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On the evaluation of matrix elements of symmetry operators in a determinantal basis
Volume 75, number
3
1 November
CHEMICAL PHYSICS LETTERS
ON THE EVALUATION
OF MATRIX
IN A DETERMINANTAL
ELEMENTS OF SYMMETRY
1980
OPERATORS
BASIS
L.C. VANQUICKENBORNE Department
of Chemstty,
Received 8 November
and A. CEULEMANS Umverslty of Leuven, 3030 Heverlee. Belgrum
1979, III tinal form 15 September
1980
A general algebraic theorem IS presented to calculate the matrLz elements of symmetry operators m a bassls of determlnantal wavefunctlons. It IS also shown how the matrix elements of complementary open-shell wavefunctions are related to each other.
The most wdely used technique m constructmg elgenfunctions of symmetry operators consists m applymg projection operato_rs to an appropnate set of basis functions. The use of projection operators leads one to expansions of the type R\Ir, = C,clk+[, where the coefficients elk are given by elk = (\IrlIf?lQ,). It 1s the purpose of this note to introduce two very simple theorems, that can be useful m the evaluatron of the matnx elements elk
for many-electron systems. (I) In evaIuatmg matnx
elements of quantum mechanical operators, a number of classical rules are of general apphcatlon. Especially the rules concerned with sums of one- and two-electron operators m an orthonormal basis of Slater determmants are very well known [I]. A symmetry operator is an all-electron operator; when operatmg m a determmantal basis, it can be considered as a product of one-electron symmetry operators
d(l,2,
. . iz) = f?( 1) d(2)
. . R(I2) .
(1)
It wdl be shown how the many-electron matrix elements can be evaluated m a very simple way by using the rnmors of an appropnate one-electron matrix. Let the result of a symmetry operator d on a set of orthonormal symmetry adapted spm orbltals q(l), ps(2), . be given by k(l)(P,(l)
= c I
r,(R),,
P,(l)
1
d(2)42)
= ~r&?)r,~t(2),
(2)
..,
._ are the representation matrices correspondmg to the Irreducible representations r,, I?,, . . . . where r,(R), rb(R), the transformation properties Let r be the r-&mensional &rect sum representation of l?,, r,, _.., mcorporating of all the spm orbltals under consideration, m some arbitrary order Due to the spin degeneraaes, I- will always be an even number. The n-electron (II
2, ..- PZ)= (12’)--1’21(pk1(1)Lpk2(2)
= W-1’2~~ where k denotes 494
n
--- Ip&f)l
(sgnr7)%k1(1)%42)
a specific selection
of n orbitals
. . . I&
(II) ) n
amongst
the set of r avllable
(3) orbltals.
ri IS the operator,
permutmg
Volume 75, number 3
the function
i=
CHEMICAL
labels (i.e. the columns
...
k,
k,
Kx_’
Kk,
K~, 1s the function
__.
1 November
1980
k,, Kx_
>n
label associated
2,. . n)l&l,
LETTERS
of the determmant)
wth
the ith electron
the permutatlon IS even or odd The matrix element of the rzelectron W/Jl,
PHYSICS
Now, any factor in the latter
product
QJl
determmants
2, --. H)l\kl(l,2,.
III the i-permutation;
,2,
as a matnx
on whether
__11) and QII(l, 2, _ _II) IS given by
- II)) = (Iz’)“‘((pk,(l)~~*(2)
can be expressed
(sgn~?) = 21 depending
element
--- ~~$Z)l_~(l,
2, -.. H)lF[(l,
2, _._ fI))
of r.
Therefore (*k(l,
2, --- Il)ld(l,
(qk(i,2,.
.12)ifZ(i,
2, . . . n)l*,(l,
2, . . . w) = T
2, . . . ~z)~\I’I(I, 2, ___Iz))=
(sgnx)
WZ)X.~,~~, T(R)k2,h,2
. . T(R)k,,AI,
,
ir~$_:-~i =M,:- .
(4)
Thus means that the many-electron matnx element can be simply reduced to an tz-square mmor I@ of the rsquare one-electron matnx r(R), obtamed by leavmg only the columns I,, I,, . . I,, and the rows k, , k, , _._k,, _ Obviously, ths matnx element is zero, whenever the seiectrons k and I differ rn symmetncaIly unrelated orbit&; If the group IS abelian, the r(R) matnx 1s diagonal, and the matrix element 1s zero unless k = 1. If the group is not abehan, on the other hand, for the matnx element of eq. (4) to be non-zero, there is In general no limltation on the number of spm orbltals that can be &fferent m the two detemlinants This 1s m marked contrast to the situation where the operator is a sum of one- or two-electron operators. Clearly, the theorem is especially useful when degeneracles are mvolved and when the construction of symmetry adapted wavefunctlons IS an otherwlse laborious
undertaking, a case in point IS !igand field theory where the construction of dn or ftl states requires a rather extenslve application of symmetry operators [2] _ If tz = r, the corresponding closed shell state IS described by e( 1,2, . . . r) and the &agonal element (*(I,
2, . . . r)lri(r,
2, .._ ~)I*(I,
Smce a closed shell is always totally
in = 1. (II)Any
2,
. r))= Ii-1
symmetnc
tzelectron detemunant *,&I, shell state function consldered m eq (5)
0)
*, eq (5) shows that the unitary
2, .. tz) can Itself be consldered
matrix
r IS also ummodular:
to be a mmor of the relectron
closed
* In the present context, the totally symmetncli nature of a closed shell can most easily be looked upon as resulting from two properties (I) the ununodular nature of the transformauon matrices of the spoor (a, p) as shown eg m ref. [2], (ii) the spatial non-degeneracy shells. one wth
of a half-ftied shell with maximal multlphclty. When the closed shell IS then constructed from two half-filed only 01 spins, the other one wth only p spms, the correspondmg function is readdy shown to be totatty symmetric.
495
Volume 75, number 3
CHEMICAL
PHYSICS
+722-a - --
rp,m
1 November 1980
LETTERS
I
*(1,2,
_.
q(2)
r) = (,!)-I/2
The rz-electron functron *,,
defined m eq (3) can now be rewrttten as
m order to stress Its minor character Q&z+
To any qk can be assoctated an (r - n)electron
funchon
_ tl)I] lE},g~~+l.~w~-
-lir, n+1,?1+2,.. r ’
_r) = {(_l)q”/[(r
l,t1+2,
(6)
(8)
whrch can be considered to be the algebraic complement of eq. (7) m the Laplace expansion, and where qk = 1 + 2 i- . . . +tz+kl+k,+_. k,, . In analogy to eq. (4) one can wnte (e&z
+ 1, . . r)lk(tt
f 1) -- r)pP[(tz f 1,. . t-)) = (-l)qk+4’Ir~~;;;
1;;
I = (-l)~“+%I;~~I
The relatronshrp between the minors m eqs. (4) and (9) can be found from the followmg standard
result of elementary
P, = (-I)41
where ~7= kl -I-IQ
matnx
IY’fv,-~t
algebra
. theorem,
(9) whrch is a
[3] :
,
(10)
_+ k, + I, + I2 + . _ + l,, obviously, (-l)Q = (-1) qk+ql In eq. (lo), P, 1s the n-square minor of the adjomt of I? (transpose of the cofactor matnx off’), whose elements occupy the same posrtron m (adj. f’) as those of itl,k*’ occupy m r Smce I rl = 1, the umtarrty of r implies that its adjomt equals its complex conjugate transpose and consequently @I:‘>* (e,(l,
+
.
= (- l)%VI;:;Z
2, . .. tt)ld(l,
,
2, . . . tt)pD,(l,
(11)
2, . . . tz))* = */Jr2 + 1, tz +2, .. . r)lk(,z + 1, tz +2, . . f)pP[(lZ + 1, t1+2,
.. r)),
Therefore, the matnx of I? based on the rz-electron wavefunctrons V!k(l, 2, . II) ISthe complex conjugate of wavefuncnons +,&I + 1, tz + 2, . . . r). It is obvious that eq (11) IS very
the matrrx of R based on the (r-rz)-electron
useful in treating problems concerned with the hole-partrcle equrvalence [2]. More specifically, rf the mntnx elements of eq. (11) are real, M,k-’ = (-l)qM~:l’,. Ln tius case, a unitary matrrx, characterized by the symmetry labels r,, Mra, I’,, transformmg the ek( 1,2, . . . n)-set mto a set of wavefunctions It+/) _._ ~111also transform the corresponding *,&z + 1, .._ r)-set mto a set of wavefunctrons characterized by the sLmze symmetry labeis. Thrs property imposes a sample but very general restnctron on the form of the wavefuncttons corresponding to half-filled shells. Indeed, m the latter case, the functions qk( 1,2, . II) and \I’x-(tz + 1, . . t-) correspond to the same configuratron. Obvrous apphcatrons of thus result can agarn be found in hgand field theory [2/V] -
References [ 11 J C. Shter, Quantum theory of atonuc structure, Vol 1 (McGraw-HI& New York, 121 J S. Gnffith, The theory of transItion-metal 1011s (Cambridge Unlv Press, London, [31 F-A- Awes Jr.. hfntnces Gchaum Pubhshmg Co , New York, 1962). [4] A-D. Llehr, J. Phys. Chem 68 (1964) 665. [5] J R. Perumaredti, 3. Phys. Chem. 71 (1967) 3144.
1960) 1971)
3
1 November
CHEMICAL PHYSICS LETTERS
ON THE EVALUATION
OF MATRIX
IN A DETERMINANTAL
ELEMENTS OF SYMMETRY
1980
OPERATORS
BASIS
L.C. VANQUICKENBORNE Department
of Chemstty,
Received 8 November
and A. CEULEMANS Umverslty of Leuven, 3030 Heverlee. Belgrum
1979, III tinal form 15 September
1980
A general algebraic theorem IS presented to calculate the matrLz elements of symmetry operators m a bassls of determlnantal wavefunctlons. It IS also shown how the matrix elements of complementary open-shell wavefunctions are related to each other.
The most wdely used technique m constructmg elgenfunctions of symmetry operators consists m applymg projection operato_rs to an appropnate set of basis functions. The use of projection operators leads one to expansions of the type R\Ir, = C,clk+[, where the coefficients elk are given by elk = (\IrlIf?lQ,). It 1s the purpose of this note to introduce two very simple theorems, that can be useful m the evaluatron of the matnx elements elk
for many-electron systems. (I) In evaIuatmg matnx
elements of quantum mechanical operators, a number of classical rules are of general apphcatlon. Especially the rules concerned with sums of one- and two-electron operators m an orthonormal basis of Slater determmants are very well known [I]. A symmetry operator is an all-electron operator; when operatmg m a determmantal basis, it can be considered as a product of one-electron symmetry operators
d(l,2,
. . iz) = f?( 1) d(2)
. . R(I2) .
(1)
It wdl be shown how the many-electron matrix elements can be evaluated m a very simple way by using the rnmors of an appropnate one-electron matrix. Let the result of a symmetry operator d on a set of orthonormal symmetry adapted spm orbltals q(l), ps(2), . be given by k(l)(P,(l)
= c I
r,(R),,
P,(l)
1
d(2)42)
= ~r&?)r,~t(2),
(2)
..,
._ are the representation matrices correspondmg to the Irreducible representations r,, I?,, . . . . where r,(R), rb(R), the transformation properties Let r be the r-&mensional &rect sum representation of l?,, r,, _.., mcorporating of all the spm orbltals under consideration, m some arbitrary order Due to the spin degeneraaes, I- will always be an even number. The n-electron (II
2, ..- PZ)= (12’)--1’21(pk1(1)Lpk2(2)
= W-1’2~~ where k denotes 494
n
--- Ip&f)l
(sgnr7)%k1(1)%42)
a specific selection
of n orbitals
. . . I&
(II) ) n
amongst
the set of r avllable
(3) orbltals.
ri IS the operator,
permutmg
Volume 75, number 3
the function
i=
CHEMICAL
labels (i.e. the columns
...
k,
k,
Kx_’
Kk,
K~, 1s the function
__.
1 November
1980
k,, Kx_
>n
label associated
2,. . n)l&l,
LETTERS
of the determmant)
wth
the ith electron
the permutatlon IS even or odd The matrix element of the rzelectron W/Jl,
PHYSICS
Now, any factor in the latter
product
QJl
determmants
2, --. H)l\kl(l,2,.
III the i-permutation;
,2,
as a matnx
on whether
__11) and QII(l, 2, _ _II) IS given by
- II)) = (Iz’)“‘((pk,(l)~~*(2)
can be expressed
(sgn~?) = 21 depending
element
--- ~~$Z)l_~(l,
2, -.. H)lF[(l,
2, _._ fI))
of r.
Therefore (*k(l,
2, --- Il)ld(l,
(qk(i,2,.
.12)ifZ(i,
2, . . . n)l*,(l,
2, . . . w) = T
2, . . . ~z)~\I’I(I, 2, ___Iz))=
(sgnx)
WZ)X.~,~~, T(R)k2,h,2
. . T(R)k,,AI,
,
ir~$_:-~i =M,:- .
(4)
Thus means that the many-electron matnx element can be simply reduced to an tz-square mmor I@ of the rsquare one-electron matnx r(R), obtamed by leavmg only the columns I,, I,, . . I,, and the rows k, , k, , _._k,, _ Obviously, ths matnx element is zero, whenever the seiectrons k and I differ rn symmetncaIly unrelated orbit&; If the group IS abelian, the r(R) matnx 1s diagonal, and the matrix element 1s zero unless k = 1. If the group is not abehan, on the other hand, for the matnx element of eq. (4) to be non-zero, there is In general no limltation on the number of spm orbltals that can be &fferent m the two detemlinants This 1s m marked contrast to the situation where the operator is a sum of one- or two-electron operators. Clearly, the theorem is especially useful when degeneracles are mvolved and when the construction of symmetry adapted wavefunctlons IS an otherwlse laborious
undertaking, a case in point IS !igand field theory where the construction of dn or ftl states requires a rather extenslve application of symmetry operators [2] _ If tz = r, the corresponding closed shell state IS described by e( 1,2, . . . r) and the &agonal element (*(I,
2, . . . r)lri(r,
2, .._ ~)I*(I,
Smce a closed shell is always totally
in = 1. (II)Any
2,
. r))= Ii-1
symmetnc
tzelectron detemunant *,&I, shell state function consldered m eq (5)
0)
*, eq (5) shows that the unitary
2, .. tz) can Itself be consldered
matrix
r IS also ummodular:
to be a mmor of the relectron
closed
* In the present context, the totally symmetncli nature of a closed shell can most easily be looked upon as resulting from two properties (I) the ununodular nature of the transformauon matrices of the spoor (a, p) as shown eg m ref. [2], (ii) the spatial non-degeneracy shells. one wth
of a half-ftied shell with maximal multlphclty. When the closed shell IS then constructed from two half-filed only 01 spins, the other one wth only p spms, the correspondmg function is readdy shown to be totatty symmetric.
495
Volume 75, number 3
CHEMICAL
PHYSICS
+722-a - --
rp,m
1 November 1980
LETTERS
I
*(1,2,
_.
q(2)
r) = (,!)-I/2
The rz-electron functron *,,
defined m eq (3) can now be rewrttten as
m order to stress Its minor character Q&z+
To any qk can be assoctated an (r - n)electron
funchon
_ tl)I] lE},g~~+l.~w~-
-lir, n+1,?1+2,.. r ’
_r) = {(_l)q”/[(r
l,t1+2,
(6)
(8)
whrch can be considered to be the algebraic complement of eq. (7) m the Laplace expansion, and where qk = 1 + 2 i- . . . +tz+kl+k,+_. k,, . In analogy to eq. (4) one can wnte (e&z
+ 1, . . r)lk(tt
f 1) -- r)pP[(tz f 1,. . t-)) = (-l)qk+4’Ir~~;;;
1;;
I = (-l)~“+%I;~~I
The relatronshrp between the minors m eqs. (4) and (9) can be found from the followmg standard
result of elementary
P, = (-I)41
where ~7= kl -I-IQ
matnx
IY’fv,-~t
algebra
. theorem,
(9) whrch is a
[3] :
,
(10)
_+ k, + I, + I2 + . _ + l,, obviously, (-l)Q = (-1) qk+ql In eq. (lo), P, 1s the n-square minor of the adjomt of I? (transpose of the cofactor matnx off’), whose elements occupy the same posrtron m (adj. f’) as those of itl,k*’ occupy m r Smce I rl = 1, the umtarrty of r implies that its adjomt equals its complex conjugate transpose and consequently @I:‘>* (e,(l,
+
.
= (- l)%VI;:;Z
2, . .. tt)ld(l,
,
2, . . . tt)pD,(l,
(11)
2, . . . tz))* = */Jr2 + 1, tz +2, .. . r)lk(,z + 1, tz +2, . . f)pP[(lZ + 1, t1+2,
.. r)),
Therefore, the matnx of I? based on the rz-electron wavefunctrons V!k(l, 2, . II) ISthe complex conjugate of wavefuncnons +,&I + 1, tz + 2, . . . r). It is obvious that eq (11) IS very
the matrrx of R based on the (r-rz)-electron
useful in treating problems concerned with the hole-partrcle equrvalence [2]. More specifically, rf the mntnx elements of eq. (11) are real, M,k-’ = (-l)qM~:l’,. Ln tius case, a unitary matrrx, characterized by the symmetry labels r,, Mra, I’,, transformmg the ek( 1,2, . . . n)-set mto a set of wavefunctions It+/) _._ ~111also transform the corresponding *,&z + 1, .._ r)-set mto a set of wavefunctrons characterized by the sLmze symmetry labeis. Thrs property imposes a sample but very general restnctron on the form of the wavefuncttons corresponding to half-filled shells. Indeed, m the latter case, the functions qk( 1,2, . II) and \I’x-(tz + 1, . . t-) correspond to the same configuratron. Obvrous apphcatrons of thus result can agarn be found in hgand field theory [2/V] -
References [ 11 J C. Shter, Quantum theory of atonuc structure, Vol 1 (McGraw-HI& New York, 121 J S. Gnffith, The theory of transItion-metal 1011s (Cambridge Unlv Press, London, [31 F-A- Awes Jr.. hfntnces Gchaum Pubhshmg Co , New York, 1962). [4] A-D. Llehr, J. Phys. Chem 68 (1964) 665. [5] J R. Perumaredti, 3. Phys. Chem. 71 (1967) 3144.
1960) 1971)