Download Unified view on multiconfigurational time propagation for systems

THE JOURNAL OF CHEMICAL PHYSICS 127, 154103 共2007兲
Unified view on multiconfigurational time propagation
for systems consisting of identical particles
Ofir E. Alon,a兲 Alexej I. Streltsov,b兲 and Lorenz S. Cederbaumc兲
Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg,
Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany
共Received 11 June 2007; accepted 18 July 2007; published online 16 October 2007兲
We show that the successful and formally exact multiconfigurational time-dependent Hartree
method 共MCTDH兲 takes on a unified and compact form when specified for systems of identical
particles 共MCTDHF for fermions MCTDHB for bosons兲. In particular the equations of motion for
the orbitals depend explicitly and solely on the reduced one- and two-body density matrices of the
system’s many-particle wave function. We point out that this appealing representation of the
equations of motion opens up further possibilities for approximate propagation schemes. © 2007
American Institute of Physics. 关DOI: 10.1063/1.2771159兴
I. INTRODUCTION
Dynamics of many-particle systems captures a central
part in both theoretical and experimental atomic, molecular,
and condensed-matter physics.1,2 In many cases, the equation
governing the systems’ dynamics is the well-known timedependent many-particle Schrödinger equation. Although
this equation is linear, rarely can it be solved exactly. Thus,
approximations are a must. One of the most simplest and
direct approaches is to expand the time-dependent manybody wave function by time-independent configurations, an
expansion which becomes already enormously expensive
with as low as a few particles.
The idea to expand and optimize the time-dependent
many-body wave function of distinguishable particles is long
known.3–5 By allowing the configurations to be time dependent, a much larger effective subspace of the many-particle
Hilbert space can be sampled in comparison with expansions
utilizing configurations that do not change in time. A particular efficient variant of this idea led to the multiconfiguration
time-dependent Hartree approach 共MCTDH兲 which has been
successfully and routinely used for multidimensional dynamical systems consisting of distinguishable particles or degrees of freedom.6–11
The MCTDH approach can efficiently treat dynamical
and—using imaginary time propagation11—static properties
of a few-particle systems. Of course, the system under investigation by MCTDH can consist of identical particles. For
instance, we mention that very recently static properties of
weakly to strongly interacting trapped few-boson systems
have been studied on a quantitative many-body level by
MCTDH.12,13 Yet, in treating a larger number of identical
particles it is essential to use their statistics properties to
truncate the large amount of redundancies of coefficients in
the distinguishable-particle multiconfigurational expansion
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
c兲
Electronic mail: [email protected]
b兲
0021-9606/2007/127共15兲/154103/6/$23.00
of the MCTDH wave function. In this case, the challenge
was first approached for fermionic systems, where
MCTDHF—the fermionic version of MCTDH—was independently developed by several groups,14–16 taking explicitly
the antisymmetry of the many-fermion wave function to permutations of any two particles into account. MCTDHF is
presently successfully employed to study correlation effects
in a few-electron systems including under irradiation by laser
fields.17–19 Very recently, we accepted the respective challenge for bosons and developed MCTDHB 共Refs. 20 and
21兲—the bosonic version of MCTDH. This is, in particular,
valuable since very-many bosons can reside in only a small
number of orbitals owing to Bose-Einstein statistics. Alternatively speaking, by explicitly exploiting boson statistics it
is possible to successfully and quantitatively attack the dynamics of a much large number of bosons with the MCTDHB theory. As a first application of MCTDHB, the manybody dynamics of splitting a condensate in a trap was studied
in Ref. 20.
Besides utilizing particle statistics to specialize the original MCTDH theory to identical particles, there is a second
important difference in comparison with MCTDH which is
the nature of interparticle interactions. Identical particles
commonly interact via a two-body interaction which is explicitly taken into account and implemented in MCTDHF
共Refs. 14–16兲 and MCTDHB.20,21 We remind that MCTDH
was designated to treat nuclear dynamics in which interactions normally involve several degrees of freedom or coordinates. In deriving MCTDHB we noticed that the combination
of indistinguishability and the two-body interaction leads to
an appealing formulation of the equations of motion in terms
of the reduced one- and two-body density matrices of the
bosonic system. It is instructive to mention that reduced twobody density matrices and their usage in electron-structure
theory is an active field or research.22,23
The purpose of this work is to expand on our recent
finding for bosons. Specifically, we show that the MCTDH,
specified for systems of identical particles, namely, MCT-
127, 154103-1
© 2007 American Institute of Physics
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154103-2
J. Chem. Phys. 127, 154103 共2007兲
Alon, Streltsov, and Cederbaum
DHF for fermions and MCTDHB for bosons, can be formulated in a unified and compact form. In particular, the equations of motion for the orbitals depend explicitly and solely
on the reduced one- and two-body density matrices of the
system’s many-particle wave function. We also point out that
representing the equations of motion in terms of the reduced
one- and two-body density matrices opens up further possibilities for approximate self-consistent-like propagation
schemes for systems comprising of identical particles.
II. THE ANSATZ
Our starting point is the field operator ⌿̂共r兲 satisfying
the usual fermionic 共bosonic兲 anticommutation 共commutation兲 relations,
⌿̂共r兲⌿̂†共r⬘兲 ± ⌿̂†共r⬘兲⌿̂共r兲 = ␦共r − r⬘兲.
共1兲
It is convenient to expand the field operator with a complete
set of time-dependent orthonormal orbitals,
⌿̂共r兲 = 兺 ĉk共t兲␾k共r,t兲,
共2兲
k
where the time-dependent annihilation and corresponding
creation operators obey the usual anticommutation 共commutation兲 relations ĉk共t兲ĉ†j 共t兲 ± ĉ†j 共t兲ĉk共t兲 = ␦kj for fermions
共bosons兲 at any time. Note that r implicitly indicates spatial
and spin degrees of freedom and k enumerates spin-orbitals.
The many-body Hamiltonian is standardly written as
Ĥ = 兺
ĉ†k ĉqhkq
k,q
1
+ 兺 ĉ†k ĉs†ĉlĉqWksql ,
2 k,s,l,q
共3兲
where the matrix elements of the one-body Hamiltonian ĥ共r兲
and two-body interaction potential Ŵ共r − r⬘兲 with respect to
the orbitals are given by
hkq =
冕
Wksql =
␾*k 共r,t兲ĥ共r兲␾q共r,t兲dr,
冕冕
共4兲
␾*k 共r,t兲␾s*共r⬘,t兲Ŵ共r − r⬘兲
兩⌿共t兲典 = 兺 Cn共t兲兩n1,n2, . . . ,n M ;t典,
n
共5兲
兩n1,n2, . . . ,n M ;t典 =
1
冑n1!n2! ¯ nM!
⫻共ĉ†1共t兲兲n1共ĉ†2共t兲兲n2 ¯ 共ĉ†M 共t兲兲nM 兩vac典,
with the appropriate permutational symmetry. For fermions
one chooses the configurations 兩n1 , n2 , . . . , n M ; t典 as Slater determinants with time-dependent orbitals, and for bosons one
employs permanents 兩n1 , n2 , . . . , n M ; t典 assembled from timedependent orbitals. The summation in Eq. 共5兲 runs over all
possible configurations generated by distributing N identical
particles over M orbitals. We collect the occupations in the
vector n = 共n1 , n2 , . . . , n M 兲, where n1 + n2 + . . . + n M = N. For
fermions each occupation nk is equal to either 0 or 1, in
accordance with Fermi-Dirac statistics, whereas for bosons
each nk can take any value between 0 and N as stipulated by
Bose-Einstein statistics.
In practical computations we have of course to restrict
the number M of orbitals from which the configurations
兩n1 , n2 , . . . , n M ; t典 are assembled. Suppose we take M ⬘ virtual
orbitals atop the minimal number of orbitals needed to host
N identical particles. For fermions M ⬘ = M − N, since the
minimal number of orbitals required to house the fermions is
obviously N. For bosons we have M ⬘ = M − 1, since all
bosons can accumulate in a single orbital. With these observations, it is attractive to find out that the total number of
configurations for identical particles coincide and is given
both in the fermionic and bosonic cases by24
number of configurations =
N + M⬘
N
冊
共6兲
.
Of course, if M ⬘ goes to infinity then the ansatz 关Eq. 共5兲兴 for
the wave function becomes exact since the set of configurations 兩n1 , n2 , . . . , n M ; t典 spans the complete N-particle Hilbert
space.
III. WORKING EQUATIONS AND DISCUSSION
To derive the set of equations of motion for the multiconfigurational time evolution of identical particles, it is useful to employ the Lagrangian formulation of the timedependent variational principle.25,26 Thus, we substitute the
many-body ansatz 关Eq. 共5兲兴 into the functional action of the
time-dependent Schrödinger equation which reads
⫻␾q共r,t兲␾l共r⬘,t兲drdr⬘ .
S关兵Cn共t兲其,兵␾k共r,t兲其兴 =
Specifying for identical particles the multiconfiguration
time-dependent Hartree approach, the ansatz for the manybody wave function ⌿共t兲 is taken as a linear combination of
time-dependent configurations,
冉
冕
dt
M
再冓 冏
⌿ Ĥ − i
冏冔
⳵
⌿
⳵t
冎
− 兺 ␮kj共t兲关具␾k兩␾ j典 − ␦kj兴 .
k,j
共7兲
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154103-3
J. Chem. Phys. 127, 154103 共2007兲
Multiconfigurational time propagation of identical particles
We recall that the orbitals 兵␾k共r , t兲其 and coefficients 兵Cn共t兲其
are independent variables 共arguments兲 of the action 关Eq. 共7兲兴.
Our strategy is first to take expectation values and only subsequently perform the variation and require stationarity of
the action with respect to the arguments 兵␾k共r , t兲其 and
兵Cn共t兲其. The time-dependent Lagrange multipliers ␮kj共t兲 are
introduced to ensure that the time-dependent orbitals remain
normalized and orthogonal to one another throughout the
propagation. Moreover, the Lagrange multipliers “compensate” for terms appearing when the variation is performed
before matrix elements are evaluated, as done within the
Dirac-Frenkel formulation of the time-dependent variational
principle;27,28 also see in this context Refs. 21 and 26.
To perform the variation of the action 关Eq. 共7兲兴 with
respect to the orbitals, we express the expectation value of
Ĥ − i共⳵ / ⳵t兲 appearing in Eq. 共7兲 in a form which explicitly
depends on the orbitals. This is done by resorting to the
reduced one- and two-body density matrices ␳共r1 兩 r1⬘ ; t兲 and
␳共r1 , r2 兩 r1⬘ , r2⬘ ; t兲 of ⌿共t兲. Given the normalized wave function ⌿共t兲, the reduced one-body density matrix reads
␳共r1兩r1⬘ ;t兲 = N
冕
i共⳵ / ⳵t兲 can be written as a one-body operator,
i
共8兲
⌿*共r1⬘,r2⬘,r3, . . . ,rN ;t兲
⫻⌿共r1,r2,r3, . . . ,rN ;t兲dr3 ¯ drN
= 具⌿共t兲兩⌿̂†共r1⬘兲⌿̂†共r2⬘兲⌿̂共r2兲⌿̂共r1兲兩⌿共t兲典
M
=
兺
␳kslq共t兲␾*k 共r1⬘,t兲␾s*共r2⬘,t兲␾l共r2,t兲␾q共r1,t兲, 共9兲
k,s,l,q=1
where the matrix elements of the two-body density matrix
␳kslq共t兲 = 具⌿兩ĉ†k ĉs†ĉlĉq兩⌿典 for fermions29 and for bosons30 are
prescribed in a unified manner in Appendix A. Now, the result for the expectation value in Eq. 共7兲 compactly reads
冓冏
冏冔
冋 冉 冊册
M
⳵
⳵
⌿ Ĥ − i
⌿ = 兺 ␳kq hkq − i
⳵t
⳵t
k,q=1
冋
kq
⳵␾q共r,t兲
dr.
⳵t
兺
册
兵␳共t兲其−1
jk ␳kslqŴsl兩␾q典 ,
k,s,l,q=1
共12兲
兺 兩␾ j⬘典具␾ j⬘兩,
Ŵsl共r,t兲 =
冕
␾s*共r⬘,t兲Ŵ共r − r⬘兲␾l共r⬘,t兲dr⬘
˙ q典 = 0,
具 ␾ k兩 ␾
are satisfied at any time. Obviously, if conditions 关Eq. 共14兲兴
are satisfied at any time, the orbitals remain orthonormal
functions at any time. This representation simplifies considerably the equations of motion 关Eq. 共12兲兴 for the orbitals, j
= 1 , . . . , M,
冋
kq
共10兲
where, when acting on the orbitals, the time-derivative
共14兲
k,q = 1, . . . ,M
˙ j典 = P̂ ĥ兩␾ j典 +
i兩␾
1
⳵C
兺 ␳kslqWksql − i 兺n Cn* ⳵tn ,
2 k,s,l,q=1
共13兲
˙ j ⬅ ⳵␾ j / ⳵t. Exare the local time-dependent potentials and ␾
amining Eq. 共12兲 we see that eliminating the Lagrange multipliers ␮kj共t兲 has emerged as a projection operator P̂ onto the
subspace orthogonal to that spanned by the orbitals. This
projection operator appears both on the left- and right-hand
sides of Eq. 共12兲, making it a cumbersome coupled system of
integrodifferential nonlinear equations.
To simplify the equations of motion 关Eq. 共12兲兴 we recall
that the many-body wave function 关Eq. 共5兲兴 is invariant to
unitary transformations of the orbitals, compensated by “reverse” transformations of the coefficients. Fortunately, there
exists one specific unitary transformation which guarantees
without introducing further constraints that6,7
M
+
␾*k 共r,t兲
M
˙ j典 = P̂ ĥ兩␾ j典 +
P̂i兩␾
where
where the matrix elements of the one-body density matrix
␳kq共t兲 = 具⌿兩ĉ†k ĉq兩⌿典 for fermions29 and for bosons30 are prescribed in a unified manner in Appendix A. It is convenient
to collect these matrix elements as ␳共t兲 = 兵␳kq共t兲其. Similarly,
the reduced two-body density matrix of ⌿共t兲 is given by
冕
=i
j⬘=1
␳kq共t兲␾*k 共r1⬘,t兲␾q共r1,t兲,
= N共N − 1兲
kq
⳵
⳵t
Representation 共10兲 is appealing because the only explicit
dependence on the orbitals 兵␾k共r , t兲其 is grouped into the matrix elements hkq, 共i共⳵ / ⳵t兲兲kq, and Wksql, whereas the elements
␳kq and ␳kslq of the reduced one- and two-body density matrices do not depend explicitly on the orbitals.
Let us now perform a straightforward variation of the
functional action 关Eq. 共7兲兴 with respect to the orbitals. Using
the fact that the 兵␾k共r , t兲其 are orthonormal functions to eliminate the Lagrange multipliers ␮kj共t兲, we obtain the following
set of equations of motion for the time-dependent orbitals in
which the particles reside, j = 1 , . . . , M:
P̂ = 1 −
M
␳共r1,r2兩r1⬘,r2⬘ ;t兲
i
M
⌿*共r1⬘,r2, . . . ,rN ;t兲⌿共r1,r2, . . . ,rN ;t兲
= 具⌿共t兲兩⌿̂†共r1⬘兲⌿̂共r1兲兩⌿共t兲典
兺
k,q=1
,
共11兲
⫻dr2dr3 ¯ drN
=
冉 冊 冉 冊 冕
⳵
⳵
= 兺 ĉ†ĉq i
⳵t k,q k
⳵t
M
P̂ = 1 −
M
兵␳共t兲其−1
兺
jk ␳kslqŴsl兩␾q典
k,s,l,q=1
册
,
共15兲
兺 兩␾ j⬘典具␾ j⬘兩.
j⬘=1
The projector P̂ remaining on the right-hand side of Eq. 共15兲
makes it clear that conditions 关Eq. 共14兲兴 are indeed met at
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154103-4
J. Chem. Phys. 127, 154103 共2007兲
Alon, Streltsov, and Cederbaum
any time throughout the propagation of the orbitals. In practice, the meaning of these conditions is that the temporal
changes of the 兵␾k共r , t兲其 are always orthogonal to the
兵␾k共r , t兲其 themselves. This property introduced by the
MCTDH developers6,7 generally makes the time propagation
of Eq. 共15兲 robust and stable, and can thus be exploited to
maintain accurate propagation results at lower computational
costs.
To complete the derivation, we perform the variation of
Eq. 共7兲 with respect to the coefficients which is easily done
after expressing the expectation value of Ĥ − i共⳵ / ⳵t兲 in a form
which explicitly depends on the 兵Cn共t兲其. The following result
then emerges:
H共t兲C共t兲 = i
冓
⳵C共t兲
,
⳵t
冏
冏
冔
⳵
n⬘,n⬘, . . . ,n⬘M ;t ,
Hnn⬘共t兲 = n1,n2, . . . ,n M ;t Ĥ − i
⳵t 1 2
共16兲
where the vector C共t兲 collects the coefficients 兵Cn共t兲其. The
matrix elements of Ĥ − i共⳵ / ⳵t兲 with respect to two general
configurations 兩n1 , n2 , . . . , n M ; t典 and 兩n1⬘ , n2⬘ , . . . , n⬘M ; t典 are
easily evaluated in the fermionic case with Slater-Condon
rules for determinants31 and with their bosonic analog for
permanents.30 In Appendix B we unite these rules and depict
together the corresponding fermionic and bosonic matrix elements. Finally, making use of conditions 关Eq. 共14兲兴 we obtain the familiar equations of motion for the propagation of
the coefficients,
H共t兲C共t兲 = i
⳵C共t兲
,
⳵t
共17兲
Hnn⬘共t兲 = 具n1,n2, . . . ,n M ;t兩Ĥ兩n1⬘,n2⬘, . . . ,n⬘M ;t典.
The coupled equation sets 关Eq. 共12兲兴 for the orbitals
兵␾ j共r , t兲其 and 关Eq. 共16兲兴 for the expansion coefficients
兵Cn共t兲其, or, respectively, Eqs. 共15兲 and 共17兲 constitute a unified and compact representation of MCTDHF 共Refs. 14–16兲
and MCTDHB 共Refs. 20 and 21兲, i.e., of the multiconfigurational time-dependent Hartree method for systems consisting
of identical particles.
The differences in particle statistics in our derivation explicitly come into play only in the first of the above equations, namely, Eq. 共1兲 for the field operator ⌿̂共r兲. Alternatively speaking, the difference in particle statistics translates
only implicitly in the form of the reduced one- and two-body
density matrices ␳共r1 兩 r1⬘ ; t兲 and ␳共r1 , r2 兩 r1⬘ , r2⬘ ; t兲 and the matrix elements of the generic Hamiltonian 关Eq. 共3兲兴 between
two determinants or two permanents, which are simple to
evaluate with the help of Condon-Slater rules or their
bosonic analog. Comfortably, the corresponding matrix elements for fermions and bosons can also be represented in a
unified manner and are collected for completeness in Appendices A and B. Another appealing point bearing a practical
perspective is that the size of the Hilbert subspace is the
same for fermions and bosons, and only depends on the number of particles N and number of virtual orbitals M ⬘, see Eq.
共6兲. In view of all the above and in principle, one could
algorithmize one efficient multiconfigurational timedependent Hartree code for both fermions and bosons.
The equations of motion for the multiconfigurational
time-dependent Hartree approach for systems consisting of
identical particles become an exact representation of the
time-dependent many-particle Schrödinger equation in the
limit where the number of virtual orbitals M ⬘ goes to infinity.
In practice, one has of course to limit M ⬘, where the employment of time-dependent orbitals which is at the heart of
MCTDH is of great advantage. Still, even with timedependent orbitals the actual size 关Eq. 共6兲兴 of the Hilbert
subspace fastly increases with the number of particles and
number of virtual orbitals employed. Thus, it is instructive to
devise strategies for further approximations atop the multiconfigurational expansions 关Eq. 共5兲兴 utilizing complete Hilbert subspaces, i.e., when all configurations resulting by distributing N particles over M orbitals are explicitly taken into
account.
In principle there are two natural strategies to devise
approximations. The first type of approximations is simply to
limit the number of configurations taken into account in expansion 共5兲. This applies, in particular, to Eqs. 共12兲 and 共16兲.
The second strategy which is directly applicable also to Eqs.
共15兲 and 共17兲 stems from the representation of Eqs. 共12兲 and
共16兲, or Eqs. 共15兲 and 共17兲, in terms of the reduced one- and
two-body density matrices and is based on the reduced density matrices themselves. Specifically, we can replace the
equations of motion 关Eqs. 共16兲 or 共17兲兴 for the coefficients
with equations of motion for the elements of the reduced
density matrices, see in this context Ref. 32, and solve the
latter set directly with Eqs. 共12兲 or 共15兲 for the orbitals. If the
entire hierarchy of equations of motion for the reduced density matrices is included, then no further approximations
have been introduced. However, when truncations of the
coupled equations of motion for the reduced density matrices
are employed, we arrive at a new type of a self-consistentlike propagation schemes. Moreover, we can directly investigate the usefulness of these reduced-density-matrix truncation schemes by comparing to the “exact” solution obtained
within the full multiconfigurational subspace.
Summarizing, we have shown that the multiconfigurational time-dependent Hartree method specified for systems
of identical particles takes on a unified and compact form,
where the equations of motion for the orbitals depend explicitly and solely on the reduced one- and two-body density
matrices of the N-particle system. This representation of the
equations of motion opens up further strategies for approximate self-consistent-like propagation schemes which are directly based on the reduced density matrices of the system’s
many-particle wave function.
ACKNOWLEDGMENTS
This paper is dedicated to Professor Hans-Dieter Meyer
on the occasion of his 60th birthday. Financial support
by the Deutsche Forschungsgemeinschaft is gratefully
acknowledged.
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154103-5
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Multiconfigurational time propagation of identical particles
APPENDIX A: MATRIX ELEMENTS OF THE REDUCED
ONE- AND TWO-BODY DENSITY MATRICES FOR
MULTICONFIGURATIONAL WAVE FUNCTIONS
OF IDENTICAL PARTICLES
In this appendix we prescribe the elements of the reduced one- and two-body density matrices for multiconfigurational fermionic29 and bosonic30 wave functions 兩⌿共t兲典
= 兺nCn共t兲兩n1 , n2 , . . . , n M ; t典 in a unified manner. We remind
that the configuration 兩n1 , n2 , . . . , n M ; t典 stands in the case of
fermions for a determinant and in case of bosons for a permanent. Given an N-particle system, the maximal number of
fermions per orbital is of course p = 1, whereas the maximal
number of bosons per orbital equals the number of particles
in the system, p = N. To represent in a unified manner the
statistics-based restriction on the number of particles in each
orbital, we introduce the notation 关n兴 which stands for
关n兴 = n if 0 艋 n 艋 p and 关n兴 = 0 otherwise. For instance,
关1兴 = 1 both for fermions and bosons, 关2兴 = 2 for systems
comprising N = 2 or more bosons, whereas 关2兴 = 0 for any
fermion system, and so on.
We also need a shorthand notation for a reference
configuration and relevant excited configurations atop. Let
the reference configuration be denoted by 兩n ; t典
Then,
the
= 兩n1 , . . . , nk , . . . , ns , . . . , nl , . . . , nq , . . . , n M ; t典.
configuration
denoted
by
兩nqk ; t典 = 兩n1 , . . . , nk
− 1 , . . . , ns , . . . , nl , . . . , nq + 1 , . . . , n M ; t典 differs from 兩n ; t典
by an excitation of one particle from the kth to the qth
ll
; t典 = 兩n1 , . . . , nk − 2 , . . . , ns , . . . , nl
orbital;
兩nkk
+ 2 , . . . , nq , . . . , n M ; t典 represents excitations of two particles
ql
; t典 = 兩n1 , . . . , nk
from the kth to the lth orbital; 兩nkk
− 2 , . . . , ns , . . . , nl + 1 , . . . , nq + 1 , . . . , n M ; t典 represents excitations of two particles from the kth orbital, one to the qth
ql
; t典
orbital, and another to the lth orbital; and 兩nks
= 兩n1 , . . . , nk − 1 , . . . , ns − 1 , . . . , nl + 1 , . . . , nq + 1 , . . . , n M ; t典 represents excitations of two particles, one from the kth to the
qth orbital and a second particle from the sth to the lth orbital. Note that we do not utilize excitation operators to define the excited configurations atop the reference configuration. Rather, it is convenient for our needs to employ a
nomenclature in which the same ordering of the orbitals
␾1 , ␾2 , . . . , ␾M as in Eq. 共5兲 is kept in all configurations. In
this nomenclature the following states are equivalent:
ql
ql
lq
lq
; t典 ⬅ 兩nsk
; t典 ⬅ 兩nsk
; t典 ⬅ 兩nks
; t典. Finally, we define the
兩nks
“distance” between the kth and qth entries, k ⬍ q, in the conq
nl , nl 苸 n.
figuration 兩n ; t典 as dnkq = 兺l=k+1
With these observations and notations, the elements of
the reduced one-body density matrix ␳共r1 兩 r1⬘ ; t兲 given the
multiconfigurational ansatz 兩⌿共t兲典 = 兺nCn共t兲兩n ; t典 are
␳kk共t兲 = 兺 Cn* Cnnk ,
n
␳kq共t兲 = 兺
n
Cn* Cnq
k
冑nk关nq + 1兴共⫿1兲兵dnkq其,
共A1兲
The elements of the reduced two-body density matrix
␳共r1 , r2 兩 r1⬘ , r2⬘ ; t兲 given the multiconfigurational ansatz
兩⌿共t兲典 = 兺nCn共t兲兩n ; t典 are
␳kkkk = 兺 Cn* Cnnk关nk − 1兴,
n
␳kkkq = 兺 Cn* Cnq关nk − 1兴冑nk关nq + 1兴,
␳kkll = 兺 Cn* Cnll 冑关nk − 1兴nk关nl + 1兴关nl + 2兴,
k ⬍ l,
kk
n
␳kssk = 兺 Cn* Cnnkns,
k ⬍ s,
共A2兲
n
␳kklq = 兺 Cn* Cnql 冑关nk − 1兴nk关nl + 1兴关nq + 1兴,
kk
n
␳kssq = 兺 Cn* Cnqns冑nk关nq + 1兴共⫿1兲兵dn 其,
kq
k ⫽ 兵l ⬍ q其,
s ⫽ 兵k ⬍ q其,
k
n
␳kslq = 兺 Cn* Cnql 冑nkns关nl + 1兴关nq + 1兴
n
⫻
再
ks
kq
sl
k ⬍ s ⬍ l ⬍ q,
共⫿1兲兵dn +dn 其 ,
共⫿1兲
兵dnkq+dnls其
,
k ⬍ l ⬍ s ⬍ q,k ⬍ l ⬍ q ⬍ s.
冎
All other nonvanishing matrix elements can be computed
due to the symmetries of the two-body operator, ␳kslq
*
= ⫿ ␳sklq = ␳skql = ⫿ ␳ksql, and its hermiticity, ␳kslq
= ␳lqks.
APPENDIX B: MATRIX ELEMENTS OF THE ONE-BODY
ĥ − i„⵲ / ⵲t… AND TWO-BODY Ŵ OPERATORS FOR
MULTICONFIGURATIONAL WAVE FUNCTIONS OF
IDENTICAL PARTICLES
In this appendix we present in a unified manner the
Slater-Condon rules for evaluating matrix elements with
determinants31 and their bosonic analog for evaluating matrix elements with permanents.30 In the equations below the
upper sign refers to fermions and the lower sign to bosons.
We use the conventions introduced in Appendix A and
the time-dependent matrix elements of the one- and twobody operators with respect to the orbitals given in Eqs. 共4兲
and 共11兲 of the main text. The nonvanishing matrix elements
of the one-body operator ĥ − i共⳵ / ⳵t兲 follow from
冓 冏 冏 冔 冋 冉 冊册
冋 冉 冊册
冓冏 冏 冔
M
⳵
⳵
n;t ĥ − i
n;t = 兺 nl hll − i
⳵t
⳵
t
l=1
,
ll
⳵ q
⳵
n;t ĥ − i
nk ;t = 冑nk关nq + 1兴 hkq − i
⳵t
⳵t
k ⬍ q,
where here and hereafter the upper sign refers to fermions
and the lower sign to bosons. From the hermiticity of the
*
共t兲.
one-body operator we readily have ␳kq共t兲 = ␳qk
k ⫽ q,
k
n
⫻共⫿1兲
兵dnkq其
,
共B1兲
kq
k ⬍ q,
and the fact that the one-body operator ĥ − i共⳵ / ⳵t兲 is selfadjoint,
冓冏 冏 冔冓 冏 冏冔
n;t ĥ − i
⳵
⳵
n⬘ ;t = n⬘ ;t ĥ − i
n;t
⳵t
⳵t
*
.
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154103-6
J. Chem. Phys. 127, 154103 共2007兲
Alon, Streltsov, and Cederbaum
The nonvanishing matrix elements of the two-body operator Ŵ follow from
M
具n;t兩Ŵ兩n;t典 =
1
2
冉
兺 n j 关n j − 1兴W jjjj +
j=1
冉
冊
M
兺
兵i⫽j其=1
niW ji兵ij其 ,
具n;t兩Ŵ兩nqk ;t典 = 冑nk关nq + 1兴 关nk − 1兴Wkkkq + nqWkqqq
M
kq
+ 共⫿1兲兵dn 其
ll
;t典
具n;t兩Ŵ兩nkk
=
1
2
兺
兵i⫽k,q其=1
冊
niWki兵iq其 ,
k ⬍ q,
冑关nk − 1兴nk关nl + 1兴关nl + 2兴Wkkll,
共B2兲
k ⬍ l,
ql
具n;t兩Ŵ兩nkk
;t典 = 冑关nk − 1兴nk关nl + 1兴关nq + 1兴Wkklq,
k ⫽ 兵l ⬍ q其,
ql
;t典
具n;t兩Ŵ兩nks
= 冑nkns关nl + 1兴关nq + 1兴Wks兵lq其
⫻
再
kq
sl
k ⬍ s ⬍ l ⬍ q,
共⫿1兲兵dn +dn 其 ,
共⫿1兲
兵dnkq+dnls其
,
k ⬍ l ⬍ s ⬍ q,k ⬍ l ⬍ q ⬍ s,
冎
where
Wks兵lq其 = Wkslq ⫿ Wksql ,
共B3兲
and the fact that the two-body operator Ŵ is self-adjoint,
具n;t兩Ŵ兩n⬘ ;t典 = 具n⬘ ;t兩Ŵ兩n;t典* .
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