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A simple algorithm for the construction of genealogical spin functions
Volume
31,
number
LSMarch 1.975
CHEhfICAL PHYSICS LETTERS
3.
.’
A SIMPLE ALGORItiM
FOR THE CONSTRUCTION
OF GENEALOGICAL
SPlN FUNCTIONS
Ruben PAUNCZ * Department
of Ptrysicol Chetktry,
UniversiIy of Basel, Base!, Swi~rerlmd
Received 30 October 1974
A sir& algorithm is given for the construction of spin eigenfunctions accordin, 0 to the genealogiical scheme. The method can deal.dtiectly with the N.electron problem without any knowledge of the (?J-l)-elecLron spin eigenfunctions. It uses the representation matrices corresponding to the transpositions (k,k + l), tic latter an bc writtea dawn from the knowlcd~c of the Young tzbleclux.
1. Introduction Whenever the hamiltonian is spin free, the wavefunction has to be a spin eigenfunction. When dealing with closed-shell systems’, this goal can be very sirnpl~ achieved, but the problem arises with open-shell systems or using the different orbitals for different spins (DODS) scheme. There are different ways of constructing spin eigenfunctions; one of them is the so called genealogical construction. Here the spin functions are obtained step by step by adding or subtracting the spin of the last electron. The advantage of this method is that one can visualize the spin coupling scheme. Fig. 1 shows
the five functions,
obtained
l?p
@I
11221
I2121
s r
:
for N = S, S = A. The
functions obtained in this way form an orthonormal and complete set of spk functions and they generate a representation of the symmetric group which can be characterized by Young tableaux having not more than two rows. The disadvantage of this method is that in , order to construct the N-electron wavefunction we have to know the (N-l)-electron spin eigenfunctions, and the calculation soon becomes quite tedious. In this note a simple algorithm is given which yields .the spin eigenfunctions, obtained by the genealogical scheme for any number of elections without any knowledge of the (N-l)-electron spin eig0nfunctions.
2. Young’s orthogonal representation Consider the Young tableaux which have p boxes in the first row and u boxes in the second row, where p and Yare related to the spin quantum number and the numb&
p=+P.,+s
’ On Sabbatical leave from the Depkment of,Chemistry, -nion, Isxel institute of Technology, Haif?, Israel.
Tech-.
N
N
x (41 x 151 Fig. 1. Young lableaux, Yamanouchisymbols and spin coupling schemes for N = 5, S = 4.
of electrons >
by the. relation
Y=&N-S_
Cl)
1~ the standard tableaux the numbers are increasing in each row (from left to right) and in each coltimn (going down). We can associate the Yamanouchi symbol
443
VoluUle3 1, nurnbar 3
CHEh!ICALPriYSICSLETTERS ..:
.I5
March
197s
with each tableau in the’followirig way: oni goes over tions obtained from the gonealagiicalconstruction aceach number from 1 id IV; if it occurs in the first row cording to me last letter sequence of their,Yamanouchi ‘.of the tableau, one w&s 1, if in the second row, one .. symbols. ‘Theie are two spin functions which can be : writes ,2- I_& us arra;lge ‘the tableaux in such a way that given exp?&itZy fur any number of electrons: the first 1 (he Yamanouchi &bols are ordered in the last letter ‘: and the last one. Let us deno& the spin eigenfunctidns by X(i); i = 1; ...f. The first’one is proportional to the ‘. sequence! rl symbol in which bhe last letter is 2 precedes projected spin functioq given by lkwdin [3] i>e one in which the last letter is .l_ If the last lettkr is the same we go to the last but one letter, and so on. One can associate with each tableau a spin eigenfun&n obtained by the gencalo&al construction. As an inte-@ate step one uses the Yamanouchi bsymbol: 1 means addition,‘2 subtraction of the n?xt spin. Accordin& the set of spin functions corresponding to a given number of electrons (IJ) and a &en spin qumHere the square brackets are defined as follows: r&p’] turn number (S), f=f(N,,S), can be divided into two sub&. The first set consists of functions where the is the sum of all primitive spin functions with k a’s and .u amanouchl symbol ha the last letter 2, which means l p1s.In the formula the first bracket refers to the first ccelectrons and the second one to the remaining ekcthat all these are obtained by subtraction v, = trons. The last function Xy-) is the so-called valenceNW, s .%3)>, whd e in the ‘second set the last letter bond spin function (McWeeny [4]) is 1, i.e., they are obtained by addition and t-h& number .is/~,.=f(hr-l,S-)).f=J1 +fi. xcf) = 2-G ~~(lMiwxMw -.. Young [l] gave a method for constructing an ortho.' . gonadrepresentation of the symmetric group from the x {~(Lc-l.)P(u)-P(u-l)olo)cu(u+l) . . .ct(N), (3) : k-.ode$ge of the tableaux. One defines the a&cl his-. where II =A’-2s = 2v. In both cases the spin eigenfuncIonce dw in tableau Ti between the number p md 4, tion is given as a certain linear combination of primiy 2,sthe number of steps needed to arrive fromp to 4. tive spin functions and the coefficients are explicitly Left arid down are countkd as positive, up and right, :.negative. Let us number the rows and columns of the given for any number of electrons and for any spin. We shall rhow that the remaining spin functions are ma-ices =cbdjng to the cableauK, The matrices corre. qonding to-&e transpositions (k, k + 1) are given as obtained by starting at either end and applying th’e follows: transpositions (k, k + 1) and using the representation, matrices wkch are given explicitly from the tableaux. -. Uii(k,k +,I) = -I/& =P,., As an illustrrltion we list some of the matrices for the ‘. caseN=5,S=t. U&k.+l)=(l -<)li2, ifTi-(k,k+l)q;
1’ = 0,
. .
- .\
1-c d .
It has been shdwn (Pauricz [2]) that if the-spin functions are arranged using the Yamanoudhi symbols as ’ intermediate symbols, then the orthogonal representstion genera&d by the spin ftiction is ddeniiccl to the Young orthbgonal represtitation. As we shall see later this fact is veti useful in ccinstructing the spin eigen-
I . .. .. 4bL
ua/
,\.
1, I ‘. ,,
. -Q
.
*.
_I
. 1.
.b.
i for the construction
of spin eigenfunctions
I_$ Us assume that we have arranged the spin func‘444 _.
..
.\
U(2,3) = I :
firnctions, 3. Aigc+th&
.
dthetiise.
I.
where Q = =&, 2 = 31t2/2, c = l/3, d = S,1/2/i.
. -
. .
. .1 . . . -1
Volume 3 1, number 3
15 March 1975
CHEMICALPHYSICSLETTERS
a = l/(X + I), b = (1 - li2)1’2, 7 is a unit matrix. Here the dimensions of the submatrices fI I ry12= f21, f,, are given by f(N-2, St l), J$V-~& end
Inspection of the matrices shows that the functions X(2), . . .X(4) are obtained as follows: X(2) = [3 (3,4)X(l).+ X(1)1/8”*, X(3) = [2 (2,3)X(2) +X(2)] /311*,
f(iV- 2, S-l), respectively. This matrix yields the following retations:
X(4) = [2 (4,$X(2)
(N-lfl)xql
+ X(2)] /3l”,
PX(i) = =& X(i) T+(P):
(41
j=I
Using the same matrices we might start from the X(5) and obtain X(4), . . .X(2j in a similar way. We shall now prove that this algorithm
is alway:
feasible. The proof will be inductive. We can easiy show that the algorithm works for N = 3,4,5. Let LISassume that it works up to (N-1). Cons;der
f&t
which do not itlvalve
the tra.mposXons
the last letter N. For these all the representation matrices are factorized 1n the following way:
u$?:/”
fl U&,/c+
l)=
0
I ,’
$-;I* -
:
where the sub-matrices correspond to orthogonal representations of the NT1 electron problem with spin (S + $1, and (S - f), respectively. We assumed that the theorem is true up to (N-l), therefore we have all the relations funtions,
wirhin each group (consisting of fl, and fi respectively) just using the (N-l) electron
matrices. Next consider the matrix corresponding to the transposition (PI- 1,N). This is given as: f 11 qy&_~,N)
=fl2
2; - OJ’O ‘0 -r__ O I -+
i O_
!0 --i-f2.2-q-o-~0 I1
fil
0;
_i
i=l,...,f,,,
fl1
VI*
t-)+b
Xl& +i), (4)
=fy
Therefore, we have the connections between functions belonging to the two subsets. Tiiis concludes the proof of the feasibility of the algorithm. In practice, if we do not need all the functionqbut only some of them, we can start from the end which is closer to the given function(s). The procedure is st:aightforward
and yields
any number
with little labour
the fiu~ctions
for
of electrons without knowledge of th:
(Ni lblectron
functions.
Acknowledgement
’
-_---+----_
f2OI:
+i)=-Qx(j=Il
by
:
The author would like to express his sincere thanks to Professor Edgar Heilbronner who made his stay in Base1 stimulating and pleasant. This work is part OF a research project supported by the U.S.-Israel Binational Science Foundation.
References [I] D.E. Rutherford,
Substitutional xnz!ysis CEdi&u& Univ. Press, Edinburgh, 1948). (2) R. Pauna, Altcrrxmt molecular orbital method (Saunders, Philadelphia, 1967) p. 216. [3) P.O. L’bwdin, CzJcul des fonctions d’onde mole’culaire (Centre National de la Recherchc ScientLfique, Paris, 1958) p. 23. [4] R. hicWeeny and B.T. Sutcliffc, Methods of molecular quantum mechanics (Aademic Press, New York 1969) p. 61.
31,
number
LSMarch 1.975
CHEhfICAL PHYSICS LETTERS
3.
.’
A SIMPLE ALGORItiM
FOR THE CONSTRUCTION
OF GENEALOGICAL
SPlN FUNCTIONS
Ruben PAUNCZ * Department
of Ptrysicol Chetktry,
UniversiIy of Basel, Base!, Swi~rerlmd
Received 30 October 1974
A sir& algorithm is given for the construction of spin eigenfunctions accordin, 0 to the genealogiical scheme. The method can deal.dtiectly with the N.electron problem without any knowledge of the (?J-l)-elecLron spin eigenfunctions. It uses the representation matrices corresponding to the transpositions (k,k + l), tic latter an bc writtea dawn from the knowlcd~c of the Young tzbleclux.
1. Introduction Whenever the hamiltonian is spin free, the wavefunction has to be a spin eigenfunction. When dealing with closed-shell systems’, this goal can be very sirnpl~ achieved, but the problem arises with open-shell systems or using the different orbitals for different spins (DODS) scheme. There are different ways of constructing spin eigenfunctions; one of them is the so called genealogical construction. Here the spin functions are obtained step by step by adding or subtracting the spin of the last electron. The advantage of this method is that one can visualize the spin coupling scheme. Fig. 1 shows
the five functions,
obtained
l?p
@I
11221
I2121
s r
:
for N = S, S = A. The
functions obtained in this way form an orthonormal and complete set of spk functions and they generate a representation of the symmetric group which can be characterized by Young tableaux having not more than two rows. The disadvantage of this method is that in , order to construct the N-electron wavefunction we have to know the (N-l)-electron spin eigenfunctions, and the calculation soon becomes quite tedious. In this note a simple algorithm is given which yields .the spin eigenfunctions, obtained by the genealogical scheme for any number of elections without any knowledge of the (N-l)-electron spin eig0nfunctions.
2. Young’s orthogonal representation Consider the Young tableaux which have p boxes in the first row and u boxes in the second row, where p and Yare related to the spin quantum number and the numb&
p=+P.,+s
’ On Sabbatical leave from the Depkment of,Chemistry, -nion, Isxel institute of Technology, Haif?, Israel.
Tech-.
N
N
x (41 x 151 Fig. 1. Young lableaux, Yamanouchisymbols and spin coupling schemes for N = 5, S = 4.
of electrons >
by the. relation
Y=&N-S_
Cl)
1~ the standard tableaux the numbers are increasing in each row (from left to right) and in each coltimn (going down). We can associate the Yamanouchi symbol
443
VoluUle3 1, nurnbar 3
CHEh!ICALPriYSICSLETTERS ..:
.I5
March
197s
with each tableau in the’followirig way: oni goes over tions obtained from the gonealagiicalconstruction aceach number from 1 id IV; if it occurs in the first row cording to me last letter sequence of their,Yamanouchi ‘.of the tableau, one w&s 1, if in the second row, one .. symbols. ‘Theie are two spin functions which can be : writes ,2- I_& us arra;lge ‘the tableaux in such a way that given exp?&itZy fur any number of electrons: the first 1 (he Yamanouchi &bols are ordered in the last letter ‘: and the last one. Let us deno& the spin eigenfunctidns by X(i); i = 1; ...f. The first’one is proportional to the ‘. sequence! rl symbol in which bhe last letter is 2 precedes projected spin functioq given by lkwdin [3] i>e one in which the last letter is .l_ If the last lettkr is the same we go to the last but one letter, and so on. One can associate with each tableau a spin eigenfun&n obtained by the gencalo&al construction. As an inte-@ate step one uses the Yamanouchi bsymbol: 1 means addition,‘2 subtraction of the n?xt spin. Accordin& the set of spin functions corresponding to a given number of electrons (IJ) and a &en spin qumHere the square brackets are defined as follows: r&p’] turn number (S), f=f(N,,S), can be divided into two sub&. The first set consists of functions where the is the sum of all primitive spin functions with k a’s and .u amanouchl symbol ha the last letter 2, which means l p1s.In the formula the first bracket refers to the first ccelectrons and the second one to the remaining ekcthat all these are obtained by subtraction v, = trons. The last function Xy-) is the so-called valenceNW, s .%3)>, whd e in the ‘second set the last letter bond spin function (McWeeny [4]) is 1, i.e., they are obtained by addition and t-h& number .is/~,.=f(hr-l,S-)).f=J1 +fi. xcf) = 2-G ~~(lMiwxMw -.. Young [l] gave a method for constructing an ortho.' . gonadrepresentation of the symmetric group from the x {~(Lc-l.)P(u)-P(u-l)olo)cu(u+l) . . .ct(N), (3) : k-.ode$ge of the tableaux. One defines the a&cl his-. where II =A’-2s = 2v. In both cases the spin eigenfuncIonce dw in tableau Ti between the number p md 4, tion is given as a certain linear combination of primiy 2,sthe number of steps needed to arrive fromp to 4. tive spin functions and the coefficients are explicitly Left arid down are countkd as positive, up and right, :.negative. Let us number the rows and columns of the given for any number of electrons and for any spin. We shall rhow that the remaining spin functions are ma-ices =cbdjng to the cableauK, The matrices corre. qonding to-&e transpositions (k, k + 1) are given as obtained by starting at either end and applying th’e follows: transpositions (k, k + 1) and using the representation, matrices wkch are given explicitly from the tableaux. -. Uii(k,k +,I) = -I/& =P,., As an illustrrltion we list some of the matrices for the ‘. caseN=5,S=t. U&k.+l)=(l -<)li2, ifTi-(k,k+l)q;
1’ = 0,
. .
- .\
1-c d .
It has been shdwn (Pauricz [2]) that if the-spin functions are arranged using the Yamanoudhi symbols as ’ intermediate symbols, then the orthogonal representstion genera&d by the spin ftiction is ddeniiccl to the Young orthbgonal represtitation. As we shall see later this fact is veti useful in ccinstructing the spin eigen-
I . .. .. 4bL
ua/
,\.
1, I ‘. ,,
. -Q
.
*.
_I
. 1.
.b.
i for the construction
of spin eigenfunctions
I_$ Us assume that we have arranged the spin func‘444 _.
..
.\
U(2,3) = I :
firnctions, 3. Aigc+th&
.
dthetiise.
I.
where Q = =&, 2 = 31t2/2, c = l/3, d = S,1/2/i.
. -
. .
. .1 . . . -1
Volume 3 1, number 3
15 March 1975
CHEMICALPHYSICSLETTERS
a = l/(X + I), b = (1 - li2)1’2, 7 is a unit matrix. Here the dimensions of the submatrices fI I ry12= f21, f,, are given by f(N-2, St l), J$V-~& end
Inspection of the matrices shows that the functions X(2), . . .X(4) are obtained as follows: X(2) = [3 (3,4)X(l).+ X(1)1/8”*, X(3) = [2 (2,3)X(2) +X(2)] /311*,
f(iV- 2, S-l), respectively. This matrix yields the following retations:
X(4) = [2 (4,$X(2)
(N-lfl)xql
+ X(2)] /3l”,
PX(i) = =& X(i) T+(P):
(41
j=I
Using the same matrices we might start from the X(5) and obtain X(4), . . .X(2j in a similar way. We shall now prove that this algorithm
is alway:
feasible. The proof will be inductive. We can easiy show that the algorithm works for N = 3,4,5. Let LISassume that it works up to (N-1). Cons;der
f&t
which do not itlvalve
the tra.mposXons
the last letter N. For these all the representation matrices are factorized 1n the following way:
u$?:/”
fl U&,/c+
l)=
0
I ,’
$-;I* -
:
where the sub-matrices correspond to orthogonal representations of the NT1 electron problem with spin (S + $1, and (S - f), respectively. We assumed that the theorem is true up to (N-l), therefore we have all the relations funtions,
wirhin each group (consisting of fl, and fi respectively) just using the (N-l) electron
matrices. Next consider the matrix corresponding to the transposition (PI- 1,N). This is given as: f 11 qy&_~,N)
=fl2
2; - OJ’O ‘0 -r__ O I -+
i O_
!0 --i-f2.2-q-o-~0 I1
fil
0;
_i
i=l,...,f,,,
fl1
VI*
t-)+b
Xl& +i), (4)
=fy
Therefore, we have the connections between functions belonging to the two subsets. Tiiis concludes the proof of the feasibility of the algorithm. In practice, if we do not need all the functionqbut only some of them, we can start from the end which is closer to the given function(s). The procedure is st:aightforward
and yields
any number
with little labour
the fiu~ctions
for
of electrons without knowledge of th:
(Ni lblectron
functions.
Acknowledgement
’
-_---+----_
f2OI:
+i)=-Qx(j=Il
by
:
The author would like to express his sincere thanks to Professor Edgar Heilbronner who made his stay in Base1 stimulating and pleasant. This work is part OF a research project supported by the U.S.-Israel Binational Science Foundation.
References [I] D.E. Rutherford,
Substitutional xnz!ysis CEdi&u& Univ. Press, Edinburgh, 1948). (2) R. Pauna, Altcrrxmt molecular orbital method (Saunders, Philadelphia, 1967) p. 216. [3) P.O. L’bwdin, CzJcul des fonctions d’onde mole’culaire (Centre National de la Recherchc ScientLfique, Paris, 1958) p. 23. [4] R. hicWeeny and B.T. Sutcliffc, Methods of molecular quantum mechanics (Aademic Press, New York 1969) p. 61.