The Chemistry Maths Book, Second Edition

258 Chapter 9Functions of several variables

subject to the constraint

where theC

ij

are constants (withC

ij

1 = 1 C

ji

). For example, forn 1 = 13 ,

f(x

1

, x

2

, x

3

) 1 = 1 C

11

x

1

2

1 + 12 C

12

x

1

x

2

1 + 12 C

13

x

1

x

3

1 + 1 C

22

x

2

2

1 + 12 C

23

x

2

x

3

1 + 1 C

33

x

3

2

with constraintg 1 = 1 x

1

2

1 + 1 x

2

2

1 + 1 x

3

2

1 = 11.

By the method of Lagrange multipliers the stationary values of the function are

obtained by forming the auxiliary functionφ 1 = 1 f 1 − 1 λgand solving equations (9.14).

For the casen 1 = 13 , we have

φ 1 = 1 (C

11

1 − 1 λ)x

1

2

1 + 12 C

12

x

1

x

2

1 + 12 C

13

x

1

x

3

1 + 1 (C

221

− 1 λ)x

2

2

1 + 12 C

23

x

2

x

3

1 + 1 (C

33

1 − 1 λ)x

3

2

Differentiation with respect to x

1

,x

2

, and x

3

and setting each derivative to zero then

gives the set of simultaneous equations

(C

11

1 − 1 λ)x

11

• 1 C

12

x

2

• 1 C

13

x

3

1 = 10

C

21

x

1

1 + 1 (C

221

− 1 λ)x

2

1 + 1 C

23

x

3

1 = 10

C

31

x

11

• 1 C

32

x

21

• 1 (C

33

1 − 1 λ)x

3

1 = 10

Equations of this kind are often called secular equations. They occur, for example, in

the ‘method of linear combinations’ in quantum chemistry, when the quadratic form

represents the energy of the system (or an orbital energy in molecular orbital theory)

and the numbersx

1

, x

2

,1=, x

n

provide a representation of the state (of an orbital in

molecular orbital theory). The significance and solution of such systems of equations

are discussed in Chapters 17 and 19.

0 Exercise 30

9.5 The total differential

Letz 1 = 1 f(x, y)be a function of the variables xand y, and let the values of the variables

change continuously from(x, y), at point p in Figure 9.5, to(x 1 + 1 ∆x, y 1 + 1 ∆y)at point

q. The corresponding change in the function is

∆z 1 = 1 z

q 1

− 1 z

p 1

= 1 f(x 1 + 1 ∆x,y 1 + 1 ∆y) 1 − 1 f(x,y)

and is shown in the figure as the displacement P to Q on the representative surface of

the function; ∆zis the change of ‘height’ on the surface.

gx x ...x x

n

i

n

i

()

12

1

2

,,, = = 1

=

∑