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Mixed-state random phase approximation
Volume
79, number
2
CHEhIICAL PHYSICS
MIXED-STATE
RANDOM
PHASE
Yu DMITRIEV
* , C PEINEL
Rscewed
I8 August 1980
1981
APPROXIMATION
and A STOFF
Sektron Pl~_vrrkder Karl-hlarr-Unlversl:t,
15 April
LIZTIERS
*
Arbeltsgruppe
UI final form 22 January
hfolehml-NhlR. DDR-7010
Letpzrg. GDR
1981
A vanant of the RPA formahsm for a nuxed state 1s proposed I1 pernuts simultaneous coupled npprowmatlons for the states of the mL\ture The resultmg equations have the normal WA form and wthm thx approwmatlon an evaluation of the second-order propertles of the esclted states 1s possible
1 Introduction
Z Procedure
The random phase approxrmatron (RPA) 1s widely used n-r the calculatron of second-order propertres m dynamrc and statrc problems Generally speakmg RPA IS an approxrmatron to the two-partrcle Green’s functron [ I] or to the reduced resolvent operator i( 1.2,~) defined by the kernel
A quantum system m penodrcahy osctllatmg fields IS convemently described m the steady-state formalrsm [I?] where the Bloch-type soluhons of the nonstationary Schrodmger equatron are mtroduced,
(a+(l)a(l’)(H
- E - o)-
b+(2)@‘))
(1)
lhs
appro\rmatron can be obtamed at the first step of the equations-of-motion method or u-r the second order of the trme-dependent Hartree-Fock theory. Here we employ the quasi-energy vanational approach [2] to underhne the steady-state properties of the approxlmatron. We reformulate the vanatronal princrple for a nuxture of steady states and from the perturbational expansion for the mean quasi-energy we get the RPA approxrmatron to the correspondmg average of the reduced resolvents. In thrs case stngulantres and residues of the average resolvent evaluate the transttron densrtres and transrtion energres from the states tn the chosen rmxture. Hence, rn this way we get addrttonal mformatron which can be usea, for example, 111 the second-order properties of a molecule m exctted states
t) = pE(r, t)e-
*(r,
(2) l&(T, t + mT) = q,(r,
0 009-26
14/S l/0000-0000/O
02.50
0 North-Holland
t) .
The perrodrc rn tune functrons trons of the hamrltonran [2]
~~(r-, t) are ergenfunc-
5c(r, t) = H(r) + V(r, t) - I a/at and they can be found dure for the functional
where j?(r, t) rs the density core r&, t)p*(~‘, f’) and
=
T-l
s’dt j-d’
(3)
m a bounded
e(p) = (ST, IX(r, 0P& rr
varratronal
fi(r, t) = c
C&C
the functional
Publishmg
Company
proce-
t>ll (“9, P&7 t) 3
(4)
r,
operator
JC(r, t)~(r,
defined
.t)p*(r’,
0 For a mixture
* Permanent address. Lerungrad State Umverslty. PhysIcal Faculty, Department of TheoretIcal Physics, Petrodvorets, USSR.
rE* ,
of the density
by the
f’)r,_r _ C--t
operators
1) ,
(4) grves the mean quasi-energy
(5)
of the 289
Volume
79. number
nurture.
CHEMICAL
1.
To provide
the orthogonahty
PHYSICS LETTERS
of the steady
states II-I the kernel of (5) non-diagonal lagrangmn multlphers have to be Introduced In special cases, wh~h are analogous to the smgle-determinant HartreeFork dpproxlmatlon, (5) IS Invariant to unitary transformations of the steady states III lts kernel and only
the diagonal pJrt of the lagralgldn nlatrlb. can be taken Into conslderatlon As an example for such a mL\ture we consider a generahzed one-electron timed state gtven by the density operator of the rlelectron system. aO=&Jri(,)(rl
i+,,o-,,.a
f)iq,)!$f)
where ,i+,)(~-~ t) IS the one-particle wrth the Dux-type kernel
density
(6)
operator
c,,&) = J&e-lW’ + Y,,,, elwf For the coefficients X,,,, and YmP a set of equations IS then obtained, wIuch has the customary form [ I] of the normal RPA where the twoelectron matrtx elements are Formalized by the factor (II - l)/(k - 1)
3 Second-order
properties
the mixed-state
RPA
sum
_I-li;(r.
functloris ~~(1. t) (P = 1. . k) dre which satisfy the orthogondllty
b(r
t) (r = I-,
. _
_ P-,,)
IS a trial
density
operator for an equally welghted rmxture of kf/nl(k - 1~)’ steady states. To get the random phase approulmatlon we consider From the variational
zeroth-order
V(r, t) as a small prrturbatlon pnnclple for (4) we get the
Euler equation
(8) w1th
/ZHF(f,)= h(q) + [(I2 - I)l(k - I)] [J(q) - K(q)]
,
where J(r,), K( r [) are the Coulomb and exchange parts bul!t from (7) m zeroth order. Though formally they correspond to a (k - II)-times charged negative Ion, eq. (5) has bounded solutrons whch build a basic set for a perturbational expansion. Followmg the ordmary random phase scheme, the first correctlon IO qE(rl) (JJ = !, -. , k) IS to be taken as a time-dependent hnear combmatlon of the virtual functions
(9) wth 290
states
within
As a result of the RPA calculation we get here the of (1) for the states of the chosen rmxture which
Ce+(l)j(l)
t)~“v(r, t) dr = A,,,
If k > )I. then
of the excited
IS
T
k one-electron the trial functions condltlons
15 April 1981
UT-w
x~@Jx*~
XT@ Y*’
Y'sx*T
YT@ Y+’
(
_
1(80 f?‘(Z)) ’
where w:, XT, Y’ are solutions of the ruled-state RPA elgenvalue problem, X’e Yr IS the direct product of the columns XT and YT, 13(1) IS the column of the transition operators /j,,,,(l) (p = 1, . . k, 112= k + 1, ) with the kernel pmP( 1,l’) = q$( 1) I,$ *( 1’) Now, to be able to separate a contnbutlon of the particular pure state in (10) we have to ldentlfy the excitation energies. Thus can be done by comparmg them with expenmental data or, perturbationally, wtth eucltatlon energes m the frozen-core Hartree-Fock calculatton U-Ithe basis I,$ functions Then for a chosen selectlon of the pole terms III (10) and the correspondmg particle-hole pars m (9), parts of the reduced resolvents for the states III the mtiture are obtamed Obvlously the spectral sum (10) does not contam the pole terms with transltlon energies among the pure states m the chosen mLxture. These terms are equal, have opposite sgn and compensate one another. Hence, they have to be calculated separately and to be added to the mdlvldual pure state resolvents (for example urlthm a truncated configuratlon mteractlon scheme m the basis of yf).
4. The relevance
to the open-shell
WA
problem
There is a relatively simpie way to apply the method Just presented to an open-shell system. We can bulld a unitary mvanant sum (5) of the density operators for the states arising from an open-shell configura-
Volume
79, number
2
CHEMICAL
PHYSICS LETTERS
tron (the Slater dragonal sum as well). In tlus case the operator (10) wluch we get from the RPA procedure wdl be a sum of the reduced resolvent operators for the open-shell states. They can be Identified and separated m the same way as was mentroned above We postpone such a drscussron for the present. A conclusron concernmg the utrhty of the procedure along with a conclusron about the correlatron energy which can be extracted from the mued-state RPA awrut more extensrve consrderatron. Note that its accuracy 1s about
15 April 1981
the same as that of the ordmary normal RPA, but it accounts for the “relaxatron” m transrtron processes and retams the common prcture
References [I] D J. Thouless. Nucl Phys 22 (1961) 78. [2] H Snmbe, Phls Rev. A7 (1973) 2203
291
79, number
2
CHEhIICAL PHYSICS
MIXED-STATE
RANDOM
PHASE
Yu DMITRIEV
* , C PEINEL
Rscewed
I8 August 1980
1981
APPROXIMATION
and A STOFF
Sektron Pl~_vrrkder Karl-hlarr-Unlversl:t,
15 April
LIZTIERS
*
Arbeltsgruppe
UI final form 22 January
hfolehml-NhlR. DDR-7010
Letpzrg. GDR
1981
A vanant of the RPA formahsm for a nuxed state 1s proposed I1 pernuts simultaneous coupled npprowmatlons for the states of the mL\ture The resultmg equations have the normal WA form and wthm thx approwmatlon an evaluation of the second-order propertles of the esclted states 1s possible
1 Introduction
Z Procedure
The random phase approxrmatron (RPA) 1s widely used n-r the calculatron of second-order propertres m dynamrc and statrc problems Generally speakmg RPA IS an approxrmatron to the two-partrcle Green’s functron [ I] or to the reduced resolvent operator i( 1.2,~) defined by the kernel
A quantum system m penodrcahy osctllatmg fields IS convemently described m the steady-state formalrsm [I?] where the Bloch-type soluhons of the nonstationary Schrodmger equatron are mtroduced,
(a+(l)a(l’)(H
- E - o)-
b+(2)@‘))
(1)
lhs
appro\rmatron can be obtamed at the first step of the equations-of-motion method or u-r the second order of the trme-dependent Hartree-Fock theory. Here we employ the quasi-energy vanational approach [2] to underhne the steady-state properties of the approxlmatron. We reformulate the vanatronal princrple for a nuxture of steady states and from the perturbational expansion for the mean quasi-energy we get the RPA approxrmatron to the correspondmg average of the reduced resolvents. In thrs case stngulantres and residues of the average resolvent evaluate the transttron densrtres and transrtion energres from the states tn the chosen rmxture. Hence, rn this way we get addrttonal mformatron which can be usea, for example, 111 the second-order properties of a molecule m exctted states
t) = pE(r, t)e-
*(r,
(2) l&(T, t + mT) = q,(r,
0 009-26
14/S l/0000-0000/O
02.50
0 North-Holland
t) .
The perrodrc rn tune functrons trons of the hamrltonran [2]
~~(r-, t) are ergenfunc-
5c(r, t) = H(r) + V(r, t) - I a/at and they can be found dure for the functional
where j?(r, t) rs the density core r&, t)p*(~‘, f’) and
=
T-l
s’dt j-d’
(3)
m a bounded
e(p) = (ST, IX(r, 0P& rr
varratronal
fi(r, t) = c
C&C
the functional
Publishmg
Company
proce-
t>ll (“9, P&7 t) 3
(4)
r,
operator
JC(r, t)~(r,
defined
.t)p*(r’,
0 For a mixture
* Permanent address. Lerungrad State Umverslty. PhysIcal Faculty, Department of TheoretIcal Physics, Petrodvorets, USSR.
rE* ,
of the density
by the
f’)r,_r _ C--t
operators
1) ,
(4) grves the mean quasi-energy
(5)
of the 289
Volume
79. number
nurture.
CHEMICAL
1.
To provide
the orthogonahty
PHYSICS LETTERS
of the steady
states II-I the kernel of (5) non-diagonal lagrangmn multlphers have to be Introduced In special cases, wh~h are analogous to the smgle-determinant HartreeFork dpproxlmatlon, (5) IS Invariant to unitary transformations of the steady states III lts kernel and only
the diagonal pJrt of the lagralgldn nlatrlb. can be taken Into conslderatlon As an example for such a mL\ture we consider a generahzed one-electron timed state gtven by the density operator of the rlelectron system. aO=&Jri(,)(rl
i+,,o-,,.a
f)iq,)!$f)
where ,i+,)(~-~ t) IS the one-particle wrth the Dux-type kernel
density
(6)
operator
c,,&) = J&e-lW’ + Y,,,, elwf For the coefficients X,,,, and YmP a set of equations IS then obtained, wIuch has the customary form [ I] of the normal RPA where the twoelectron matrtx elements are Formalized by the factor (II - l)/(k - 1)
3 Second-order
properties
the mixed-state
RPA
sum
_I-li;(r.
functloris ~~(1. t) (P = 1. . k) dre which satisfy the orthogondllty
b(r
t) (r = I-,
. _
_ P-,,)
IS a trial
density
operator for an equally welghted rmxture of kf/nl(k - 1~)’ steady states. To get the random phase approulmatlon we consider From the variational
zeroth-order
V(r, t) as a small prrturbatlon pnnclple for (4) we get the
Euler equation
(8) w1th
/ZHF(f,)= h(q) + [(I2 - I)l(k - I)] [J(q) - K(q)]
,
where J(r,), K( r [) are the Coulomb and exchange parts bul!t from (7) m zeroth order. Though formally they correspond to a (k - II)-times charged negative Ion, eq. (5) has bounded solutrons whch build a basic set for a perturbational expansion. Followmg the ordmary random phase scheme, the first correctlon IO qE(rl) (JJ = !, -. , k) IS to be taken as a time-dependent hnear combmatlon of the virtual functions
(9) wth 290
states
within
As a result of the RPA calculation we get here the of (1) for the states of the chosen rmxture which
Ce+(l)j(l)
t)~“v(r, t) dr = A,,,
If k > )I. then
of the excited
IS
T
k one-electron the trial functions condltlons
15 April 1981
UT-w
x~@Jx*~
XT@ Y*’
Y'sx*T
YT@ Y+’
(
_
1(80 f?‘(Z)) ’
where w:, XT, Y’ are solutions of the ruled-state RPA elgenvalue problem, X’e Yr IS the direct product of the columns XT and YT, 13(1) IS the column of the transition operators /j,,,,(l) (p = 1, . . k, 112= k + 1, ) with the kernel pmP( 1,l’) = q$( 1) I,$ *( 1’) Now, to be able to separate a contnbutlon of the particular pure state in (10) we have to ldentlfy the excitation energies. Thus can be done by comparmg them with expenmental data or, perturbationally, wtth eucltatlon energes m the frozen-core Hartree-Fock calculatton U-Ithe basis I,$ functions Then for a chosen selectlon of the pole terms III (10) and the correspondmg particle-hole pars m (9), parts of the reduced resolvents for the states III the mtiture are obtamed Obvlously the spectral sum (10) does not contam the pole terms with transltlon energies among the pure states m the chosen mLxture. These terms are equal, have opposite sgn and compensate one another. Hence, they have to be calculated separately and to be added to the mdlvldual pure state resolvents (for example urlthm a truncated configuratlon mteractlon scheme m the basis of yf).
4. The relevance
to the open-shell
WA
problem
There is a relatively simpie way to apply the method Just presented to an open-shell system. We can bulld a unitary mvanant sum (5) of the density operators for the states arising from an open-shell configura-
Volume
79, number
2
CHEMICAL
PHYSICS LETTERS
tron (the Slater dragonal sum as well). In tlus case the operator (10) wluch we get from the RPA procedure wdl be a sum of the reduced resolvent operators for the open-shell states. They can be Identified and separated m the same way as was mentroned above We postpone such a drscussron for the present. A conclusron concernmg the utrhty of the procedure along with a conclusron about the correlatron energy which can be extracted from the mued-state RPA awrut more extensrve consrderatron. Note that its accuracy 1s about
15 April 1981
the same as that of the ordmary normal RPA, but it accounts for the “relaxatron” m transrtron processes and retams the common prcture
References [I] D J. Thouless. Nucl Phys 22 (1961) 78. [2] H Snmbe, Phls Rev. A7 (1973) 2203
291