490 Chapter 17Determinants
This follows from the expansion of the three determinants along the relevant row or
column; column 1 in (17.43). Thus,
- Interchange of rows (or columns). Antisymmetry
If two rows (or two columns) of a determinant are interchanged the
value of the determinant is multiplied by (−1):
(17.44)
(17.45)
The determinant is said to be antisymmetricwith respect to the interchange of rows
or columns. It is this property of determinants that has made them so useful for the
construction of electronic wave functions.
EXAMPLE 17.11Determinant of order 2.
For interchange of columns,
ab
ab
ab ba ba ab
ba
ba
1
12212 12 12 121
122=−=− − =−()
abc
abc
abc
acb
acb
acb
1
112223331
11222333=− (inter rchange of columns and23)
abc
abc
abc
abc
abc
abc
1
112223332
22111333=− (inter rchange of rows and12)
=+
abc
abc
abc
dbc
dbc
dbc
1
112223331
11222333+−+d
bc
bc
d
bc
bc
d
bc
bc
12
23321
13331
122=−+a
bc
bc
a
bc
bc
a
bc
bc
12 23321 13331 122ad b c
ad bc
ad bc
ad
bc
bc
11
11222233 33112
233=+() −−+() ()ad ++
bc
bc
ad
bc
bc
221
133331
122