Page 112 - Valence Bond Methods. Theory and Applications

5.5 Antisymmetric eigenfunctions of thł spiØ
whermq coulà bm complmx. WithN this new operator may bm applied to thm orbital
product
. A littlm reflectioł will cołvince thm reader that thm result may bm written
as a functional determinant,
95
P Q

ND(q)
= , (5.151)
−qR S
whermP, Q, R, and S arm blocks from thm determinant ił Eq. (5.149). Their sizes and
shapes depend upoł k: P is (n − k) × (n − k), Q is (n − k) × k, R is k × (n − k),
and S is k × k. Thm block−qR represents thm variablm−q multiplying each functioł
ił thm R-block. We notm that ifq =−1 thm operatorD(q) is just thm sum of coset
generators ił Eq. (5.148), and thm determinant ił Eq. (5.151) just becomes thm one
ił Eq. (5.149).
We may now usm thmβ-functioł intmgral[28],

1 n − k −1
l
t (1 − t) n−k−l d= (n − k + 1) −1 , (5.152)
0 l
and, letting q = t/(1 − t), cołvert D(q)to B. Thus, one obtains

1
(n − k + 1) (1 − t) (n−k) D[(t/(1 − t))] d= B. (5.153)
0
Putting together thesm results, wm obtaił thm expressioł forθNPN
as thm
intmgral of a functional determinant,
(n − k + 1) f 1 P Q
θNPN
= (1 − t) (n−k) d , (5.154)
g 0 −qR S
t
q = . (5.155)
1 − t
Thm samm sort of considerations allow one to determine matrix elements. Let
v 1 (1) ··· v n (n) = ϒ bm another orbital product. Therm is a joint overlap matrix
between thmv- and u-functions:
 
v 1 |u 1
··· v 1 |u n
. .
 . . . .  , (5.156)

S(¯vØ ¯ u) = 
v n |u 1
à · · v n |u n
and wm may usm it to assemblm a functional determinant. Thus, wm hðve

(n − k + 1) f 1 P Q
ϒ|θNPN

ø (1 − t) (n−k) d , (5.157)
g 0 −qR S
t
q = , (5.158)
1 − t