- Email: [email protected]
Quasi-degenerate rayleigh-schrödinger perturbation theory with hermitean model hamiltonian
Vo!“me
32, numtier 3’
..” ‘.
..
_, ‘.
‘,
CHEMICAI’PNYSICS LETTERS
: ,, ._ “.
.
_,
.. :..,
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‘.
1 May l.975
‘.
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A methbd for the construction of ;h& hermitaan’tiodel hakiltonian in &a framework of thequasidegenerate Rayleigh@ertuxbationtheory is kuggesterj. The approach of a model hamiltonian is bati on the assunption that ifit
SchCdinpr is ~kqmE.=d
in a chosen fmitedimension~
model space it will yield eigenvalues of the origin&l
hzmiltonian
in the entire
space, For a unitary operator tra’nsformingihe unperturbti~stite vectors onto the prturbed rtite vectors, md P model-interaction operator, the pcrh\rbo~oncxp3nsionformSe are developed. H&ert
The calculation of the eigenvaIu6s of a hamilt~~~ which describe some’molecular system, is one of the basic prdblems of quantuq chemistry. If we attempt to solve this problem using the standard Rayleighh-Ritz variational method, then we meet some difficulties originating from the diagonalization of a hermitean matrix ofenotious dimensiondity. An alternative approach fir overcoming these difficulties is the perturbation theory, namely, its idea of a model hamiltonian defined in a unite-Dimensions model space. The approach of a model ham~tonian is based on the assumption. that if it is diagon~zed in a model space it will yield eigenv@ues of the originat perturbed hamiltonian defmed in the entire Hilbert space. It s,ho&be emphasized that the method of a’model hamiltonian in tfie present short communication is noting else than a natural extension of the non-degenerate perturbation
theorv to a more generaleither degenerateor quasi*de~eRerat~ pe~~rbation theory. Genera& speaking,the model h~~ton~ans, sj~arly as the standard perturbation theoq, CXLbe cIassif%d into two types: Those which are dependent on the pertukbed eigenenerm E and those’which are nor: dependent OR E. An Emdependent model hamiltonian was formulated as an extension of the Bril.lo&-Wigner perturbation theory-by Bloch and Horowiti [I] tid L6wdin (21. A construction of a.~E-independent model ham&or&n, as zn extezision of the ~ylei~-Schr~d~ger p~r~rbatio~ theory i;grascarried out i~t~y by E&to. f31, Spiesman [4], Bloch.f!Q, Des Cloizaux- [ti] an$_Primas [7], Rece&Iy, the con$uction of&independent made1 hamiItonians has been studied and treated from different point of views and by’differe’nt appioaches @--251. It shouftj be noted that .ELS was said by Elir~chfelder arid @ertti [2iJJithere is nothing basically tiew to be discovered in the Rayleigh-SchrGiing~r ~r~rbatio~.theory. An extensive review of different fo~u~ations of ~yIe~~chr~dinge~ perturba~on theory as we&as intkrrelation between these diffe‘rent approaches was given by K$ein [24] .’ .The aim bf [email protected] short communication 6 tdsugg&t an operator extensioti of.the standard Rayie$G Scfir6dinger pkiturbation theory inio a quasi-degenerate case producing a her@tean E.indepkndetit model hamiltp&n. This problem has been sofved 1255 by t&k van Vleck ~r~sfo~ation~ and $.so by the Rimas <‘s~Ipe~o~e~ ator”,technique 1221. We believe that thk present for&disni’&. &nipler and easier to use th& the mentioned approaches. It $&ld,be tinderstood as a gener~~~~n of,some ideas’of the HirschfeIder and Certain method E19,,201;into,a’qiuasi-deg~nerate syieig;?-SchiBdingei pe$urb,ation theory. ‘Let is assume that a.totd hamikonia~ of m &&ctron byste,riz may be writ& k t&e split: form
‘.’
-.y
;, y
..,
. :f k+&jP”P&~ .f ., :_Y*. : ” ;,, : ., ._ ,, .,; ,,;.,, ;I c-3 .:. ..::. ,‘.,... (’ .: . :’$Mcondition .meBns that the space ~2i~n+ tb tk.model spat+ !$, when .the. r&b&ion $1 is “swit&ed-off’. .; I+othFr’words, there exist d perturbed eigenvalues {E,; )\uMj tending to: {E, 6 ; aElM 1 ihrhenHI.+ $4 I_:,:. .: thti piesent apijroach is a utiitary wave operator defined. as follows .: ., ._ I.,.’:l3e ._ basic cqkept.of ‘,,
‘_.
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’ $&ch m&@s that the’ b&ator ‘_ .: maps !J.ontd ino:, A.&ogdkly. .: ,.
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;
:.:wheretheprojectolsqh.gre.deter~~~tiby’,_’..
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~~‘~~ul&ying
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gives the @ojkto~,~o
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Volume 32, number 3
_’
,@J =;(g&@J+);
._
.’ *
.; ‘. ..
:
:,
.:
.‘.
The algebraic structure of the above,recurrent formulae (17) and (20)$3 is suitable-for their simtiltane6us iterative solution. Indeed, for the constructibn Ucn)PO and @$ @I2.2) only the lower-&de; contdbutions uWg, cmP(), ..:, U+-ljPo and c#;;@& .a., ~~$‘)~shoul~l be kyvwn,.Ftiran illustratidll of the presen’t method we give, the perturbationckpansioti ,f~r&la fdr the model hamiltonian up to the.fcqrth ordei, I-fRS = g$
f R&j,
- ._
xRS = @‘!! ‘,; ,.gwO
..
:
‘.
(23a)
.’
,.’ 1-PO 1 !f&l~J)kE(oj H N&(or) 0 0 y- 0
pd-I1 &)_;-;$p~~ a Lo!.
‘.
: :
where the non-hermite&
(23b)
IF@ is BIOCII’s moael hamiltonian generalized for the quasigdegeneiate theory hamiltonian may be constructed by a simple symmetrization [like (23a)] of Bloch’s model hamiltonian c&y up to the third order., Starting from the fourth order there are appearjng such terms which cznilot b&43ained in -the framework of this symm&ization proce.dure. .In tht degenerate theory, as was explicitly’shown by,Soliverqz [2l],.such symmetritition is -true up to the fourth ordq. Soliverez also noted, that these new terms are tiutually.cticelled for .tie non-degenerate perkrbaiioh t@j. The present m&hod has been applied in sotie ruditientary for&to a direct diagrammatic calculatiorI &f thejonization potentials [263 as ., well .& excita$on energies 1273 .. operator
1221.Thus,ourhem&sapmodel
,’
‘.
.
[1] C. Blbch and ;. Horowitz, Nucl. Phyi 8 (1958) 911”.’ [2] P.-O.Lowdin, J. Math. whys:3 (1962) 969_ [3] T: Kato, Rogr. Theoret whys. 4 (1949) ISP. .. [4]‘G. Spiesman, F’hys Rev. 107 (1957) 1180. [5]-C. Bloch, NucL F%ys:6 (1958) 329. [6] J. Dei C!oi&w, Nud. Phyr 20 (1960) 321., [7] .H.Primaq’Rev. Mail. Phys, 35 (i963) 710. ‘..’ is] B. Brandow, Reu. M& Phys. 39 (lYG7) 771;. (P] G. Oberlechner,‘F- Owono-$-Guema and J. Richter, Noovo Cimento 869 (1970) 23, [lo] M. Johnson and hi. Baranger, Ann. Fhys. 62 (1971) 172.
[ll] T.T.S. Kuo, S.Y. ke,and K.F. R~tcliff~ Nucl..E?~ys. A176 (1971) 65. 1121 P.G.H. Sandak, Advti. Chem. Fhys. 14 (1969).36S. ,, ‘. ; i Teor..@iz. ST (1966)‘230; Soviet Phyn J&P [14].v. Kvasnii and 1. Hub$, J..Chem. phys 60 (1974) 4483. .’ ‘.[15] J.O. Hirs-chfelder,~ntent Jr Quantum Chem. 3 (1969).731; .-
.:. (161 H.J. Silverstotie, J; ahem. fiys.54 (197lj 2325. [!7] EJ; Silverstone and T.T.,Hol&wtiy, whys Rev.-+ (1971) 2191.
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Le’lters 24.(1974)-36&y
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..
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,.
-’ ‘.’ ;’ .. -I ;.I.. .’ ‘, ., .. :,.f. ‘. [24] D.1. K$in,_J. Cl-&n._I’hy~ 51 (1974) 786. : -‘.-, ‘.[25] F. Jdfgetingld T. Pedersen, MqL l%y$. 2+.‘(1974) 33,954;: _. [26] I. Hhba*c,V. Kvasniia and A. Ho!ubec, Cheti. Phyk L;??teis 2! (19731’3BL. ‘A. ,Hol&/c
. :-
24 (19$7)‘lS4.
... (19I.P.p. Cer@in&d J.O. I$r&feider’, J. Chem. Whys; 5i[i970) 5977. (201 J.O. Hirschfelder andP.R. Certain, J. Chhem. whys 60 (1974) 1118., -[21] C.E. Soliverez. J. phy&‘C 2 (1969),2161.
~~~[27~:V.,l&snii~
.’
;”
’ [18] J.H. Choi, J. Math;F%ys. 10(1969)2141.
18. ..
.’
:
[ 13) LN. &laevgki, Zh. Eksperiti.
[22] V. Kvakiiiki, Czech, J, Phys, B 23 (1974) 605. [23] V. ~ratiiFka;-Czech, I, Phys, B, to be published.
,’
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: -1 _’‘, .. :
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32, numtier 3’
..” ‘.
..
_, ‘.
‘,
CHEMICAI’PNYSICS LETTERS
: ,, ._ “.
.
_,
.. :..,
”
,.
‘.
1 May l.975
‘.
,. ,.
.‘,.
..
A methbd for the construction of ;h& hermitaan’tiodel hakiltonian in &a framework of thequasidegenerate Rayleigh@ertuxbationtheory is kuggesterj. The approach of a model hamiltonian is bati on the assunption that ifit
SchCdinpr is ~kqmE.=d
in a chosen fmitedimension~
model space it will yield eigenvalues of the origin&l
hzmiltonian
in the entire
space, For a unitary operator tra’nsformingihe unperturbti~stite vectors onto the prturbed rtite vectors, md P model-interaction operator, the pcrh\rbo~oncxp3nsionformSe are developed. H&ert
The calculation of the eigenvaIu6s of a hamilt~~~ which describe some’molecular system, is one of the basic prdblems of quantuq chemistry. If we attempt to solve this problem using the standard Rayleighh-Ritz variational method, then we meet some difficulties originating from the diagonalization of a hermitean matrix ofenotious dimensiondity. An alternative approach fir overcoming these difficulties is the perturbation theory, namely, its idea of a model hamiltonian defined in a unite-Dimensions model space. The approach of a model ham~tonian is based on the assumption. that if it is diagon~zed in a model space it will yield eigenv@ues of the originat perturbed hamiltonian defmed in the entire Hilbert space. It s,ho&be emphasized that the method of a’model hamiltonian in tfie present short communication is noting else than a natural extension of the non-degenerate perturbation
theorv to a more generaleither degenerateor quasi*de~eRerat~ pe~~rbation theory. Genera& speaking,the model h~~ton~ans, sj~arly as the standard perturbation theoq, CXLbe cIassif%d into two types: Those which are dependent on the pertukbed eigenenerm E and those’which are nor: dependent OR E. An Emdependent model hamiltonian was formulated as an extension of the Bril.lo&-Wigner perturbation theory-by Bloch and Horowiti [I] tid L6wdin (21. A construction of a.~E-independent model ham&or&n, as zn extezision of the ~ylei~-Schr~d~ger p~r~rbatio~ theory i;grascarried out i~t~y by E&to. f31, Spiesman [4], Bloch.f!Q, Des Cloizaux- [ti] an$_Primas [7], Rece&Iy, the con$uction of&independent made1 hamiItonians has been studied and treated from different point of views and by’differe’nt appioaches @--251. It shouftj be noted that .ELS was said by Elir~chfelder arid @ertti [2iJJithere is nothing basically tiew to be discovered in the Rayleigh-SchrGiing~r ~r~rbatio~.theory. An extensive review of different fo~u~ations of ~yIe~~chr~dinge~ perturba~on theory as we&as intkrrelation between these diffe‘rent approaches was given by K$ein [24] .’ .The aim bf [email protected] short communication 6 tdsugg&t an operator extensioti of.the standard Rayie$G Scfir6dinger pkiturbation theory inio a quasi-degenerate case producing a her@tean E.indepkndetit model hamiltp&n. This problem has been sofved 1255 by t&k van Vleck ~r~sfo~ation~ and $.so by the Rimas <‘s~Ipe~o~e~ ator”,technique 1221. We believe that thk present for&disni’&. &nipler and easier to use th& the mentioned approaches. It $&ld,be tinderstood as a gener~~~~n of,some ideas’of the HirschfeIder and Certain method E19,,201;into,a’qiuasi-deg~nerate syieig;?-SchiBdingei pe$urb,ation theory. ‘Let is assume that a.totd hamikonia~ of m &&ctron byste,riz may be writ& k t&e split: form
‘.’
-.y
;, y
..,
. :f k+&jP”P&~ .f ., :_Y*. : ” ;,, : ., ._ ,, .,; ,,;.,, ;I c-3 .:. ..::. ,‘.,... (’ .: . :’$Mcondition .meBns that the space ~2i~n+ tb tk.model spat+ !$, when .the. r&b&ion $1 is “swit&ed-off’. .; I+othFr’words, there exist d perturbed eigenvalues {E,; )\uMj tending to: {E, 6 ; aElM 1 ihrhenHI.+ $4 I_:,:. .: thti piesent apijroach is a utiitary wave operator defined. as follows .: ., ._ I.,.’:l3e ._ basic cqkept.of ‘,,
‘_.
:. ~~~2$,‘_‘..
‘.. .. y
_
’ $&ch m&@s that the’ b&ator ‘_ .: maps !J.ontd ino:, A.&ogdkly. .: ,.
:_‘ ., :
..’
_y .‘.
“:
;
:.:wheretheprojectolsqh.gre.deter~~~tiby’,_’..
b&.(6) :,
~~‘~~ul&ying
:
gives the @ojkto~,~o
_,
: .. (6)
in the,f+wi~g
‘_ ,., ,, :_ : ..:. :
f&m
.-..:._:.,.‘_
(7) froth @& s&t
‘_ ,:, ..
., .. ._., :. ,: .., ‘1,,I, : I:. .. ‘: : 1 ‘. .. :. ‘., ., ‘,‘__, ., ,: ,~ ,_ .. : ,‘. ; 1” .. . ‘_ .‘,:: ,;’ -:.i., )_. 1,. ;
:
.,Subsiitution,(&)
‘,j,‘., ,.‘
C!Po gaps’ the model space 6,-, ‘onto spa& S2, qd c&Gersely , thd’opeiatof VP to’(6) let us defme’k hermitkan mbdel ~a~i~~o~~n ‘. ,i :’ .’
..A,
:
;
by the prajec,t&
.
:
I,.
(7). :.
._ ,’
..’ . ,.,. &&&p+ob!em, ;
,yh WC arr;;ve ‘iit a &b&l:
. .. ,’
” ‘.
1.‘. ,_
: -.
_,
‘. :
”
,‘.
:
;
:,
..I.
Volume 32, number 3
_’
,@J =;(g&@J+);
._
.’ *
.; ‘. ..
:
:,
.:
.‘.
The algebraic structure of the above,recurrent formulae (17) and (20)$3 is suitable-for their simtiltane6us iterative solution. Indeed, for the constructibn Ucn)PO and @$ @I2.2) only the lower-&de; contdbutions uWg, cmP(), ..:, U+-ljPo and c#;;@& .a., ~~$‘)~shoul~l be kyvwn,.Ftiran illustratidll of the presen’t method we give, the perturbationckpansioti ,f~r&la fdr the model hamiltonian up to the.fcqrth ordei, I-fRS = g$
f R&j,
- ._
xRS = @‘!! ‘,; ,.gwO
..
:
‘.
(23a)
.’
,.’ 1-PO 1 !f&l~J)kE(oj H N&(or) 0 0 y- 0
pd-I1 &)_;-;$p~~ a Lo!.
‘.
: :
where the non-hermite&
(23b)
IF@ is BIOCII’s moael hamiltonian generalized for the quasigdegeneiate theory hamiltonian may be constructed by a simple symmetrization [like (23a)] of Bloch’s model hamiltonian c&y up to the third order., Starting from the fourth order there are appearjng such terms which cznilot b&43ained in -the framework of this symm&ization proce.dure. .In tht degenerate theory, as was explicitly’shown by,Soliverqz [2l],.such symmetritition is -true up to the fourth ordq. Soliverez also noted, that these new terms are tiutually.cticelled for .tie non-degenerate perkrbaiioh t@j. The present m&hod has been applied in sotie ruditientary for&to a direct diagrammatic calculatiorI &f thejonization potentials [263 as ., well .& excita$on energies 1273 .. operator
1221.Thus,ourhem&sapmodel
,’
‘.
.
[1] C. Blbch and ;. Horowitz, Nucl. Phyi 8 (1958) 911”.’ [2] P.-O.Lowdin, J. Math. whys:3 (1962) 969_ [3] T: Kato, Rogr. Theoret whys. 4 (1949) ISP. .. [4]‘G. Spiesman, F’hys Rev. 107 (1957) 1180. [5]-C. Bloch, NucL F%ys:6 (1958) 329. [6] J. Dei C!oi&w, Nud. Phyr 20 (1960) 321., [7] .H.Primaq’Rev. Mail. Phys, 35 (i963) 710. ‘..’ is] B. Brandow, Reu. M& Phys. 39 (lYG7) 771;. (P] G. Oberlechner,‘F- Owono-$-Guema and J. Richter, Noovo Cimento 869 (1970) 23, [lo] M. Johnson and hi. Baranger, Ann. Fhys. 62 (1971) 172.
[ll] T.T.S. Kuo, S.Y. ke,and K.F. R~tcliff~ Nucl..E?~ys. A176 (1971) 65. 1121 P.G.H. Sandak, Advti. Chem. Fhys. 14 (1969).36S. ,, ‘. ; i Teor..@iz. ST (1966)‘230; Soviet Phyn J&P [14].v. Kvasnii and 1. Hub$, J..Chem. phys 60 (1974) 4483. .’ ‘.[15] J.O. Hirs-chfelder,~ntent Jr Quantum Chem. 3 (1969).731; .-
.:. (161 H.J. Silverstotie, J; ahem. fiys.54 (197lj 2325. [!7] EJ; Silverstone and T.T.,Hol&wtiy, whys Rev.-+ (1971) 2191.
..
%rid ?. Huba;; $cm;,Phys;
_.
,:- .,__- ,.. -.: I,,. ,,.’ : :_. ., .-.:.
..’ . ..‘. . I . ... ‘,’ ,, ._
_:
,..
‘.,.’
,-
‘,
: -.
‘. ,.
:
_, ,. ,’ .. - “- .
,_ ., _,
,, ,”
,. ,, :.:,
:,,‘.I .,.,:.
! _.’.. ., :.,
...
.’ -
_’
:
-’
:.,
: : ..
Le’lters 24.(1974)-36&y
,‘.
..
:, .:. ..
,.
-’ ‘.’ ;’ .. -I ;.I.. .’ ‘, ., .. :,.f. ‘. [24] D.1. K$in,_J. Cl-&n._I’hy~ 51 (1974) 786. : -‘.-, ‘.[25] F. Jdfgetingld T. Pedersen, MqL l%y$. 2+.‘(1974) 33,954;: _. [26] I. Hhba*c,V. Kvasniia and A. Ho!ubec, Cheti. Phyk L;??teis 2! (19731’3BL. ‘A. ,Hol&/c
. :-
24 (19$7)‘lS4.
... (19I.P.p. Cer@in&d J.O. I$r&feider’, J. Chem. Whys; 5i[i970) 5977. (201 J.O. Hirschfelder andP.R. Certain, J. Chhem. whys 60 (1974) 1118., -[21] C.E. Soliverez. J. phy&‘C 2 (1969),2161.
~~~[27~:V.,l&snii~
.’
;”
’ [18] J.H. Choi, J. Math;F%ys. 10(1969)2141.
18. ..
.’
:
[ 13) LN. &laevgki, Zh. Eksperiti.
[22] V. Kvakiiiki, Czech, J, Phys, B 23 (1974) 605. [23] V. ~ratiiFka;-Czech, I, Phys, B, to be published.
,’
:
: -1 _’‘, .. :
;.I’. ‘. ,‘-_I _:,; .:,
._‘. ‘.._.., I,‘.,
__ : I’. ,. ..c._,,,
“.. ‘:..
_ ,, :
: .,
: ”
j’
‘, ,.. ;’ : _,’ .,_,:: y : .:.‘_;.;
..