of the molecule and b is in the direction of the molecular symmetry

axis. They use the following traceless de¬nition of the quadrupole

moment:

1

ρ(r)[3x± xβ ’ r2 δ±β ] dr.

θ±β =

2

From these data, derive the quadrupole moment Q±β as de¬ned in

(13.119) on page 356, expressed in “molecular units” e nm2 (see

Table 8 on page xxv), and in a coordinate system with its origin

in the position of the oxygen atom. Use the following data for the

transformation: OD-distance: 0.09584 nm, DOD-angle: 104—¦ 27 ,

dipole moment: 1.85 Debye, oxygen mass: 15.999 u, deuterium mass:

2.014 u. An esu (electrostatic unit) of charge equals 3.335 64 — 10’10

C; the elementary charge e equals 4.8032 — 10’10 esu. Give the

accuracies as well.

14

Vectors, operators and vector spaces

14.1 Introduction

A vector we know as an arrow in 3D-space with a direction and a length,

and we can add any two vectors to produce a new vector in the same space.

If we de¬ne three coordinate axes, not all in one plane, and de¬ne three

basis vectors e1 , e2 , e3 along these axes, then any vector v in 3D-space can

be written as a linear combination of the basis vectors:

v = v1 e1 + v2 e2 + v3 e3 . (14.1)

v1 , v2 , v3 are the components of v on the given basis set. These components

form a speci¬c representation of the vector, depending on the choice of basis

vectors. The components are usually represented as a matrix of one column:

⎛ ⎞

v1

v = ⎝ v2 ⎠ . (14.2)

v3

Note that the matrix v and the vector v are di¬erent things: v is an entity

in space independent of any coordinate system we choose; v represents v

on a speci¬c set of basis vectors. To stress this di¬erence we use a di¬erent

notation: italic bold for vectors and roman bold for their matrix represen-

tations.

Vectors and basis vectors need not be arrows in 3D-space. They can

also represent other constructs for which it is meaningful to form linear

combinations. For example, they could represent functions of one or more

variables. Consider all possible real polynomials f (x) of the second degree,

which can be written as

f (x) = a + bx + cx2 , (14.3)

where a, b, c can be any real number. We could now de¬ne the functions 1, x,

379

380 Vectors, operators and vector spaces

and x2 as basis vectors (or basis functions) and consider f (x) as a vector

with components (a, b, c) on this basis set. These vectors also live in a real

3D-space R3 .

14.2 De¬nitions

Now we wish to give more general and a bit more precise de¬nitions, without

claiming to be mathematically exact.

• A set of elements, called vectors, form a vector space V over a scalar

¬eld F when:

(i) V is an Abelian group under the sum operation +;

(ii) for every v ∈ V and every a ∈ F : av ∈ V;

(iii) for every v, w ∈ V and a, b ∈ F:

a(v + w) = av + bw

(a + b)v = av + bv

(ab)v = a(bv)

1v = v

0v = 0

A scalar ¬eld is precisely de¬ned in set theory, but for our purpose it

su¬ces to identify F with the set of real numbers R or the set of complex

numbers C. An Abelian group is a set of elements for which a binary

operation + (in this case a summation) is de¬ned such that v +w = w +v

is also an element of the set, in which an element 0 exists for which

v + 0 = 0 + v, and in which for every v an element ’v exists with

v + (’v) = 0.

• A vector space is n-dimensional if n vectors e1 , . . . , en exist, such that

every element v ∈ V can be written as v = n vi ei . The n vectors

i=1

must be linearly independent, i.e., no non-zero set of numbers c1 , c2 , . . . , cn

exists for which n ci ei = 0. The vectors e1 , . . . , en form a basis of V.

i=1

• A vector space is real if F = R and complex if F = C.

• A vector space is normed if to every v a non-negative real number ||v||

is associated (called the norm), such that for every v, w ∈ V and every

complex number c:

(v) ||cv|| = |c|||v||;

(vi) ||v + w|| ¤ ||v|| + ||w||;

(vii) ||v|| > 0 for v = 0.

• A vector space is complete if:

14.3 Hilbert spaces of wave functions 381

(viii) for every series v n with limm,n’∞ ||v m ’ v n || = 0 there exists a v

such that limm,n’∞ ||v ’ v n || = 0. Don™t worry: all vector spaces

we encounter are complete.

• A Banach space is a complete, normed vector space.

• A Hilbert space H is a Banach space in which a scalar product or inner

product is de¬ned as follows: to every pair v, w a complex number is asso-

ciated (often denoted by (v, w) or v|w ), such that for every u, v, w ∈ H

and every complex number c:

cv|w = c— v|w ;

(ix)

(x) u + v|w = u|w + v|w ;

v|w = w|v — ;

(xi)

(xii) v|v > 0 if v = 0;

||v|| = v|v 1/2 .

(xiii)

• Two vectors are orthogonal if v|w = 0. A vector is normalized if

||v|| = 1. A set of vectors is orthogonal if all pairs are orthogonal and the

set is orthonormal if all vectors are in addition normalized.

14.3 Hilbert spaces of wave functions

We consider functions ψ (it is irrelevant what variables these are functions

of) that can be expanded in a set of basis functions φn :

ψ= cn φn , (14.4)

n

where cn are complex numbers. The functions may also be complex-valued.

We de¬ne the scalar product of two functions as

φ— ψ d„,

(φ, ψ) = φ|ψ = (14.5)

where the integral is over a de¬ned volume of the variables „ . The norm is

now de¬ned as

ψ — ψ d„.

||ψ|| = (14.6)

These de¬nitions comply with requirements (viii) - (xii) of the previous

section, as the reader can easily check. Thus the functions are vectors in a

Hilbert space; the components c1 , . . . , cn form a representation of the vector

ψ which we shall denote in matrix notation by the one-column matrix c.

382 Vectors, operators and vector spaces

The basis set {φn } is orthonormal if φn |φm = δnm . It is not manda-

tory, but very convenient, to work with orthonormal basis sets. For non-

orthonormal basis sets it is useful to de¬ne the overlap matrix S:

Snm = φn |φm . (14.7)

The representation c of a normalized function satis¬es1

c† c = c— cn = 1 (14.8)

n

n

on an orthogonal basis set; on an arbitrary basis set c† Sc = 1.