The Chemistry Maths Book, Second Edition

17.7 Alternating functions 495

If two variables are equal the function is zero,

f(x

1

, x

1

, x

3

, =, x

n

) 1 = 10 (17.53)

A determinant is an alternating function of its rows (or columns). More importantly, a

determinant that is an alternating function of nvariables,x

1

, x

2

, x

3

=, x

n

, has the form

(17.54)

wheref

1

,f

2

, =,f

n

are arbitrary functions. The interchange of any pair of variables

leads to the interchange of two columns and, therefore, to a change of sign.

Forn 1 = 12 ,

(17.55)

Forn 1 = 13 ,

(17.56)

The expansion of the determinant has n! products of the functionsf

1

,f

2

, =,f

n

, each

with a distinct ordering of the nvariablesx

1

, 1 x

2

,1=, 1 x

n

; these orderings are the n!

permutations of nobjects. Thus, in (17.56), the3! 1 = 16 permutations ofx

1

,x

2

, andx

3

are

x

1

x

2

x

3

, x

1

x

3

x

2

, x

2

x

3

x

1

, x

2

x

1

x

3

, x

3

x

1

x

2

, x

3

x

2

x

1

(17.57)

Each term contributes to the sum with +sign if the permutation is obtained from

x

1

x

2

x

3

by an evennumber of transpositions, and with – sign for an oddnumber of

transpositions.

0 Exercise 29

Alternating functions in the form of single determinants or sums of determinants

are important in quantum mechanics for the construction of electronic wave

=

−

• 
  

fxfx fx fxfx fx

fx

112 2 33 112332

1

()()() ()()()

(

2 22331 122133

132

)()() ()()()

()

fxfx fxfxfx

fx f

−

• (()() ()()()xfx fxfxfx

132 132231

−















fx fx fx

fx fx fx

fx f

11 12 13

21 22 23

31

() () ()

() () ()

()

3 32 33

() ()xfx

fx fx

fx fx

fxfx fx

11 12

21 22

112 2 12

() ()

() ()

=−()() ())()fx

21

fx fx fx fx

fx fx fx

11 12 13 1 n

21 22 23

() () () ()

() () ()







fx

fx fx fx fx

fx

n

n

n

2

31 32 33 3

1

()

() () () ()

() ffx fx fx

nn nn

() () ()

23