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Submitted on 1 Jan 1995
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On Some Attempts to Generalize the Effective
Hamiltonian Approach
Jean-Louis Heully, Stefano Evangelisti, Philippe Durand
To cite this version:
Jean-Louis Heully, Stefano Evangelisti, Philippe Durand. On Some Attempts to Generalize the
Effective Hamiltonian Approach.
Journal de Physique II, EDP Sciences, 1995, 5 (1), pp.63-74.
Classification
Physics
Abstracts31 15 31.20 33.10
On
Some
Attempts
to
Generalize
the
Elllective
Hamiltonian
Approach
J. L.
Heully (~),
S.Evangelisti
(2)
and Ph. Durand(~)
(~) Laboratoire de
Physique
Quantique
(*)
Universitd PaulSabatier,
118 route de Narbonne 31062 ToulouseCedex,
France(~)
Dipartimento
di Chimica Fisica eInorganica,
Universita diBologna,
VialeRisorgimento
4,40136
Bologna, Italy
(Received
18July
1994, received in final form lo October 1994,accepted
17 October1994)
Abstract, A clear and
simple
method to derive various intermediate Hamiltonians ispro-posed.
Several intermediate Hamiltonians usable for various purposesare
presented.
The newintermediate Hamiltonians have the same
properties
as the traditional effective Hamiltoniansbut
they
should present better convergence properties, and should avoid any intruder stateproblem
P'1. Introduction
Until
recently,
thepredominant
correlation methods:many-body
perturbation
theory
(MBPT)
11, 2]
configuration-interaction
(CI)
[3] andcoupled-cluster
theories(CC)
[4] wereapplied
almostexclusively
assingle-reference
basedprocedures.
This is due inpart
to the increasedcomputational
complexity
of the multireference-basedprocedures.
For MBPT and CC one more reason is therequirement
on the reference space to becomplete
yielding
ratherlarge
reference spaces.Although
one-dimensional MBPT or CC have been shown toprovide
highly
accurateener-gies
andspectroscopic
parameters,
they
are notuniversally
applicable,
particulary
when exact or neardegeneracies
are encountered.They
areusually
insufficient forpotential
surfacesin-volving
the dissociation ofmultiple
bounds,
andthey
are notordinarily
applicable
to excitedstates other than the lowest ones of each
symmetry.
Arigourous
treatment of suchproblems
necessitates a multidimensional
procedure.
Quasidegenerate
perturbation
theory
(QDPT)
[5] has beenreceiving
increasing
attention in recent years. Itprovides
theperturbational
ana-logue
of multireferenceconfiguration-interaction
techniques
(MRCI
orCIPSI)
[6] and it may beexpected
toprovide
faster convergence and moregeneral
applicability
thansingle
referenceperturbation
expansions.
The earliestapplications
ofQDPT
ii,
8](and
its CCanalogues)
were all based oncomplete
active spaces orcomplete
model reference spaces. Such a reference space(*)U.R-A.
505 du C.N-R-S.is a space where all
configurations
that can begenerated by
distributing
the valence electrons among the valence orbitals are included. In many cases such a model space may beunneces-sarily large
and leads tocomputational
difficulties. To face these difficultiesincomplete
modelspaces have been advocated
[9-13],
whereonly
the moststrongly
interacting
configurations
are included and the
remaining
ones are treatedby
perturbation.
Such a reference space has a lowercomputational
requirement
and could reduce the intruder-stateproblem,
when statesoriginating
from the model space interferestrongly
with states of thecomplementary
space. TheQDPT
methods face a dilemma: very often alarge
model space isrequired
torepresent
correctly
the zeroth-order functions and at the same time a small model space is needed toavoid the intruder-state
problem
and savecomputational
timeAlthough
theincomplete
model spaceapproach
isobtaining
several successes it leads to a reformulation of thetheory
and stillrequires
alarge
computational
effort.A method which should solve the above
problems
has beenproposed by
Malrieu et al.[14]
They
claimed that themajor
difficulties withQDPT
(and
CCanalogues)
do not come from the size of the model space but from the conditionimposed
to themethods,
ma the effectiveHamiltonian,
togive
exact results for all the states of the model space. A much more reasonableapproach
is to pursueonly
a subset of truesolutions,
the other onesbeing
approximated
orneglected.
This also leads to a newwave-operator,
the sc-called intermediate waveoperator.
Because some conditions on the effective Hamiltonian are relaxed the intermediate Hamiltonian is not
uniquely
defined and several different forms can be formulated.One
particular
form has been used several timesby
our group with verypromising
results in difficult cases[15-17].
Anotherapplication
with another intermediate Hamiltonian has beenpublished
[18].
Intermediate Hamiltonian schemes inFock-space,
based on theopen-shell
coupled-cluster
formalism,
have beenproposed
by
Mukherjee
[19]
and Koch[20].
However,
noexhaustive theoretical
presentation
of the intermediate Hamiltonianconcept
has beenpresented
up to now. This is thus the aim of this article: to
give
a better theoretical foundation to the intermediate Hamiltonians and to present different forms ~vhich can beadapted
for differentproblems.
An effort is made togeneralize
the traditionalQDPT
to theconcept,
keeping
as much aspossible
of the structure of the standardapproach.
Obviously,
theproblem
is verycomplicated
in its fullgenerality,
but it is found that newequations
can bederived,
whichshow
great
similarity
with the standardequations.
Only
the formalaspects
of the method willbe discussed here. We expect to discuss their
many-body
realization in terms ofdiagrammatic
expansions
in future contributions.Very recently
we received apreprint by
Meissner et al. [21] whose purpose is also togeneralize
the
approach
of Malrieu. Their method is based on effective Hamiltonians whereas ours isbased on wave
operators
but the results are very similar.However,
to avoid the intruderstate
problem
Meissner et al. use a new iterativeprocedure
which is not considered in thiswork. Our
opinion
is that this new scheme shouldgive
excellent convergenceproperties
to theintermediate Hamiltonian
perturbation
theories.2. Basic Formalism
2.i. THE MODEL SPACE. The
concept
of model spaceplays
a central role in thetheory
ofeffective Hamiltonians. It is a finite d-dimensional
subspace
Sm
of the entire Hilbert space. Itsorthogonal
complement
is the outer spaceS(.
Theorthogonal
projection
operators associatedwith
Sm
andS(
arePm
andQm, respectively:
d2.2. THE WAVE OPERATOR AND THE EFFECTIVE HAMILTONIAN. A wave operator fi is
defined
by
~§~ =
Q~§(
(a
=1,
2...d)
(2)
where ~§~ are exact solutions of the
Schr0dinger
equation
and ~§( are the
corresponding
zeroth-orderfunctions,
confined to the model space. The latter functions areeigenfunctions
of an effective HamiltonianHeR@~I "
E~~§I,
(4)
the
eigenvalues
being
thecorresponding
exactenergies.
By
combining
equations
(2), (3)
and(4),
the basicequation
relating
H~~
and Q can be derivedHa
=
QH~I
is)
Many
effective Hamiltonians can be derived fromequation
is).
Thesimplest
way is obtainedby
multiplying
equation
is)
by
Pm
on the left and on theright
H~i
=(PmQPm)~~PmHQPm
(6)
If,
asusually done,
the intermediate normalization isadopted,
PmQPm
=
Pm,
(7)
equation
(6)
yields
the usual Bloch effective Hamiltonian [22]H~i
=
Pr~HQPr~
(8)
The intermediate normalization has the
disadvantage,
among many others[22, 23],
that the effective Hamiltonian(8)
is non-Hermitian and thecorresponding
zeroth-order wavefunctions~§a p ~§a
(~)
0 n1
are
non-orthogonal.
This can be avoided
by using
other normalizations of the zeroth-order functions.However,
the connection between different normalizations is rather well-documented[24, 25]
and we shallonly
consider the intermediatenormalization,
thusworking
in Bloch's scheme.With this normalization it can be shown that the
wave-operator
satisfies thegeneralized
Blochequation
[26]in,
Ho)Pr~
=(VQ
QPr~VQ)Pr~
(io)
where the Hamiltonian has been
partitioned
into a zeroth-order HamiltonianHo
and apertur-bation V.
This
equation
can beconveniently
used togenerate
aperturbative
expansion
of theoperator
3. Intermediate Hamiltonians and Intermediate
Wave-Operators
To the
previous
model spaceSr~
considered in Section2,
we now add a secondsubspace,
the socalled intermediate space noted
S,,
containing
all theconfigurations
strongly
interacting
withSr~but
which are of no direct interest. The sum ofSr~
andS,
defines our new model space ofdimension D. We shall not pursue all the D states
emerging
from the model space butonly
arestricted number d < D states
corresponding
to the main model spaceSr~.
We thus have twonew
projectors
P
=
Pr~
+Pi
and = i P(ii)
is the
projector
on the new outer space. In order tokeep
thepresentation
clear,
we shallfollow the same scheme as in the
preceding
paragraph.
An intermediate waveoperator
R,
isdefined such that
~§~ =
Rja
(a
=
1,
2...d)
(12)
where the ~§~ are d exact solutions of the
Schr0dinger
equation
andja
thecorresponding
zeroth-order functions
belonging
to the model space These functions areeigenfunctions
of anintermediate Hamiltonian:
HintP
"
E~p
a #1,
2...d(13)
It should be noted that
Hint
acts in a space of dimension D > d.The
id's
aredefined,
as in the normalQDPT,
as theprojections
of the exact functions ~§~onto the model space.
ja
=
P~§~
a =1,
2...d(14)
We are now
looking
for a relation similar to the relation(7)
which was necessary for the derivation ofH~i
(8)
and theequation
for Q(io).
Unfortunately
such a relation cannot beobtained
directly
because theid's
span a space of dimension D and we haveonly
d relationsto define them.
By
combining
equations
(12)
and(14)
we obtain the newgeneral
relationRP~§~
= ~§~ a =
1, 2...d,
(is)
By using
equation
(2),
the aboveequation
can be transformed into the basicequation
RPQPm
=
QPm.
(16)
This
operator
equation
is true when it acts on theright
in the main model space withprojector
Pm.
This relation can be rewritten asRPr~
+RPzQPr~
=QPr~
(ii)
RPr~
=(i
RP,)QPr~
(18)
This relation is the fundamental relation that will be used in this paper for
deriving
interme-diate Hamiltonians. The relation does not
give
all thepossible
intermediate Hamiltoniansbut,
as we shallshow,
gives
alarge
class ofHint
closely
related to the normalQDPT.
Inequation
(ii),
theright
hand sideQPr~
iscompletely
defined and thus if wegive
apriori
aparticular
form to
RPz,
RPr~
iscompletely
definedby
equation
(ii).
We shall continue to define Rby
projecting
equation
(16)
on the left onto the various parts of P and then onQ.
Theprojection
onPr~
gives
Pr~RPr~
+Pr~RPzQPr~
=Pr~
(19)
and it seems convenient to choose
Pr~RPz
= o m order to havePr~RPr~
multiplying
equation
(16)
on the leftby
Pi
gives
P,RPr~
+P,RP,QPr~
=P,QPr~
(21)
As in the
previous
case, it seems convenient to choosePzRPr~
= oyielding
PzRP~
= P~(22)
These two conditions can be
gathered
into the normalization condition PRP= P
(23)
In the normal
QDPT,
this relation is a direct consequence of relation(9)
whereas in our case it is notcompulsory.
Multiplying
both sides ofequation
(ii)
on the leftby
theprojector
on the outer spacegives
QRPr~
+QRPzQPr~
=QQPr~
(24)
where
tiRPz
has to be defined in some way.The choice of
QRP~
can be based on severalarguments
such as:.
computational
difficulties;
.
Hermiticity
of theresulting
Hint
. inclusion of some correlation to accelerate the convergence of
Hintl
. inclusion of some correlation in order to have the non-exact intermediate
energies
E~ and intermediate functionsR~§i
not too far from the exact ones...There are also other
arguments
such as thesize-consistency
ofHint
butthey
are much moredifficult to
specify
In the nextparagraph
we shallanalyze
andpresent
differentpossible
choicesfor R and the
resulting
H]~~s.
In the conventional
QDPT
the effective Hamiltonian is derived from the Blochequation
(5)
which is a natural result of
equations
(2-4).
Such anequation
cannot be obtained in our case,at least if we want to
keep
equations
defined on aprojector
P built from theunperturbed
projectors
Pr~
andPi.
Of course, an
equation
similar toequation
(5)
can be stated with thehelp
of theperturbed
projector
PM
"~
@11@~
HRiM
=RH,ntPM
a =1, 2,
...dPi
= PPM
(25)
The
j~~'s
are thebiorthogonal
of thej[s.
But the use of the unknownprojectors
PM
andPi
leads to a
quite
different formulation[20,
28].
We shallpostulate
the existence of a pseudo-Bloch effective HamiltonianHint
= PHRP.(26)
It is easy to see that this
operator
gives
d exacteigenvalues
and deigenvectors
which are theprojections
upon the model space of the exact vectors. On the otherhand,
nothing
can be said about the other solutions. This effective Hamiltonian cannot be derived from theequation
HRli'
=
RHintP
(27)
4. Different Choices of R
We have seen that with various choices of
QRPz
thepseudowave
operator
R is solution ofequations
(16, 17).
We shall look at different choices fortiRPz.
4.I. THE MINIMAL CHOICE. The most obvious and
simplest
choice forQRP~
inequation
(24)
isQRP~
= o
giving
QRP
=
QQPr~
(28)
and thus since PRP
= P
R = P +
QQPr~.
(29)
Multiplying
equation
(28)
on the left then the sameequation
on theright by
Ho
and thensubtracting
bothequations
andusing
equation
(io)
leads toQ[R,
Ho)Pm
=Q[Q, Ho)Pm
=Q(VQ
QVQ)Pm
(30)
and
Hint
=
PHP +
PVQQPm
(31)
This Hamiltonian is
strongly
non-Hermitian because theperturbation
actsonly,
on theright,
in the main model space.4.1.1. Perturbational 7Featment If this Intermediate Hamiltonian is built
by
perturbation
care must be taken in the choice of the intermediate space. If we assume that acomplete
setof
unperturbed
orthonormaleigenfunctions
(#~)
andcorresponding
eigenvalues
are availablejf
ja
j~aja
ja
jb)
§j~~)
0 a,b,
then the first and second-order
expressions
for the Roperator
aretiR(~~Pm
= ~~
VPm
(33a)
oHo
tiR(~~Pm=
~
~~~
+~~~~
~
PmVj
Pm
(33b)
Eo
Ho
Eo
Ho
Eo
Ho
Eo
Ho
In the two above
expressions
it has been assumed thatHo
isdegenerate
in the main modelspace with all its
eigenvalues
equal
toEo.
For the first-order term nospecial
problem
can occur but at the second-order somedangerous
terms appear. The second term in theright
hand side of
equation
(33b)
gives
large
contributions if the intermediateconfigurations
arestrongly coupled
to those of the main model space. This will make theperturbation
expansion
quite
unreliable and thus this intermediate Hamiltonian should be usedonly
to second-order.4.1.2. Alternative ~eatment. In order to avoid the
problem
of thecoupling
between the main and the intermediate space, one should isolate thePzQPr~
part
of Q. Let us rewritea
=
p,+p~ap,+x
=
Pap,
+ X(34)
With this notation
equation
(30)
can be written asQ(R,
Ho)Pr~
The second term P Q
Pm,
which could bedivergent
with aperturbational
treatment,
can be obtainedby
diagonalizing
the intermediate Harniltonian. With the d exact zeroth-ordereigenfunctions
and theirbiorthogonals
we can obtainPzQPm
as dPzaPm
=
~j
Pi
il~)(ill~
Pm
(36)
a=i
In this way the most
dangerous
terms are treated to infinite order whereas the normal termsare included
by
usualperturbational
methods and thus can beapproximated
at a certain order.Consider,
forinstance,
a model space ofsingle
and doubleconfiguration
interaction(SDCI)
type
where the fundamental state defines the main model space and allsingle
and double excitations define the intermediate space. In such a caseequation
(30)
gives
R°
= P H)/~~ = PVP =AESDCI
Rl~~
= oH)j/
= oR(~)
= ~~
~V~
~
~VPm
H)11
=PV~
~
V~
VPr~
(37)
o o o o oHo
Eo
Ho
where the
projector
ti
isgenerated by triple
andquadruple
excitations andPi
by single
and double excitations. The fact thatR(~)
is zero shows that the SDCI energy is correct tothird-order. The term
R(2)
shows that the SDCI wave function should be corrected at secondorder,
the effective Hamiltonian at third order and hence the energy(diagonalized)
at fourth order.By
using
equation
(35)
instead ofequation
(30)
somehigher
orders caneasily
be included.4.2. AN EXTENDED TREATMENT. The
previous
choice ofQRPz
provided
a verysimple
intermediate Hamiltonian but it could be more
useful,
in view ofbuilding
more HermitianH]~~s
orhaving
transferable effectiveinteractions,
to include some interactions between theintermediate states and the outer states, hence
choosing
QRPz #
o.Equation
(24)
can berecast in commutator form. This is very convenient since it ensures the invariance of the
wave-operator
upon a translation in energy of thediagonal
energies
ofHo-Q[R,
Ho)Pr~
=Q[Q, Ho)Pr~
Q[RPzQ, Ho)Pr~
(38)
OnceQRPz
has beenchosen,
QRPm
is determined in aunique
wayby equation
(38).
We shallpresent
two ways ofchosing
theQRPz
part.
The first one willyield
a neworiginal
intermediateHamiltonian,
the second one willgeneralize
theapproach
proposed by
Malrieu et al. in their firstpresentation
of the intermediate Hamiltonianapproach.
a)
In the samespirit,
as in theprevious
paragraph,
we shall isolate the termscontaining
PzQPm.
For this purpose,by
using equation
(io),
equation
(38)
can be written in the formQ[R, Ho)Pm
=Q(V0
0Pr~V0)Pm
+Q(VPzQ
QPmVPzQ)Pm
-QRP~
in,
Ho)Pr~
Q(RPz, Ho)QPr~
= A + B + C + D(39)
where
a
=(p,
+Q)a
(40)
A =Q(Vfl
0Pr~Vfl)Pr~
B =Q(VPzQ
fIPmVPzQ)Pm
C =-QRPz[Q,Ho)Pm
D =-Q[RP~,Ho]QPr~
One
particular
solution is to putQ[R, Ho)Pr~
= A + C(41)
which
implies
that the condition B + D = o must also be satisfied.By
this way,equation
(39)
is transformed into a system of two
equations
that will allow aunique
definition ofQRPr~
andQRP~.
Q[R, Ho)Pr~
=Q(Vfl
fIPr~VQ)Pr~
tiRP~[Q,
Ho)Pr~
Q[R,Ho)P~QPr~
=
Q(V-fIPr~V)P~QPr~
(42)
The second
equation
(42)
does not defineQRP~
in aunique
way. This can be doneby
trans-forming (42)
into:Q[R, Ho)Pr~
=Q(V0
OPr~VOPr~
QRPz[Q, Ho)Pr~
Q[R,Ho)Pz
=
Q(V-fIPr~V)P~
(43)
It can
immediately
be checked that(43)
implies
(42)
by
multiplying
both sides of the secondequation
(42)
on theright
by QPr~.Up
to first orderR(~)
isequivalent
to the waveoperator
given
by
the standardQDPT
on the whole model space. This means that the second-orderHamiltonians are also
equivalent
and that information on the intermediate statesbegins
to belost at the third-order
only.
Unfortunately,
in the case of intruder states, theseequations
haveonly
a formal interest because the secondequation
(43)
willdiverge
in the same way as theQDPT
would do.However,
it ispossible
to define a new zeroth-order Hamiltonian such asho
=
Ho
+Pi WPz
=
(Pn~
+Q)Ho
+~j
ii
i) fi
(44)
where W is a
diagonal
operator
changing
theenergies
of the intermediate space.Q[R,
Ho)Pr~
=Q(V0
flPmvfl)Pm
QRP~[Q,
fro]Pm
Q(R,fro)P~
=
Q(V-flPmV)P~
(45)
V in
equation
(45)
has not been modifiedby
equation
(44).
It should be noted that W doesnot act as a
shifting
operator
because it does notchange
thepartitionning
of H intoHo
and V in the main model space. No
particular
care isrequired
in the choice of W and thus any Wavoiding
the intruder states would be convenient.However,
the final results for theintermediate states and
only
for them woulddepend
strongly
on this choice. Beforelooking
atsome
particular
cases we would like to summarize ourapproach.
I)
we have defined apseudo
waveoperator
Racting
in a model spacecontaining
the main and intermediate states[Eq.
(12)];
it)
a wave operator Q has been introduced to discriminate between the main and intermediatestates
[Eq.
(16)];
iii)
TheRP~
part of R which is apriori
arbitrary
in thetheory
of intermediate Hamiltonians is chosen in such a way that it cancels thedangerous
terms ofequation
(39)
associated withthe interaction between the intermediate states and the model
state;
iv)
aspecial
shift on the intermediate states is introduced in order to avoid the intruderFurther
simplifications
are obtained if the main model space isdegenerate
to the valueEo.
In this case we have:
QRPm
=~
((Vfl
flPmV0)Pm
~j(Eo
fz)R
i)(i
QPm)
Eo
Ho
~Q[R,
ko)P~
=Q(V
fIPr~V)Pz
(46)
If W is chosen in such a way that all intermediate
energies
Ez
are moved to the energyEo
of the main model space,expression
(46)
reduces toQRPr~
=~
(V0
0Pr~V0)Pr~
Eo
Ho
QRPz
=~
(V
fIPr~V)P~
(47)
Eo
Ho
The intermediate Hamiltonian obtained from this wave
operator
is Hermitian up to thesecond-order and should
give
much better results than theprevious
H,nt.
Furthermore,
if one is concerned with the construction of transferable effectiveHamiltonians,
such an intermediateHamiltonian is well-suited because it ensures a
continuity
of the effective operators around avoided crossings.If we take
again
theexample
of an SDCI model space withonly
theground
state in the mainmodel space, the
expansions
of R andH,nt
are thefollowing:
R(°)
= PH)11
= PVPR~~)Pr~
=o;
R(~)P~
= ~~
~VPz
H)j/
= PV ~~
~VPz
o o o.- o R~~) = oH)11
= o(48)
Already
at second-orderH,nt
contains corrections but the next corrections appearonly
at thefourth-order. One can see in this
example
that thisH,nt
istotally
different from theprevious
one.Though
in both cases one includes corrections from the tri- andquadri-excitations,
their numbers arequite
different;
D in theprevious
case,D(D
i)
in this case.However,
at firstorder,
Hint
is in both cases a very sparse matrix where most of the terms are zero. Thecorrections towards the second order will fill up the whole matrix
representing
Hint
which could be inconvenient.b)
In both schemespresented
above,
amajor
defect was that the convergence rate of R wassolely
determinedby
Q.Knowing
that standardQDPT
has a very low(if any)
convergencerate, this is a bad feature which should be avoided as much as
possible.
For this purpose wereturn to
equation
(38)
and useequation
(30).
Q[R, Ho)Pr~
=Q(V
QQPr~VQ)Pr~
Q(RPzQ, Ho)Pr~
(49)
In the first term on the
right-hand
side ofequation
(49)
some termsQPr~
can bereplaced
by
RPr~
+RP~QPr~
(see
Eq.
(17)).
By
this means two results can be obtained.
equation
(49)
becomesapproximately
anequation
giving
R as a function ofR;
this should accelerate the convergence;We shall
apply
this method to the first and thirdQPr~
inequation
(49)
and the other cases will be discussed later on.This transformation
yields
Q[R,
Ho]Pr~
=Q(VR
QPr~VR)Pr~
+Q(VRPz
QPr~VRPz)QPr~
-QRPz[Q,
Ho)Pr~
ti(RPz,
HOIQPr~.
(So)
We can see that this
equation,
made up of fourterms,
has the same structure asequation
(39)
of the
previous
paragraph.
We shalladopt
the same method aspreviously by
adding
the first and third terms andby
cancelling
the second and fourth ones.Q(R,Ho)Pr~
=Q(VR-QPr~VR)Pr~-QRPz[Q,Ho)Pr~
Q[R,Ho]P~
=Q(VR-QPr~VR)Pz
(51)
We have
again
asystem
of twoequations
giving QRPr~
andQRPz
as a function of R. However the left-hand sides of theseequations
are stillsuffering
from the intruder-stateproblem
in thesecond
equation
(37)
and hence the same shift method used in theprevious
paragraph
hasto be
applied.
The introduction of the transformed zero-order Hamiltonian ofequation
(44)
allows us to pass from
equation
(51)
toQ[R, ko)Pr~
=
Q(VR
QPr~VR)Pr~
RPz
in, ko)Pr~
Q[R,ko)P~
=
Q(VR-QPr~VR)P~
(52)
This system represents the most direct
generalization
of the intermediate Hamiltonian schemepresented
by
Malrieu et al. which assumeddegeneracy
within the main model space. It is valid for any model space,degenerate
or not, in the main or intermediate model space. In thesimplest
case, when the main model space isdegenerate,
one canalways
choose the shiftoperator
W in such a way that all the intermediate states becomedegenerate
with the mainmodel space. In such a case
RP~[Q,
ko)Pm
= o and the twoequations
(52)
can bemerged
intothe
following
equationQRP
=
~
(VR
QPmVR)P
(53)
Eo
Ho
where
Eo
is the energy in the main model space. This is the relationproposed
in theoriginal
formulation of thetheory by
Malrieu et al. It was derived in a very different way andEo
waspresented
as a freeparamete~
whereas in our formulationEo
is the energy of the main modelspace.
If we return to our SDCI
example
we have with thissystem
ofequations
thefollowing
expansion
R~°)
= PH~~)
= PVP =AESDCI
R~~)=
~
VPz
H~$~=PV
~
VPz
Eo
Ho
~~'Eo
Ho
R~~)=
~
V~
VP~
H~$~=PV
~
V~
VPz
(54)
Eo
Ho
Eo
Ho
'~' Eo Ho Eo HoComparison
withequation
(48)
shows that this schemegives
a correction at third-order on theintermediate
subspace
of the intermediate Hamiltonian.Let us return to the most
general
case, in which the transformation ofQPm
inequation
(49)
can beapplied
to anyQPr~.
This leads to severalsystems
ofequations
forQRPr~
andQRPz
looking
likeequations
(51)
or(53).
All areequivalent
at first order but differ in third or fourthorder. When the main model space is
degenerate
one can use the verygeneral
equation
QRP
=
~
(VX
YVZ)
(55)
Eo
Ho
where two of the three operators
X,
Y and Z can be either Q or R.5. Conclusion
This work
presents
several classes of effectiveoperators
whichgeneralize
thewave-operator
andeffective Hamiltonian of the traditional
QDPT.
We havekept
the Bloch formalism as much aspossible
while the usual condition that all the solutions of theH~i
have to be exact has beenrelaxed. Since there are several ways to
get
approximate
solutions,
there are manypossibilities
forconstructing
intermediate Hamiltonians. We havepresented
asystematic
method to obtainintermediate Hamiltonians
ranging
from a verysimple
and economical one where almost no corrections are introduced for the intermediatestates,
to moresophisticated
ones, one of thembeing
theoriginal
H,nt
Proposed by
Malrieu et al.[14].
Concerning
theapplication
of the intermediate Hamiltonian toQuantum
Chemistry
one should admit that these intermediate Hamiltonians haveonly
a formal or mathematical interest. Onthe one
hand,
the use ofH,nt
tohigh
perturbational
order wouldimply
the same amount of work as asimple
diagonalization
andsurely
will not be faster than a standard Davidson scheme.On the other hand the use of the second or third order
H,nt
inmany-body problems
would not result in anyimprovement
because none of thepresented
Hint
have a correctsize-consistency
or areseparable.
TraditionalQDPT
wasseparable
However,
Malrieu et al. have shownthat,
if one leaves the orderby
orderconcept
andinstead,
for agiven order,
includes a littlepart
of the next
order,
size-consistency
could be restored. This will bepublished
in aforthcoming
paper.
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