F. A. MATSEN,A. A. CANTU,AND R. D. POSHUSTA
1558
Spin-Free Quantum Chemistry.l a 111. Bond Functions and the Pauling Rules
by F. A. Matsen, A. A. Cantu,lb and R. D. Poshusta10 Molecular Physics Group, The Univereity of Tezas, Austin, T e r n 78712 (Received November 12, 1966)
The Pauling rules which evaluate matrix elements for a spin-free Hamiltonian over antisymmetric Slater bond functions are derived in the spin-free formulation of quantum chemistry. In this formulation a vector bond diagram is replaced by a two-columned Young tableau. The Slater bond function for this bond diagram is replaced by a spinfree bond function projected from a spin-free primitive function by a “structure projector.” The structure projector is an element in the permutation algebra which is constructed with reference to the two-columned tableau. Matrix elements over two spinfree bond functions contain an algebraic element called the “bracket element’’ which is a product of the adjoint of the structure projector for the first bond function and the structure projector for the second bond function. The coefficients (the Pauling numbers) of the N ! permutation operators in the expansion of the bracket element are evaluated by group theoretical methods.
1. Historical Introduction The bond function, introduced by Slater2 in 1931, has been one of the most important tools of quantum chemistry. It embraces the concepts of the electron pair, the chemical bond, the pure bond state, and resonance among pure bond states. We review briefly the construction of Slater bond functions and the evaluation of matrix elements for a spin-free Hamiltonian over these functions using a method due to Pauling.a We first^ construct a vector bond diagram, denoted by D,, in which the first N integers are arranged in order in a circle. A certain number, say p , of pairs of integers are connected by arrows. See Table I. The spin state of the system is given by S = N/2
-p
Next we select a spin-free primitive function which we denote by 17). For simplicity, we may take 17) to be an orbital product with all orbitals distinct, but this is not necessary. A permutation* on the electron coordinates is denoted by
Ir;P>= P l y >
(1.4)
We complete the construction of a Slater bond function forming the antisymmetrized function
If Iy) is an orbital product, (1.5) can be expressed as a linear combination of Slater determinants.
(1.2)
We designate by RID,, those permutations which permute paired spin orbitals and which thus reverse the arrows in D,. Next we form a spin-bond function The Journal of Physical Chemtktry
I.
(1.1)
We next construct a spin-orbital product, denoted by 12), with a /3 orbital assigned to the tail and an CY orbital assigned to the head of each arrow in D,. We denote a permutation on the spin coordinates in 12) by
p ; P >=PIX)
where the sum extends over all possible arrow reversals and (- l ) Ris plus or minus one depending on whether R is an even or an odd number of reversals. See Table
(1) (a) Supported by the Robert A. Welch Foundation of Houston, Texas. and the National Aeronautics and Space Administration: (b) Jefferson Chemical Co. Fellow: (c) National Science Foundtion Postdoctoral Fellow. (2) J. C. Slater, Phys. Rev.,38, 1109 (1931). (3) L. PBdhg, J . C h .Phy8., 1, 280 (1933).
SPIN-FREE QUANTUMCHEMISTRY
Table I:
1559
Spin-Bond Functions for N = 4, p = 2 (S = 0)”
Table In: Bond Orbital Diagrams
17) =
a(l)b(2)~(3)d(4) Dx
x
I
a+b
d+c
I1
i i d
I11
See Table 11. If 17) is an orbital product, D, can be represented by an orbital bond dia,gram in which the integers in D , are replaced by the orbitals to which the electrons are assigned in D,. See Table 111. ~~
Table 11: Slater Bond Functions N = 4, p = 2 (S = 0) (Orbital Product Primitive Functiony
(YZX
=
17) = a(l)b(2)c(3)d(4) abcd 12) = a(] ) ~ ( 2 ) @ ( 3 ) ~5 ~ (@CX&Y 4 ) for x = I and 11 for x = I11 12) = p(l)a(Z)a(3)j3(4)
lrzx;A)
X
I
111
a
4(2:) abcd) Baa8
-
(E;: )
- (abcd)
(;:2)
-
(abcd)
+
+
(abed)} ff8aB
(abed)} a8Sa
Note that by Rumer’s rule, I./ZIII;A) = I y2II;A)
Any Slater bond function can be expressed as a linear combination of f N p linearly independent bond functions. In particular for N even and p = N/2, a Slater bond function for a diagram containing crossed bonds can be expressed as a linear combination of Slater bond functions for diagrams with uncrossed (or canonical) bond diagrams. The latter is known as Rumer’s rule. See Tables I and 11. A matrix element for a spin-free Hamiltonian over Slater bond functions, has the form
- I y2I;A).
The number of independent Slater bond functions which can be obtained from a given spin-free primitive function 17) is
I
;A HI rZx’;A) =
c (P>,x4YlH I P
Y ;p>
(1.9)
Here ( ~ l H l r ; P )is a spin-free matrix element. (P),,, is a numerical coefficient, which we call a “Pauling number.” It is independent of 17) and depends only on D,,D,,, and P. Pauling has given rules for the determination of (P),,, for the case of N even and p = N/2 (S = 0). He obtained the (P),,t for N odd and p = [(N - 1)/2] (S = 1/2) from the case of p = (N 1)/2 by deleting permutations involving the (N 1) th (or ghost) orbital. The Pauling rules for obtaining (P),,t are given in the following paragraphs. (a) Form the vector bond diagram D,* -= PD,, by permuting the integers in D,, according to P and returning the integers to their original order carrying along the new head and tail assignments. Superimpose D, and D,* and label the superposition [D,
+ +
,
D,*l.
(b) Reverse arrows in [D,D,*] so that the arrows are head to head and tail to tail. Label the new superposition [DPDk*] and denote the number of arrow reversals by T .
or
+ 1) + l)!
N!(N - 2 p fNP = p!(N - p
(1.8)
:
(4) A permutation P = L t l ) &) . . P k ) ) On the e’ectron coordinates of an orbital product function takes an electron coordinate i and converts it to P ( i ) all within the same orbital.
Volume 70, Number 6 May 1966
1560
F. A. MATSEN,A. A. CANTU,AND R. D. POSHUSTA
(c) Count the number of islands (the number of disjoint sets of linked integers) in [DkDEr*].Denote the number of islands by i. (d) Apply Pauling’s rule ( n x d
=
P+r 1
( /2>
(-1)
(N/2)-i
Table IV Partitions and some tableaux for N = 4 11’ ~
See Table VIII. In this paper we have derived, besides a formula for the Pauling numbers for arbitrary N and p , the Pauling rules using theorems from the permutation group algebra. While the derivation is more complicated than Pauling’s original derivation, it demonstrates clearly the role of permutational symmetry which is implicit in the Pauling development. The general formulation of spin-free quantum chemistry was presented in paper Is of this series. In paper 11,6three electron problems were considered in detail.
2. A Brief Introduction to Spin-Free Quantum Chemistry A spin-free Hamiltonian for an N-electron system commutes with the permutations on the coordinates of all the N electrons. We denote this group of permutations by SN. The eigenstates of the spinfree Hamiltonian are grouped in sets, called permutation states, each characterized by a partition of N. We designate a partition of N by [A] = [AI,
12,
. . ., A N ]
(2.1)
where Xi is a positive integer or zero such that X’
3
A2
2 ... 2
AN
20
(2.2)
and
(2.3)
ZAi = N
See Table IV. The set of all linear combinations of permutation operators of S N form an algebra7 called the permutation algebra which we denote by AN. An element of AN is
x = CP ( X ) . P
(2.4)
where (X),is a numerical coefficient. AN can be d e composed into a direct sum of invariant subalgebras AN[’]each indexed by a partition [ A ] of N. That is
AN
=
CAN1’]
(2.5)
[’I
The invariant subalgebras have the important property that if X[’l is an element of AN[’]and Y[”]is an element in A N [ ” ]then
x~’]Y[’’]=
o for [A] z
The Journal of Physical Chemistry
[A’]
TI[’]
(1.10)
(2.6)
4
Li
= 1 [1,1,1,11
Excluded partitions and tableaux
A function projected by an element of AN[’] is said to lie in the [Alth permutation state. For example, the functions
(y;X[’l)= X[”Iy)
(2.7)
and
Ir;Y [’’I ) = yh’lIT> (2.8) lie in permutation states [XI and [A’], respectively. A matrix element between two such functions is, by (2.6)
(y;X[’]IHIy ;Y [”I)
= (y /HX[’’‘Y [’I 17) = 0 for [ A ] # [A’]
(2.9)
Here X[’lt is the adjoint of X[’l which also lies in ANIh1. Equation 2.9 shows that a spin-free Hamiltonian does not mix different permutation states. Young’ has devised a simple method for the construction of elements in A N [ ’ ] . We arrange the first N integers in rows of length A’. See Table IV. This array is called a Young tableau and is designated by (5) Paper I: F. A. Matsen, “Advances in Quantum Chemistry,” Vol. I, P. 0. Lowdin, Ed., Academic Press, New York, N. Y . , 1964. (6) Paper 11: F. A. Matsen, J . Phys. C h a . , be, 3282 (1964). (7) For a thorough discussion of permutation algebras, see D. E. Rutherford, “Substitutional Analysis,” Edinburgh, 1948. See also F. A. Matsen and R. D. Poshusta, “Algebras, Ideals and Quantum Mechanics with Applications from the Bymmetrics Group,” Molecular Physics Group Technical Report, University of Texas, Austin, Texas, to be published.
SPIN-FREE QUANTUMCHEMISTRY
1561
the symbol T,[’]. We next construct the algebraic element
E,[’] E N,[’Ip,[’l
A Young operator for [A]
(2.10)
Here N,[’] is the antisymmetric sum of the permutations on the integers in the columns and is the symmetric sum of the permutations on the integers in the rows of T,lA1. See Table V . Young has shown that for any element X of A N ,the element
Xrl’l
Table V
E,‘XlX
l3(4( &[XI
N1I’1pr[’1 = [ ( e - (13))(d - (24)]][(d
+ (12)](4 + (34))l
- (13) - (24) + (13)(24) + (12) - (123) (142) + (1423) + (34) - (134) - (243) + (1324) + (12)(34) - (1234) - (1432) +
=
(2.11)
(14)(23)
is an element in A ~ [ ’ ] . We note that
A Young operator for [A] = [2,12]
U,f,E,~’]= E,%,r,
(2.12)
TI[’] =
TI,[’] = UIllT,[’l
(2.13)
From the fact that there are two, and only two, spin states for an electron and as a consequence of the antisymmetry principle, the only permutation states that occur in nature are those for which [ X I = [2p,lN-2p] 0
Q
p
Q
N/2
-S
-
-
-
+
+ (134) + (143)I[d + + (134) + (143) +
- (12)(34) +
with Tx[” any element in AN[’] called a structure projector x
(2.15)
Here p is called the permutation quantum number. It is related to the spin quantum number by the expression
p = N/2
-
(13) (14) - (34) (1211 = 4 (13) - (14) (34) (12) (123) (124) (1234) (1243)
El[’] = [d
(2.14)
Ur+t
I
i4
The a’s have the following rule of combination =
11(2) r3
where urtr is the permutation* that converts Tr[’]into TrJ[’]. That is
Urr‘U,‘,”
= [22]
E
~(o,)EI[’I orx
(3.1)
= (-1)U’X
(3.2)
where e(u1,)
is the parity of
(2.16)
Here
TI[’] =
The set of permutation states characterized by [A] = [2p,1N-2p]replaces, in spin-free quantum chemistry, the spin states in conventional theory. The tableaux associated with permutation states [A] = [2p,1N-2p]are two columned tableaux containing p rows of length two and N - 2p rows of length one. See Table IV.
I
I
1 3
121 141
12P
- 1l2Pl
(3.3)
3. Spin-Free Bond Functions In the spin-free formulation of quantum chemistry, the vector bond diagram D, is replaced by a Young tableau, Txlh1. A bond diagram with p bonds is replaced by a tableau containing p rows of length two and N 2p rows of length one. The partition of such a tableau is [ A ] = [2p,lN-2p1. Each pair of integers connected by an arrow in D, is placed in the same row of TXlx1 with the tail integer in the first column. The paired integers are placed in any order in the rows of length two and the unpaired integers in any order in the rows of length one. See Table VI. We associate
-
(8) The object6 in a tableau are numbers. A permutation P on a tableau takes a number i and converta it to P ( i ) all within the same poeition in the tableau. (9) The proof that x is independent of the order of rows of length two or one is straightforward. Let Ti[’] be a tableau that differs from TX[’]in the arrangement of the rows. Then (suppressing [XI) N x = N x , P i Px,r x k N k = r ( a x f ) N kand , oxk = SIX OX^. Thus by (2.12) and (2.14) i .
2
^x c(U1k)NrPrura = e(urfi)urcNd’k c(urxuxt)urxuxfiNkPk= e(urx)urxNxPx = u Volume 70. Number 6 May 1966
F. A. MATSEN,A. A. CANTU,AND R. D. POSHUSTA
1562
and
Table VI Structure projectors for [A] = [2*]; p = 2 (S = 0)’fNP = 2
The structure projector projects spin-free bond functions from a primitive function. Thus Projector,
rx[Xl
Dx
X
I 1 * - * 2 a
c- .3
4.
(-
rlx
x
I)OIX
(3.5)
EP]
1
4
13/41 -
where
c = 2Pp!(N - p ) !
4.
*3
4.
.3
I = 9
(3.6) Spin-free functions for N = 4 are given in Table VII. Matrix elements computed with functions 3.5 are identical with those obtained with Slater bond functiors6
- (13) - (24) + (13)(24) + (12) - (123) -
+ +
+
-
-
(142) (1423) (34) (134) (243) (1324) (12)(34) (1234) (1432) (14x23)
-
+ +
-
-
+
+
+
$ (13)(24) - (24) (132) (23) (1342) - (234) - (143) (14) (1243) (124) - (1432) (14)(23) (12)(34) (1234)
11 = -(13)
+
+
+
+
+
-
-
+ +
Table VII: Spin-Free Bond Functions, [A] 17) =
+ +
+ -
-
-
+
I
1. ---f - 2
1y;II) =
-
-3
1.
-2
j3/41
(13)(24)
1
I =
+
-
+
+
+
+
+
1y;II)
- I7;I).
-
+ 1y;III) = l/d(acbd - bcad - adbc + bdac + cabd - baed cdba + bdca + acdb - dcab - abdc + dbae + cadb - dacb - cbda + dbcaf’ -
=
E1[’](13)(24)
4. Matrix Elements over Spin-Free Bond Functions The matrix elements for a spin-free Hamiltonian over spin-free bond functions 1r;x) and 1y;x’) can be conveniently expressed in terms of an element ANIx1 which we denote by Zxx/and which we call the “bracket operator.” Thus (Y;xlHIY ; K O = (71H I Y ;Zx,J)
I1 I
4. c- - 3
+
- cabd -
+ dcab +
E P
1
-
__
+
-cbad abed cdab - adcb - bcad bdac adbc - cbda dbca cadb acbd dacb bcda dcba bade dabc)
*/4(
‘Note that by Rumer’s rule Iy;III)
J.1
4.
111
(1121
+
+
+
Structure projectors for [A] = [2,12]; p=I(S=l),f9=3
-
a(l)b(2)c(3)d(4)3 abed
Iy;I) = l/a(abcd - cbad - adcb cdab baed bdca cdba abdc - dbac - acdb bade - dabc - bcda idcba)
+
111 = +(23) - (132) - (234) (1342) (123) (12) (1423) (142) (243) (1324) - (34) (134) (1243) (124) (143) (14)’
= [2*]
If_!
I2 I
- (13) - (14) - (34) + (134) + (143) + (12) - (123) - (124) - (12)(34) + (1234) +
(4.1)
where the bracket operator is by (3.5) and (2.9)
+$
( 1243)
+
+ + + + 111 = +(13)(24) - (24) - (1342) - (1423) + (142) + (234) + (1324) - (243) - (132) - (14)(23) + (1432) + (23)b ‘Note that by Rumer’s rule, JII = 11 - I. Note t h a t for II =
-
+(13) - 9 - (134) (143) (14) (34) (132) - (23) (1324) (1432) (14)(23) (243)
-
-
[A] = [2,12],I, 11, and I11 are linearly independent.
The Journal of P h y s M Chmdstry
Herelo +.
=
=
UXIEI+€(UXI)
QxIPINIE(UxI)
(4.3)
By (2.11), (2.9), and (2.15) Zxx,= 0 unless p = p’ (10) Hereafter, we suppress [ A ] in Tx[X], Px[X],Nx[Xl, & [ X I , (see (5.3)).
Ex[’]
(4.4) and
SPIN-FREE QUANTUM CHEMISTRY
1563
i.e., unless D, and D,I contain the same number of bonds. We can obtain the Pauling numbers (P),,, for all permutation states and for all permutations by direct expansion of (4.2). We note that Z,,t is independent of the form of the spin-free primitive function 17). The values of the Pauling numbers for the states of N = 4 are given in Table VIII. An off -diagonal bracket element can be expressed in terms of a diagonal bracket element. By (4.2), (3.1),and (2.12) we have
zxn! = E(U,r)
t(~rx~~xr>ErtEi~rx~/c
z,
ZXd
5. Group Theoretical Formulation of the Pauling Numbers The diagonal bracket element Z, is an element in an invariant subalgebra ANIX1and as such can be exr,s = panded in an orthogonal matric basis5 (e,,[”]; 1 tof[’] of ANIX1.Here er8l’1 = cf[”/N!)E[P],,[’]P P
(5.1)
where [ P ] l s [ xisl the (r,s)th element in the matrix representing P in the orthogonal irreducible representation I?[’], Thus /[XI
z,
=
jhl
C C [ z ~ ~ I ~ (5.2) ~ [ T
8
For T X L xwe 1 define the idempotent element (see (-43)) E,
= (‘/Z”>PX
C
= P 4Qxd)
=
&,Z,,E,
=
(5.8)
[Plll[XIP~,,~
mJ,d) [P~,~,lll[XIP P
(5.9)
We compare (5.9) with (4.2) to obtain =
4UXd)
~PQx~xlll[xl
(5.10)
Equation 5.10, which is true for a given x and any x’, relates the Pauling numbers for arbitrary N and p (spin state) to an irreducible representation of was assumed to be the permutation group SN. an orthogonal representation with condition (5.5). It can be shown that the orthogonality requirement is not necessary and only condition (5.5) is needed to obtain (5.10). In the next section we go on to the special case treated by Pauling, p = N/2, and derive his rule from eq 5.10.
6. Explicit Evaluation of the Pauling Numbers for p = N / 2 For p = N/2 (N even) we have an explicit form for [ P c T , ~ , ]and ~ ~ [hence ~ ~ for (P),,/. The case for p = (N - 1)/2 (N odd) is obtained by deleting permuta~tions ~ involving ~ ~ ~the [( N~ ~ 1)th integer in the formula1)/2. We begin by considering tion for p = (N tableaux T , and T,I and a permutation P. We define a tableau
+
(5.3)
and we note that
P
= ~(~,,OZ,,~,,~
(P),,, (4.5)
c [Plll[xlP
where [P][xlGI’[X1. We obtain an off-diagonal bracket element by substituting (5.8) into (4.5)
= E(u,,r)Ex+Exuxwt/C = 4~,,’)Z,,Qxx~
=
T,*
+
PT,, = Pu,,,T,
(6.1)
PU,’,
(6.2)
Thus
&,P,N,N,P,E, = P,N,N,P, = Z,,
(5.4)
It is noted in Appendix A that for a given T , there = exists a form of the irreducible representation such that [E,I,,[X1
=
6,1681
Then by (5.4) and (5.5) we have [Zxxlrs[X1=
=
~ ~ x ~ n n ~ n l ~ s ~ A 1
&16,1
Next we construct new tableaux as
T ; E u;,T, T;* E u;*,*T,*
(6.3) where ukXtransposes along rows in T,, up*,* transposes along rows and rearranges rows in T,* so that T ; and T;* have identical first columns. Next we permute on the second column of T; so as to make it identical with Ti*. This is done with ut*p.
[Z,,111
On the substitution of (5.6) into (5.2) we obtain
z,
u,*, =
=
[Z,,I~~[%I~~’~
On the substitution of (5.1) and (B6) into (5.7)
T;*= u;,*;Tp
(6.4)
BY (A411 [ ~ x * , l l l [ x= l b;*klll[hl
=
(-‘/Zp
(6.5) where q is the minimum number of transpositions on volume 70,Number 6 May 1966
F. A. MATSEN,A. A. CANTU,AND R. D.
1564
POSHUST.4
Table VIII: The Pauling Numbers ( P ) x , ffor the Structures in Table VI xx'
9
(12)
(13)
(23)
-'/2
(123)
(132)
(14)
(24)
(34)
(124)
(142)
1
-'/z
-'/z
'/z
-1
p = 2(5 = 0 )
I1 I I1 I1 I I1 I1
1
1
-'/z
'12
'/2
'/Z
'/a
-1 -1
'/z
-1
-I/%
1
-'/z
I1 I I1 I I11 I1 I I1 I1 I1 I11 I11 I I11 I1 I11 I11
1
1
-'/z
-l/z
'/Z
-1
'/2
'/2
-'/z
-l/z
'/z '/z
-1 -1
1
-'/z
'/2
'/2
-'/z
'/2
-1
-'/2
p = 1 ( S = 1) -'/z
1
-'/z
1
-'/z
0 -'/z
1
0
1/2
'/Z
0
0
- '/z
'/2
-1
1
1
-'/z
-'/2
'/2
- '/2
-' / a
'/2
'/2
0
-'/z
'/2
0
'/2
-'/2
'/2
'/z
-'/2
0
'/z
-112
the second column of T k into which u p k can be decomposed. I n terms of the three tableau operations defined and by (2.14)
(6.6)
-'/2
-'/z
-'/2
0
-'/2
1 -'/2
0 -'/2
'/z
'/2
'/z
-'/2
1
'/2
'/2
-'/2
'/z
0
'/2
-1
-'/z
-1
0
'/z
0 -'/z
u,*, = u,*k*uk*ku~x
-1/2
1 '/2
-'/z
-'/z
-' / a 1
-1/2
- '/z 1
-'/z -'/z
-'/z -'/z
-'/2
-'/a
0 -'/2
1
-'/2
-'/z
-1 '/Z
-'/z '/Z -1/z
0 '/z
-1 -'/2
- '/2 '/Z
note further that u p i rdetermines the number of islands. For uk*$ = 8 , [DkDk*] contains N/2 islands. Each transposition in ui** reduces the number of islands by one. Consequently if u p k contains q transpositions
i = (N/2) - Q (6.10) Note that uxk= ukx. We substitute (6.10) into (6.9) and make use of the Consider the parity e(rk*,*)e(ukx) = E(uk*,*ukx). relation between Young tableaux and vector bond Since the permutations in ak*,* that rearrange rows diagrams to obtain the Pauling rule, eq 1.10. are of even parity, then the parity is just ( - l ) r , where r is the sum of the number of transpositions on rows 7. Conclusions contained in u p x * and ukx. By (6.2) and (6.6) We have formulated spin-free bond functions whose (-1)Ux*x = (-l)p+uxx' = (-1)'fq (6.7) matrix elements with a spin-free Hamiltonian are identical with those obtained with the spin-containing Then by (6.2) Slater bond functions. We have given a spin-free C ( b X X ! ) = (- 1)Uxx' = (- 1)P + r + n group theoretical derivation of the Pauling rules for (6.8) these matrix elements. Finally by (5.10), (6.2), (6.5),and (6.8) weobtain The results of this paper together with the results obtained in papers I and I1 of this series further establish the existence of a valid, operative spin-free quanI n summary (P)x,, is computed in the following tum chemistry. A belief in the existence of a spinPT,,. (b) Permute manner. (a) Form tableau T,* free quantum chemistry can prevent improper use of integers along rows in T , and permute integers along spin concepts in discussions of electron dynamics rows and rearrange rows in T,* so that the resulting of chemical systems and can develop a clearer view tableaux T k and TE*have identical first columns. Deof the forces which operate in these systems. note the sum of the transposition along rows by r. The mathematical tools [the theory of the permuta(c) Determine the minimum number of transpositions tion (symmetric) group] used in the spin-free formulaon the second column of T Enecessary to make it idention of quantum chemistry were in existence before tical with T,*. Denote the number of these transpothe discovery of spin. It would have been possible, sitions by q. (d) Apply eq 6.9. therefore, for quantum chemistry to have been deTo obtain Pauling's rule we replace the tableaux by veloped without knowledge of the existence of spin. The use of spin and the antisymmetry principle is an their equivalent vector bond diagrams. We note first that r determines the number of arrow reversals indirect way of introducing permutation group theory We required to obtain the superposition [DkDk*]. into quantum chemistry. The Journal of Physdcal Chemistry
SPIN-FREE QUANTUM CHEMISTRY
(134)
(143:
(234)
(243)
-'/2
-'/2
-'/z -1
-'/2
-1
'/z
-1 -'/z
-'/z
-'/2
'/t
'/2
-1
-'/2 '/2
- '/2
-'/2
'/Z
0
-'/z
-'/a
0
1
0
'/2
'/2
-1
'/2
(1423)
1
1
1
'/2
'/Z
'/2
'/e
'/2
'/Z
'/Z
'/Z
'/2
'/Z
1
1
1
1
1
'/E
-1
'/z
-'/z
z/'-
-'/2
'/2
-'/z '/a
'/2
0
'/2
'/2
-1
1
'/2
0
'/a
'/Z
0
The Evaluationll of (P),x, For a tableau T , with partition D] = [P', P2, . . .,PN1 we define a subgroup of SN composed of the elements which permute the integers along the rows of T,.
-'/2
0
-'/z
PI!
PZ!
. . . /P"
(AI)
and its index is
n, = N!/nB
E, = (l/ns)Ps
'/a
-'/z
0
1
-1
-I/z
'/Z
'/2
-'/2
0
0
0
-1
'/2
-'/t
0
1
-'/z -'/z '/Z
-'/z
'/2
'/2
-1 '/a
0
-1
'/z
'/2
-'/2
0
We can choose an irreducible representation of SN which is canonical to the symmetric idempotent E, of G,. That is, we can choose a representation such that
[ E ~ ] , ~=[ ~a,,I for T , S =
f [ ~ l [ ~ l
We designate this representation by ghan12has shown that13
(A2)
We denote by the symbol IIjs ( j = 1 to nu) a generator of the j t h left coset of G,. The symmetric idempotent of G, is
0
'/2
-'/z
We designate the group by G, and note that its order is
n@ =
0
-1
'/2
'/Z
-1/2
-'/z
Appendix A
(14)(23)
'/Z
1
'/2
(13)(24)
'/2
'/Z '/2
(12)(34)
'/Z
- 1/z
-1
(1243)
'/Z
0
'/2
(1342)
p = 1 ( S = 1) 0 0
-1
'/2
1 -'/z
-'/t
-I/?
-1
-'/z '/Z
'/Z
'/Z
0
-'/2
I/:!
-'/2
-'/z
'/Z
-
'/l
-1 -1
0 1/2
I/,!
'/2
0
-1
-1 -1
'/2
(1324)
p = 2 ( S = 0) 1 1
-'/2
'/Z
'/2
(1432)
-'/2
'/Z
-
-1
(1234)
'/z '/z
'/Z
1565
f[pl[']
=
o for
=
1 for [p] = [A]
> [A]
(A7) (A8) Murna-
(A91 (-410)
Consider first a tableau with [PI = [3,1N-3]
(-43)
With E, and the rIjs we generate a set of elements t;: =
II'%,j
=
1 ton,
(-44)
which are the basis of a subalgebra, AN5of A N . These of elements are also the basis for a representation S ., The representation is generally reducible. That is
where f[Pl[Al =
(1/N !>Ex @'(P>XIXI(P) P
(A61
is the number of times occurs in rBand x[@'(P)and x'@'(P)are characters in rB and I?['], respectively. We note that x'@'(P)and hence f[B1[rl depend only on the partition [ p ] associated with G, and not with G, itself.
Here (11) M. Kotani, "Calculations Functions d'Onde Moleculaire," CNRS, Paris, 1958; Encyclopedia of Physics, Vol. XXXVII/P, Springer-Verlag, Berlin, 1961. In these references Professor Kotani has developed eq A19 and a form of (A38). In the first reference he invoked the antisymmetry principle and then used representation theory. In the second reference his argument is based on the properties of spin eigenfunotions. Our derivation of these relations, while exploiting the Kotani approach, is entirely group theoretical and spin free. (12) E'. D. Murnaghan, "The Theory of Group Representations," Johns Hopkins University, Baltimore, Md., 1938. (13) Partitions are orders according to the dictionary order. Thus for N = 4, El'] < [2,121 < [2*1 < [3JI < [41.
Volulne 70,Number 6 M a y 1966
F. A. MATSEN,A. A. CANTU, AND R. D. PO~HUSTA
1566
Since [PI = [3,1N-3] > [x] = [2N’2], we have by (A8) and (A10)
and
+ (ef) + (4+ Crs) +
E , = ‘/of$
=
(efd + (esfll
g
l/Of
[Gg]tl[h’= 0
+ (es) + 0Pg)l { + (4
(-413)
In representation form (A14) becomes [ ~ ~ l r s [ ’ ]=
+ [(es) + [(fg) ([glt,”] +
‘/6c( [ g l , t [ ”
x
lrt”]>
1,t”I
t
Then by (A23)
[Ctltl[’]
(A14)
$
[(ef>1t8[’1)
(A241
+ [c:h[xl =
-at1
(A25)
We make use of the above relations to evaluate [Q,*,]I~[’~ of (6.5). By analogy to (3.1) and referring to (6.1) and (6.3) we have
(A151
x* = e(Q,,*)xu,,*
k* =
We consider now an irreducible representation with = [zNi2]. We wish to make the representation canonical to the idempotent G , of G, with
[XI
€( Q,*G*)
I; =
K* Q,*p
4Qxk)~Qxk
The 2’s (with carets) differ a t most by sign from the Tkand u,*p transposes along rows and rearranges rows of Tk*.
x’s because uXktransposes along rows of
Iz
and [x] = [z”2]. Let A t be a transposition on a row; Le., A: = (mm’), . . .. Then
AtP, = P,A: = P,
= E(U,k)X
f*
=
=
4Q,k)4Q,*k*)Zkk*
E(Q,**)X*
(-427) (A281
Then
z,*
(ZZ’),
(-429)
and
(A171
and thus
z,
=
Zkk
(A301
r k 1x1
(A311
Note that since E , = GP
zxx
(AW
[At]1f[’] = [A~]tl[’]= 6 t l
(A19)
AtZxx = Z , J : = Then by (5.6)
Let Ct be a transposition between rows; Le., C t = (I’m), (lm’), etc. Given a C,, consider an At which contains one of the integers in Ct. Then there is a CQ which consists of the remaining integers in A, and Ct and is given by
rxb1=
By analogy to (5.9) and by employing (A29) and (A30) we have
z,*
(Zm),
AtCtt
=
4Qxx*)C [PQ,*,I1l[A1p P
and
z,,*= 4 Q , k ) 4 Q , * k * ) Z k k *
(A201
=
4 Q w k ) 4Q,*k*)
The tableau in (All) is completely arbitrary; thus we take it to be the one with
=
E(Q,,*)ZkkQkk*
=
CIA,
(ef) = At Cfg) = CV
(A21)
€(Q,,*)C [Pak*kln[X1p P
[$]rt[xl= 6r f
(A221
then by (A15) and (A19) [ & g I t ~ [ ’= ~ 2/6(6tl
+ [Ci]t~[’]f [C:r]t~[”)
The Journal of Physical Chemistry
(A23)
(A331
On equating (A32) with (A33) and for P = 9 we have [~,*,lll[Xl =
Since
4“kk*)ZLkQPk*
= 4Q,,*)Z,,Qkk* =
(es) = Ct
(-432)
t Qk*k lll[xl
(-434)
Now uk+&can be decomposed into a product of some minimum number, say q, of transpositions on the second column of Tk. These are transpositions of the type Cr but on Tk.
SPIN-FREE QUANTUM CHEMISTRY
U&*&
=
1567
ClCZ. . .c,
(A35)
For C, consider the A , which contains the integer of C, not found in C1, CB, . . ., and C,-l. Then by (A20) and since A , will commute with C1, CZ,. . ., and Cq-l, we have C1Cz.
. .C,A,
=
AqC1C2. . . Cqf
Appendix B Evaluation of [2n,]ll[’1 By (4.5) and (4.2) Z,
=
E,‘E,/C
6436)
Thus by (AlY) and (A31)
[CiCz. . .Cq]ll[’l = [ClCz. . . CP~]11[’I
(A37)
Further by (A37) and (A25)
[ClCZ. . .c,Ill [’I
+
[CICZ. . .c,]ll[xl [ClCZ.. . c p ~ ] l l [ x l )
=
1/2(
=
‘/2C [ClC,. . .Cq-lllt[hl{[C,ltl[’] t
=
‘/zC [ClCZ.. .Cq-lllt[xl (-6t1) t
=
- ’/z [ClC,. . .Cn-l]11[’1
+
[c,’ltl~x~~
(A38)
Similarly, we have the existence of an A,-1 such that
CiCz. . . Cq-iAq-i = A,-iCiCz.
. .C(,-i)t
Then by (B3) and (B4)
(A39)
Thus by the above procedure we have
[CiCz. . .C~7]1i[’1 = (-1/z)2[C1Cz. . .C,-2]11[’]
(A40)
Doing this q times, by (A35), (A34) becomes ~ Q , * , l l l [ x= l
(-‘/d9
(-441)
Volume 70, Number 6 May 1966
