Order-by-order expansions for intermediate Hamiltonians by the shift technique

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Order-by-order expansions for intermediate Hamiltonians by the shift technique

A. Zaitsevskii

To cite this version:

A. Zaitsevskii. Order-by-order expansions for intermediate Hamiltonians by the shift technique. Jour-

nal de Physique II, EDP Sciences, 1993, 3 (4), pp.435-441. �10.1051/jp2:1993101�. �jpa-00247844�

Classification Physics Abstracts 31.15

Short Communication

Order-by-order expansions for intermediate Hamitonians by the shit technique

A-V- Zaitsevskii

Laboratory of Molecular Structure and Quantum Mechanics, Chemical Department, M. Lomonossov Moscow State University, 119899 Moscow GSP, Russia

(Received

lo December 1992; accepted 16 February

1993)

Abstract The level shift technique is applied to derive rigorous order-by-order intermediate

Hamiltouiau QDPT expansions. The novel approach offers _an insight into the relations between

different intermediate Hamiltonian theories. The direct comparison with the shifted QDPT expansions for conventional effective Hamiltouiau is also presented.

1 Introduction.

Although

the effective Hamiltonian

quasidegenerate perturbation theory (QDPT)

is

usually

considered _{as a}

quite general approximate

method in atomic

physics

and quantum

chemistry

[1, 2], ^its

applicability

to

highly degenerate

systems is

seriously

limited

by

the convergence

difficulties in the presence of intruder _states [3]. ^These difficulties _are

usually

associated with the appearance of small energy denominators in

QDPT expansions;

it _seems, however [4], that

they

stem from the fundamental

properties

of effective Hamiltonians defined

according

to Bloch [5] and des Cloizeaux [6]. ^{To solve the}

problem,

Malrieu _et al. [7] had

proposed

the intermediate effective Hamiltonian

approach

where the

requirements imposed

_on several

"intermediate"

eigenstates

of the effective Hamiltonian had been relaxed. For _a model _space of dimension N, an intermediate Hamiltonian should

provide only

M _< N

eigenvalues coinciding

with those of the total

Hamiltonian,

the

remaining (M-N)

roots

being only

_{more or} less reliable

approximations

for exact _ones. Over the last few _years several different versions of the intermediate Hamiltonian

QDPT

have been

developed. They

fall into two distinct

categories:

I) ^those

deriving

the intermediate Hamiltonian from _some reference Bloch _wave operator assc-

ciated with _a

pre-selected subspace

of the model space

(sc-called

main model

space) [7-9];

it)

those which do not need _{any a}

priori

subdivision of the model _space and _use the intermediate level shift

technique

to eliminate numerical instabilities

[10-12].

436 JOURNAL DE PHYSIQUE II N°4

The methods of the first _group offer the

possibility

to

expand

the intermediate Hamiltonian

as a strict power series in

perturbation

and to reduce

significantly

convergence difficulties in

comparison

with the conventional

QDPT.

Some of them _are able to

reproduce semiquantita- tively

the effective interactions of the intermediate

eigenstates (I.e.

to dress these _states [7,

9]).

Unfortunatelyi

in the _case of

strong coupling

of main model _{states to} intermediate _ones the

QDPT

series _converge

asymptotically

_{or even}

diverge.

It should be also noted that the

rigor-

ous derivation of

working

formulae

(especially

for

non-degenerate

model _spaces [9]) ^{is not} too

transparent.

Recent versions of the shift

technique completely

eliminate the intruder state

problem

and

guarantee

^{the true} convergence of

QDPT

series in rather

complicated

situations.

Moreover,

the

approaches

of this _group have the

advantage

of

conceptual simplicity

and enable to

produce

_a

wide

variety

of intermediate Hamiltonians and to control the

dressing

of intermediate _states in

a

physically

transparent _manner. These

advantages

are offset somewhat

by

the fact that the

resulting expansions

_are

only approximate

_power series in

perturbation

and thus _seem to be unsuitable for the

many-body

treatment.

In the present communication _we undertake to

develop

the

rigorous order-by-order

inter- mediate Hamiltonian

expansions

in the frame of the shift

techniquei introducing

the

auxiliary

Bloch _wave operator ^{associated with} a fixed main model _space. The

analysis

^{of this} _expan-

sion should offer _us

insight regarding

the relations between the _two above mentioned _groups of methods. The choice of the shift parameters ^and convergence

properties

_are ^discussed. To

demonstrate the

generality

and

flexibility

of the level shift

approach,

_we

study

its

ability

to gen- erate the

analogs

of

previously

described intermediate Hamiltonian theories. Furthermore, we

compare

directly

the shifted intermediate Hamiltonian

approach

with the level shift

technique

used to

improve

the convergence of the conventional effective Hamiltonian

QDPT.

2. Basic formalism.

Consider _a quantum system ^described

by

the

time-independent Schr6dinger equation

H

j~~)

₌ E~

j~~) (i)

Suppose

that the Hamiltonian H is

split

into the zerc-order part Ho and the

perturba-

tion V

H=Ho+V (2)

and the zerc-order

problem

Holk)

₌

6klk) (3)

is solved. Let _us divide the functional _space into _two

subspaces,

the

(extended)

model _space

Lp

with the

projector

P and its

orthogonal complement _LQ (outer space) projected by Q

_" I P.

The model space is assumed to be

spanned by

_an

appropriate

set of zerc-order

eigenvectors.

We

are

going

to search for _an intermediate Hamiltonian

ji

_{[1, 7]}

acting

^within Lp and

describing properly

M < dim Lp

eigensolutions

of

equation (I).

We

adopt

the

following

Ansatz for

ji

:

fl

= PHR

(4)

where R is the wave-like

operator obeying

^the intermediate normalization

R ₌ P +

QRP (5)

and

satisfying

the relations

RP i~bm) = i~bm), m =

ii

i

M

(6)

This Ansatz _{appears as a}

straightforward generalization

of the Bloch effective llamiltonian [51

7].

^M

eigenvectors

of

ji

_{should coincide with the}

projections

of the

corresponding

exact

eigenstates

onto Lp.

Introducing

the

orthogonal projector

P onto the

subspace spanned by

these

"good" eigenvectors [Pi§m),

_m

= I,

,

Mi

_{one can} write the basic

equation

of the intermediate Hamiltonian

theory [10-13]

HAP ₌

RjiP

= RHRP

(7)

Substituting

the

partitioning (2)

into

equation (7),

we get

Q (R, Ho]

^P =

Q(VR RVR)P (8)

This

equation

is

obviously

insufficient to define the wave-like operator

unambiguously.

Fol-

lowing ill,

_{12] we} eliminate the redundant

degrees

of freedom

by postulating

Q (R,

_HOI

(P P)

₌

Q(VR

RVR

RS)(P P) (9)

where S is the shift operator.

Equation (9)

is

compatible

with

equation (8)

for _an

arbitrary S;

the choice S

= 0 reduces

equations (8)

and

(9)

to the usual

equation

for the Bloch _wave operator ^Q [5] associated with the model

subspace Lp

Q

_IQ,HOI P ₌

Q(VQ QVQ)P (10)

[14].

^{For that}

follows,

it is convenient to define the M-dimensional main model _space LM E

Lp by

its

projector

M

PM ₌ Jim P

=

£ _(m

_><

_{ml (II)}

V-o

The

subspace

Li

Projected by

dim Lp

Pi

₌ P PM ₌

~

_{Ii >< I(}

₍₁₂₎

I=M+I

is to be referred to _as the intermediate space. Now _we

specify

the shift operator as

S=PISPI=~(i>s;<I( ₍₁₃₎

(s;)

_are fixed real numbers.

Taking

into account that for this choice

Q (R,

_HOI P +

QRS

₌

Q [R, (Ho

+

S)]

P

(14)

we can convert

equations (8, 9)

into the form

Q [R, (Ho

+

S)]

P ₌

Q(VR RVR)

+ RS

(P

PM

(15)

438 JOURNAL DE PHYSIQUE II N°4

As it _was shown earlier

[10,

12], ^similar

equations

are

readily

solved

by combining

the

approximate perturbation expansion

of R with _a

non-perturbative (iterative)

treatment of the

V-dependent projector

P. However, the aim ofthe present work is to

study

the

rigorous (Taylor- type) order-by-order expansion

of the wave-like operator. ^To construct P

perturbatively,

_we

introduce the

auxiliary

Bloch _wave operator _w associated with the main model _space

w = w PM ₌

PM

+

(Q

+

Pi)w PM

QJPM

film)

"

film) (16)

and

expand

this operator as a power series in V

w = PM +

w(~l

₊ w(~) _+..

(17)

where

w(")

are

given by

the well-known recursion formula [14]

n-I

(Pi

+

Q) _(w("~, Hoj

^{PM =}

(Pi

+

Q) Vw~"~~)

₊

£ w~'~vw~"~"~~~

_PM

(18)

i=1

Since P _may be

expressed

in terms of _w and P _as

P = Pw

(wtPw)

^~~

^wtP (19)

(the

verification of this formula is

straightforward),

_{one can} write down the

corresponding expansion

for P

p ₌ p~ +

p(1)

_~

p(2)

_~

p(1)

~

p~(1)

_~

~(i)t

_p

p(2)

~

p~(2)

_~

~(2)t

_{p +}

p~(1)~(i)t

_p

~(i)t~(1)

_{~~~ ~~~j}

Inserting (20)

into

equation (15)

and

setting

R _{= P +} R~~~ + R~~) ₊

, one

readily

obtains the recursion formula

n-I

Q (R("), ^(Ho

⁺

S)j

^{P =}

^Q VR("~~) ~j (R(~)VR("~~~~) R(~)SP("~~))

^P

⁽²¹⁾

k=1

which

yields

the desired

QDPT

series for R. The

perturbation expansion

for

k

_is

immediately

obtained

using equation (4).

3. Discussion.

Now consider the

problem

of

choosing

the shift parameters S; and discuss the convergence of

our

expansions. Taking

into account the subdivision of the model _space it is convenient to rewrite _our zerc-order Hamiltonian in the form

Ho = PM Ho PM + Pi Ho Pi +

QHOQ

=£(m>em<m(+£(I>e;<I(+£(a>ea<a( ₍₂₂₎

m ; _«

Let _us first notice that _once the main model _space is

specified,

_one is

always

able to choose the intermediate _space in such _{a way} that the PM HOPM and

QHOQ

spectra are well

separated

in energy, I-e- the differences

(em eo)

_are not small. On the other

hand,

in

highly degenerate

system ^{it is often}

impossible

to eliminate

quasidegeneracies

_{em ~-} e; and _e;

~- ea with _any

physically

reasonable choice of Ho ^Since the resolvent associated with the Liouvillian _super- operator [15] ^{in the I-h-s- of}

equation (10)

involves _energy denominators

(e; co),

the latter group of

quasidegeneracies

should

yield

ill-defined terms in the

perturbation expansion

of Q and

destroy

its convergence [1, 3]. ^{This is the usual} manifestation of the well-known intruder

state

problem. Similarly,

small _energy differences

(em e;) entering

the Liouvillian in the I-h-s- of

equation (18)

_can

provoke

_{a poor} convergence of the series

(17)

and

(20).

In contrasti _one is

always

able to prevent the _appearance of small _energy differences in the shifted Liouvillian

Q Ill, (Ho

+

S)I

P ₌

L ~ _la

_>

_{(em 6a)}

_<

_{al Rlm}

_><

_ml

+

~ L _la

_{> 16; +}

S>

6«)

<

al Rli

><

il (23)

by

_{a proper} choice ofthe shift parameters s;. Since the

perturbation

V remains

unchanged,

_we

might

expect to

improve

the convergence of the shifted

expansion. Nevertheless,

the intruder

state

problem

is not

completely

avoided. It may be

brought

back into

higher-order

terms via the

expansion

of the

projection

_operator

Pi poorly converging

in the _presence of "main

intermediate~'

quasidegeneracies.

One _can

easily verify

that the ill-defined _{terms may}

generally

appear in

RI"),

n >

2,

and

jil"),

n > 3. We _are

going

to show that the destructive role ofthese terms _may be reduced

by

_a careful choice of the shift.

Assume that Ho is

strictly degenerate

within the main model _space

PM

HOPM

₌ eoPM

(24)

and S is chosen _to lift all the intermediate zerc-order levels to co

S;

_{= co e;}

(25)

The first-order correction takes the form

R(~)

=

Go

VP

Go

+

~

~

(26)

60 _o

440 JOURNAL DE PHYSIQUE II N°4

corresponding

to the conventional "shifted Bk" scheme

[16].

^{R~~) should contain the} terrrts

stemming

from the first-order correction to P.

Although

the

dangerous

denominators do enter

P~~),

_we note that

R~~)SP~~)

= R~~)SPW~~)

= GOV

£

^{~ ~ ~' ~ ~} _VPM

# GOVPIVPM

(27)

; ~° ~i

and arrive at _a

remarkably simple expression R(~)

=

GOVGOVP G(VPVP

₊

G(

VPIVPM

(28)

I-e- ill-defined terms do not appear up to the second order

(third

order for

ji) inclusively.

This result admits _a

simple interpretation. Being sufficiently large

to avoid the intruder state

prob- lem,

the shift values

(25)

_are still small

enough

to _cause instabilities in R due to inaccuracies in P. For

non-degenerate

model _spaces the rule

(25)

^should be

generalized

_{as s; = max}

((em ))

_-e;.

It is

interesting

to note that

an

analogous

choice of the shift

was also advocated in the frame of the

semiperturbative approach involving

the iterative treatment of the

projector

P

[10,

12].

The

expressions (27, 28)

are similar to those

appearing

in the

generalized degenerate

pertur- bation

theory (GDPT)

of Malrieu _et al. [7,

9], although

their derivation is

completely

different.

It _seems that the present

approach essentially

_recovers the GDPT and related theories [9] ^while

the

flexibility

and

physical

transparency of the shift methods

[10-12]

are still retained.

Finally,

it _seems to be instructive to compare the present

approach

with the level shift

technique widely

^{used in} the frame of the conventional effective Hamiltonian

QDPT.

In the latter _case the shift appears as a modification of the zerc-order

problem:

Ho-H[=Ho+S

V - V'= _V S

(29)

While small denominators _are

readily

avoided

by

_{a proper} choice of

(s;),

the

divergence

often survives because of the

corresponding

increase of the numerators of the

higher-order QDPT

corrections

[17].

^{With the}

assumptions (24, 25), agreeing ^perfectly

^{with the} common

practice

of

lifting

all the zero-order model levels to the lowest _one [2,

18],

we

immediately

obtain the

following expressions

for the Bloch _wave operator ^Q:

Q(~)

= GOVP

(30)

Q(~)

=

GOVGOVP G(VPVP

₊

G(V £ _ii

_>

_{(co e;)}

_{< I(}

₍₃₁₎

(see

also

[18]).

^Once the shift is chosen in _a consistent _way, Q(~) coincides with the first-order correction to the wave-like operator

(26).

The second-order formula

(31)

differs from its _coun- terpart ⁱⁿ the intermediate Hamiltonian

theory (28) only by

the last term. In both _cases there should be _no

problem

^{with the} denominators. The numerators in

equation (28)

_are of the _same order

ofmagnitude

_as those in the

corresponding

non-shifted formula. In contrast, if the _energy range

spanned by

the model _space

Lp

is

broad,

the last term in

equation (31) containing

the

differences

(co e;)

_may be _enormous. This situation

frequently

_occurs in

many-body theory

when

complete

model _spaces

(e.g.

valence _spaces in quantum

chemistry)

_are used. As _{a conse-} quence, the second-order effective Hamiltonian associated with the _wave operator

(30)

should deviate

significantly

from the third- and

higher-order approximations. 9n

the other

hand,

the

same operator _may be also considered _{as a} second-order shifted intermediate Hamiltonian. The latter

point

of

view, being formally equivalent

to the former _one, is

physically

_more reasonable because

higher-order

corrections for intermediate Hamiltonian _{are not}

expected

to be _as

large

as for the conventional effective Hamiltonian.

4. Conclusions.

The level shift

technique

is used to

expand

intermediate Hamiltonians _as strict _power series in

perturbation.

^The

resulting

version of the intermediate Hamiltonian

QDPT

^{fills the} _gap ^be-

tween the

previously

formulated level shift

approaches

and the theories

deducing

intermediate Hamiltonians from the reference Bloch _wave operators. The

analysis

of

explicit

second-order wave-like operator

expressions

advocates the

lifting

of zerc-order intermediate levels to main

ones. With this

particular

choice of the

shift,

the present ^method bears strong resemblance to

the

generalized quasidegenerate perturbation theory

of Malrieu _et al. Our results suggest that the shift

technique

offers the

possibility

to

inspect

various intermediate Hamiltonian theories from _a

general

and unified

point

of view.

Being important

for better

understanding

of thec- retical

backgrounds,

the novel

approach provides

also _a

potentially

useful

computational tool;

applications

to several

problems

in quantum

chemistry

_{are now} in _progress.

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