latter problem,
which has been addressed by
Gould
and
Chandler
41
.
45
and
more recently by Gould
and
PaId us,
39
is also essential for the
evaluation
of
the single-electron density matrix, Eq.
(6'),
in
the
SO basis, Eq.
(14),
to
which we now
tum.
Following
Gould
and
Chandler
41
.
45
(see also Ref. 39),
we note
that
the
V(2n)
generators transform, under com-
mutation with the
V
(n)
X V
(2)
generators, Eqs.
(2)
and
(3),
as the representation
AdU(n)
®Ad
u
(2p
where
Adu(k)
denotes the adjoint representation
of
V(k).45.50
It
follows
that
the V
(2n)
generators, Eq.
(1),
may be resolved into
spin-shift components
41
.4
5
EJ:;
=
E(
-
)J:,
+
E(O)J:,
+
E(
+
)J:;,
(15)
where
E(
± ) increases (respectively decreases)
the
total
spin
S
of
the SO states, Eq.
(14),
by one unit, while E(O)
leavesSunchanged.
TakingtheV(2)
traceofEq.
(15)
and
equating spin-shift components, we immediately obtain
that
Ej
=
'LEJ:,
= 'LE(O)J:"
(16a)
P P
(16b)
This shift component resolution
ofEq.
(15)
induces the
following partitioning
of
the single-electron density matrix,
Eq.
(6):
PI(r,r')~
=pj-)(r,r')~
+p\O)(r,r')~
+p\+)(r;r')~,
(17)
where
n
p\e)(r;r')~
=
'L
¢Jr(r)¢Jj(r')E(E)J:" E =
0,
±.
(18)
iJ=
I
In
view
ofEq.
(16) we obtain
tr
[p\O)
(r,r')]
=
tr
[PI
(r;r')]
=
p~
(r,r'),
tr[pj
± )
(r,r')]
=
0,
(19)
where
p~
(r,r')
is the charge-density operator, Eq.
(11).
We
may
thus
write, in view
ofEq.
(12),
P
A(O)(r:r')P = I
&::pAC
(r:r')
+ R(O).u
v
I , v
'2
vI,
JL'
P
Al
± )
(r:r')P
= R( ±
)·u
V
1
'v
It'
(20a)
(20b)
where
R(e)
is given by Eq. (13) with E
J:,
replaced by the
corresponding shift
componentE(E)J:,
(E
=
0,
±
).
A corre-
sponding spin-shift resolution
of
the vector density
of
Eq.
(12')
also applies.
J. Chern. Phys., Vol. 93,
No.6,
15 September 1990
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