3. Mathematical tools for quantum chemistryMathematical tools for quantum chemistry 4. The postulates of quantum mechanics 5. Atoms and the ‘periodic’ table of chemical elements

Definition of the geometrical structure of a molecule

Methane

CH4

Ammonia

NH3Water

H2O

Dimethyl

ether

C2H6O

Ethyne

C2H2

Ethene

C2H4

Ethane

C2H6

Ethanol

C2H6O

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 2/ 42-3

1. Historical introduction

2. The Schrodinger equation for one-particle problems

3. Mathematical tools for quantum chemistry

4. The postulates of quantum mechanics

5. Atoms and the ‘periodic’ table of chemical elements

6. Diatomic molecules

7. Ten-electron systems from the second row

8. More complicated molecules


Coordinates for ethanol, C2H5OH:

Cartesian coordinates: 3N

data for an N-atomic molecule

C 0.000000 0.000000 0.000000

C 0.000000 0.000000 1.450000

H 1.026719 0.000000 1.813000H 0.513360 -0.889165 -0.363000

H -0.513360 0.889165 1.813000

H -1.026719 0.000000 -0.363000

H -0.513360 -0.889165 1.813000

O 0.659967 1.143095 -0.466667

H 0.659967 1.143095 -1.413667

Z matrix: 3N − 6 data for a

non-linear N-atomic molecule

zmat angstroms

c

c 1 cc2h 2 hc3 1 hcc3

h 1 hc4 2 hcc4 3 dih4




o 1 oc8 2 occ8 3 dih8h 8 ho9 1 hoc9 2 dih9

variables

cc2 1.450000

hc3 1.089000

hcc3 109.471

...constants

oc8 1.400000

...

end


The definition of molecular structure requires knowledge about type

(i.e. charge number Z) and position (x, y, z) of the nuclei.

Only the relative position of the nuclei is important (translation and

rigid rotation are excluded).

Two frequently used systems of coordinates are

(i) cartesian coordinates, and

(ii) Z matrix coordinates.

The Z matrix definition uses a (suitably chosen) sequence of spherical

coordinate systems, and gives values for radius r, polar distance θ and

azimuthal angle ϕ.

A position vector RK, pointing from the origin of the coordinate

system to the atomic nucleus, is thus known for each nucleus K.


Operators

Operators O represent mathematical instructions, or operations. These

operations can be applied to suitably chosen mathematical objects,

to yield new mathematical objects of the same or different kind.

Operators can be divided into classes, depending on the number of

objects required for their application:

1. Unary operators require a single mathematical object:

O(a) = z

Examples:

• f in f(x), e.g. f = ( )2, f = sin (), f = exp ()

• Important unary operators are linear operators L:

L(x+ y) = Lx+ Ly


In our typical representations for molecules (crystals, etc.) only the

nuclear coordinates have some significance (we ignore, for the mo-

ment, the nuclear motion). Everything else, like balls, sticks, ribbons,

etc. is merely an eye-guide, though a very useful one.


• ‘+’ and ‘·’ in a+ b and a · b = a b

• scalar product and vector product in R3:

a · b = a b cos (ϑ) = b · a a = |a| , b = |b| , ϑ = ](a,b)

a × b = a b sin (ϑ)n = − b × a n · a = 0 , n · b = 0 , |n| = 1

• matrix multiplication:

A(n×m) · B(m×k)= A(n×m) B(m×k) = C(n×k)

A

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=

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3. Ternary operators require an ordered triple of objects:

O(a, b, c) = z

Examples:


- L =d

dxin f ′(x) =

df

dx

- L =

∫dx in

∫dx f(x) =

∫f(x) dx

- the matrices in matrix-vector products:

A(n×m) x(m×1) = b(n×1)

(A

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) x

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=

(b

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)

xT(1×m) B(m×k) = c

T(1×k)

(xT

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) B

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)

2. Binary operators require an ordered pair of objects:

O(a, b) = z

Examples:


Groups

Given one set of elements S = {a, b, . . .} and one binary operation ◦.

A group G = (S, ◦) is formed, if the following axioms hold:

1. closure: a ◦ b = c with a, b, c ∈ G2. existence of a neutral element e: e ◦ a = a ◦ e = a

3. existence of inverse elements a−1: a−1 ◦ a = a ◦ a−1 = e

4. associative law: a ◦ (b ◦ c) = (a ◦ b) ◦ c

Subgroup:

A subset of group elements, which constitutes a group (according to

the criteria given above).

Order of the group:

The number of elements in the group is known as the order g = |G|of the group (g ∈ N, or g =∞ for continuous groups).


• triple scalar product in R3:

a · (b × c) =

∣∣∣∣∣∣∣

a1 a2 a3b1 b2 b3c1 c2 c3

∣∣∣∣∣∣∣

• triple vector product in R3:

a × (b × c) = λb− µ c , λ = a · c , µ = a · b


Permutation groups

The permutations of n objects form a group. The rule of combination

(binary operation) is ‘subsequent application’.

This group is the permutation group, or symmetric group Sn, which

has order g = n!.

When the objects are simply the first n positive integers, a general

notation for a permutation is

Pk =

(1 2 3 4 . . . ni1 i2 i3 i4 . . . in

), ij 6= il , 1 ≤ k ≤ n! (74)

The neutral element in a permutation group is the ‘identity permu-

tation’, which leaves all n objects at their places:

P1 =

(1 2 3 4 . . . n1 2 3 4 . . . n

)= e (75)


Abelian groups:

If, in addition to the criteria given above, the commutative law a◦b =

b ◦ a holds (i.e. all elements of the group commute), the group is

called a commutative (or Abelian∗) group.

Some well-known examples for Abelian groups:

integers : (Z,+)

real numbers : (R,+) , (R \ {0} , ·)

(the order g is denumerably infinite in the first case, the last two cases

represent continuous groups)

Generators of a finite group:

A subset of group elements from which all group elements can be

formed (usually, there are several possible choices for a set of gener-

ators).

∗N. H. Abel (1802-1829)


A subset of the transpositions (ij) can be taken as generators for

the permutation groups, i.e. all permutations can be expressed by a

sequence of transpositions. Depending on the number of transposi-

tions involved, the permutations can be separated in even and odd

permutations. The former require an even number of transpositions,

the latter require an odd number.

Some examples for n = 3:

e =

(1 2 31 2 3

)= (1)(2)(3) 0 (even)

(1 2 32 1 3

)= (12) 1 (odd)

(1 2 32 3 1

)= (123) = (23)(13) 2 (even)

(1 2 33 1 2

)= (132) = (13)(23) 2 (even)

Note: Start at the rightmost cycle, and work from right to left!


The next-to-trivial permutations are the permutations of pairs, or

transpositions, e.g.

P2 =

(1 2 3 4 . . . n2 1 3 4 . . . n

)(76)

Another permutation, which involves already three objects, is

P3 =

(1 2 3 4 . . . n2 3 1 4 . . . n

)(77)

A shorter notation for the permutations is the notation with so-called

cycles. This gives for our examples

P1 =

(1 2 3 4 . . . n1 2 3 4 . . . n

)= (1)(2)(3)(4) . . . (n) = e (78)

P2 =

(1 2 3 4 . . . n2 1 3 4 . . . n

)= (12)(3)(4) . . . (n) (79)

P3 =

(1 2 3 4 . . . n2 3 1 4 . . . n

)= (123)(4) . . . (n) (80)

Cycles of length 1 are usually omitted.


Thus, Ψa(1,2) changes sign, or in other words, it is antisymmetric

under this operation ‘transposition of 1 and 2’, whereas Ψs(1,2) is

not changed, in other words, it is symmetric under this operation.

The function Ψa(1,2) can be written in the form of a determinant:

Ψa(1,2) =

∣∣∣∣∣∣f(r1) f(r2)

g(r1) g(r2)

∣∣∣∣∣∣=

∣∣∣∣∣∣f(r1) g(r1)

f(r2) g(r2)

∣∣∣∣∣∣(85)

Such a ‘determinant representation’ is always possible for a totally

antisymmetric many-particle wave function, if it is approximated by

products of single-particle functions.

Real- or complex-valued single-particle functions, which are square

integrable, i.e.∫f∗(r) f(r) dr = N2 <∞, are called ‘orbitals’. Single-

particle functions which include the spin coordinate, i.e. f(x) with

x = (r, σ), are called ‘spin orbitals’. A determinant built from spin

orbitals is known as Slater† determinant.† J. C. Slater (1900-1976)


Permutations and functions

Given two functions, depending on the position coordinates of two

particles:

Ψa(1,2) = f(r1)g(r2)− g(r1)f(r2) (81)

Ψs(1,2) = f(r1)g(r2) + g(r1)f(r2) (82)

How are these functions affected by the permutation P = (12), i.e.

the interchange of the particles (or particle coordinates)?

(12)Ψa(1,2) = Ψa(2,1) = f(r2)g(r1)− g(r2)f(r1) = −Ψa(1,2)

(83)

(12)Ψs(1,2) = Ψs(2,1) = f(r2)g(r1) + g(r2)f(r1) = + Ψs(1,2)

(84)


Symmetry groups (point groups, space groups)

A set of symmetry operations (covering operations, which may be ap-

plied to a rigid body in 3-dimensional space, e.g. a molecule with fixed

structure) constitutes a symmetry group. The rule of combination

(binary operation) is ‘subsequent application’.

Symmetry groups are, in general, non-Abelian groups, i.e. the se-

quence of application of symmetry operations is important.

At least one point remains fixed in space under all point group sym-

metry operations, while space groups include also translations as sym-

metry operations.

Operation (in connection with symmetry groups):

A transformation of coordinates, or alternatively a transformation of

a molecule to a new position.


A Slater determinant for three particles (n = 3):

Ψ(1,2,3) = N

∣∣∣∣∣∣∣

φ1(x1) φ1(x2) φ1(x3)φ2(x1) φ2(x2) φ2(x3)φ3(x1) φ3(x2) φ3(x3)

∣∣∣∣∣∣∣(86)

N denotes a normalization constant. The choice φ1(x) = 1s(r)α(σ),

φ2(x) = 1s(r)β(σ), and φ3(x) = 2s(r)α(σ) for the spin orbitals yields

a valid approximate state function for the ground state of the Li atom,

1s2 2s1 2S.

Note that ‘every particle uses every function’, i.e. there is no relation

or association between the coordinates (of a particle) and the single-

particle functions which have these coordinates as arguments.

A Li atom in the ground state ‘has an occupied 2s orbital’, but there

is no ‘2s electron’.


Symmetry element Symmetry Definition of

Symbola Name operations symmetry operations

— — E Identity (neutral element)

Proper symmetry operations

Cn n-fold rotation axis Ckn Rotation through φ = k · 2π/n,

(usually assumed to 1 ≤ k ≤ n, about the principal axisbe in the z direction)

C ′2, C′′2 2-fold rotation axis C ′2, C

′′2 Rotation through φ = π

perpendicular to the about the axisprincipal Cn axis

Improper symmetry operations

Sn n-fold rotation- Skn Rotation through φ = k · 2π/n,reflection axis combined with reflection k times

in a plane normal to the axis,n even: 1 ≤ k ≤ n, n odd: 1 ≤ k ≤ 2n

i (= S2) inversion center i (= S2) Inversion through the origin

σ (= S1) mirror plane σ (= S1) Reflection in a planeσv, σh, σd (vertical, horizontal, σv, σh, σd

dihedral planes)

aNotation due to A. Schoenflies (1853-1928)


Symmetry operation:

An operation (not necessarily a physically feasible one) that carries

a molecule into a new position which is indistinguishable from (or

equivalent to) the original position.

Proper operations:

Pure rotations about a specified axis (these are physically feasible).

Improper operations:

These may be regarded as rotations-reflections (or alternatively rota-

tions-inversions, these are not physically feasible).

Symmetry element:

A geometrical entity (point, line or plane) related to a symmetry

operation.


Symmetry operations and the ammonia molecule

3

1

2

2

3

1

φ = 2π/3 σ

σ’

1

2

3

. x

y

2

1

2

3

σ φ=2π/3

3

1

3

2

1

σ’’


The symmetry operation C2 and the water molecule

21

1

2

12φ = π/2φ = π/2

φ = π. yx


Determination of point groups Cnv‡ (Examples: H2O [n = 2], NH3 [n = 3])

C1

Cs

Ci

C∞v D∞h

Cn

S2n

Cnv

CnhDn

Dnh

Dnd

D2dT Th

Td O Oh

I Ih

Linear?

Unique Cn of highest order?i?

i?

i?

i? i?

6C5? S2n ‖Cn?

4C3? nσd?

3C4? 3C2? nC2 ⊥ Cn?

3S4? σ? σh? nσv?

2σd? σh?

y

y y

y y

y y y

y y

y y y y y

y y y y

n

n n

n n

n n n

n n n

n n n n n

n n n n

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‡J. A. Salthouse, M. J. Ware: Point group character tables and related data. Cambridge, 1972.


An algorithm to determine the point group from symmetry elements‡

C1

Cs

Ci

C∞v D∞h

Cn

S2n

Cnv

CnhDn

Dnh

Dnd

D2dT Th

Td O Oh

I Ih

Linear?


i?

i?

i? i?

6C5? S2n ‖Cn?

4C3? nσd?

3C4? 3C2? nC2 ⊥ Cn?

3S4? σ? σh? nσv?

2σd? σh?

y

y y

y y

y y y

y y

y y y y y

y y y y

n

n n

n n

n n n

n n n

n n n n n

n n n n

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Point group Generating Symmetry Order Commentssymbol operations elements g

C1 E none 1 no symmetry

Cs σ σ 2 Cs= C1h = C1v = S1

Ci i i 2 Ci= S2

Cn Cn Cn n

S2n S2n Cn, S2n 2n

Cnh Cn, σh Cn, σh, Sn 2n if n odd: Cnh= Sn

Cnv Cn, σv Cn, nσv 2n n-gonal regular pyramid

Dn Cn, C′2 Cn, nC

′2 2n

Dnh Cn, C′2, σh Cn, nC

′2, Sn, σh, nσv 4n n-gonal archimedian prism

Dnd Cn, C′2, σd Cn, nC

′2, nσd, S2n 4n n-gonal archimedian antiprism

C∞v C(z)∞ , σv C∞, ∞σv ∞

D∞h C(z)∞ , C ′2, σh C∞, ∞σv, S∞, ∞C ′2 ∞

T C(xyz)3 , C(z)

2 4C3, 3C2 12

Th C(xyz)3 , C(z)

2 , i 4C3, 3C2, 4S6, 3σv 24

Td C(xyz)3 , S(z)

4 4C3, 3C2, 3S4, 6σd 24 regular tetrahedron

O C(xyz)3 , C(z)

4 4C3, 3C4, 6C2 24

Oh C(xyz)3 , C(z)

4 , i 4C3, 3C4, 6C2, 3S4, 48 regular octahedron4S6, 3σh, 6σd

I C(ico)3 , C(z)

5 6C5, 10C3, 15C2 60

Ih C(ico)3 , C(z)

5 , i 6C5, 10C3, 15C2, 120 regular icosahedron12S10, 10S6, 15σ

K C∞ ∞C∞ ∞Kh C∞, i ∞C∞, ∞S∞ ∞ sphere


Determination of the point group D2h‡ (Example: C2H4)

C1

Cs

Ci

C∞v D∞h

Cn

S2n

Cnv

CnhDn

D2h

Dnd

D2dT Th

Td O Oh

I Ih

Linear?


i?

i?

i? i?

6C5? S2n ‖Cn?

4C3? nσd?

3C4? 3C2? nC2 ⊥ Cn?

3S4? σ? σh? nσv?

2σd? σh?

y

y y

y y

y y y

y y

y y y y y

y y y y

n

n n

n n

n n n

n n n

n n n n n

n n n n

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The new position vector r′, which results from the action of the

operation R(φn) on the old position vector r, is now representable as

a linear combination of these basis vectors:

r′ = R(φn) r = a r⊥+ bn × r+ c r‖= a r+ bn × r+ (c− a) (n · r)n= (a+ bn ×+(c− a)nn ·) r

with a = cos (φ), b = sin (φ), and c = +1 for pure rotations (or c = −1

for rotations-reflections).

For the application of any symmetry operation R in a point group to a

point r in space we may thus write r′ = R±(φn) r, or r′ = R±(φn) r in

matrix-vector notation. Explicitly, with coordinates (or vector com-

ponents):

x′

y′

z′

=

a+ (c− a)n21 (c− a)n1n2 − bn3 (c− a)n1n3 + bn2

(c− a)n1n2 + bn3 a+ (c− a)n22 (c− a)n2n3 − bn1

(c− a)n1n3 − bn2 (c− a)n2n3 + bn1 a+ (c− a)n23

xyz

(87)


Transformation of vectors in R3

Introduce a suitable basis to describe the

(active) rotation of a position vector r

around an axis through the origin in di-

rection of the unit vector n by an angle φ

into a new position vector r′ = R(φn) r:

1. The component of r parallel to n:

r‖ = (n · r)n = n (n · r)

2. The orthogonal complement to r‖:

r⊥ = r − r‖3. The vector normal to the plane

spanned by n and r:

n × r = n × (r‖+ r⊥) = n × r⊥

z

x y

rr’

n

n=

n1n2n3

|n| = 1


What happens to the function f(r) = f(x, y, z) = x y exp (−r2) un-der the counterclockwise rotation around the z axis (n = (0,0,1)T)through an angle φ = π/4 = 2π/8?

C8 → R =

1√2

1√2

0

− 1√2

1√2

0

0 0 1

C−18 → R

−1 =

1√2− 1√

20

1√2

1√2

0

0 0 1

R−1

r = R−1

xyz

=

1√2(x− y)

1√2(x+ y)

z

ORf(r) = f(R−1r)

= f(1√2(x− y), 1√

2(x+ y), z)

=1

2(x2 − y2) exp (−r2)

−2 −1 0 1 2

−2

−1

0

1

2

x

y

z = 0

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With φ = 2π/n and suitably chosen n:

R = Cn −→ R+(φn) , R = Sn −→ R−(φn) (88)

For every point group, the resulting set of matrices R±(φn) forms a

group under matrix multiplication, which is isomorphic to the point

group.

Transformation of scalar functions

With knowledge about the tranformation of position vectors, r, the

transformation law for scalar functions of the coordinates, f(r), can

be derived.

The condition of equality of function values, i.e. the transformed

function ORf shall have at the transformed position r′ = Rr the

same function value as the original function f at the original position

r = R−1r′, leads to:

ORf(r′) = f(r) = f(R−1r′) ⇒ ORf(r) = f(R−1r) (89)

since this relation should be valid for every argument r in R3.


Linear vector spaces

Given two sets (in fact not only sets, but algebraic structures):

(1) an abelian group G = (S,⊕), generated from a set of ‘vectors’§

S = {| a 〉, | b 〉, . . . , | v 〉, . . .}, and a binary operation ’⊕’ called ‘vector

addition’ (with the ‘null vector’ | o 〉 as neutral element),

(2) a field F = (K,+, ·) (usually K = R or K = C) with elements α, β,γ, . . ., called ‘scalars’.

A linear vector space V over the field F is formed, if in addition to

the above

1. a scalar multiplication (S multiplication) is defined as:

α | a 〉 = | a 〉α = | v 〉 ∈ V (90)

2. distributive and associative laws hold:

(α+ β) | a 〉 = α | a 〉 ⊕ β | a 〉α (| a 〉 ⊕ | b 〉) = α | a 〉 ⊕ α | b 〉 (91)

(αβ) | a 〉 = α (β | a 〉)§ These are called ‘ket’ vectors in the Dirac notation used here.


Fields

Given one set of elements S = {a, b, . . .} and two binary operations ◦and ∗.

A field F = (S, ◦, ∗) is formed, if the following axioms hold:

1. (S, ◦) is an Abelian group (with neutral element 0)

2. (S \ {0} , ∗) is an Abelian group (with neutral element 1)

3. distributive laws: a∗(b◦c) = (a∗b)◦(a∗c), (a◦b)∗c = (a∗c)◦(b∗c).

Some well-known examples:

rational numbers : (Q,+, ·)real numbers : (R,+, ·)

complex numbers : (C,+, ·)


i.e. there exists only the trivial solution, then the set of vectors

{| ai 〉} is called linearly independent (and linearly dependent

otherwise).

(b) a basis (or basis set), i.e. a set of vectors {| bk 〉} which is linearly

independent and capable of representing an arbitrary vector | v 〉through linear combination:

n∑

k=1

βk | bk 〉 = | v 〉 for any | v 〉 ∈ V (94)

The number n ≥ 1 (n ∈ N) of basis vectors is the dimension

of V . This dimension can be finite (n < ∞) or denumerably

infinite (n = ∞), and even the case of a continuum (n = ∞)

could be included, if we change the discrete summation in eq.

(94) to an integration in a suitable way. For n = ∞, however,

the convergence of the expansion in eq. (94) cannot be taken

for granted. If convergence (pointwise or in the mean) holds for

n =∞, the set {| bk 〉} is called complete.


Usually, the binary operations ’+’ and ’⊕’ (addition of scalars and

vectors, respectively) are not distinguished any further, and only the

symbol ’+’ is used to denote both.

For K = R the linear vector space V is called a real linear vector space,

whereas for K = C it is called a complex linear vector space.

These definitions include already, or are easily extended to include:

(a) linear combinations, i.e. a weighted sum of an arbitrary finite

number l of vectors (the limit l→∞ requires further study):

| v 〉 =l∑

i=1

αi | ai 〉 =l∑

i=1

| ai 〉αi (92)

If, for the special case | v 〉 = | o 〉,l∑

i=1

αi | ai 〉 = | o 〉 ⇒ αi = 0 (for all i) (93)


- triangle inequality:

‖u+ v‖ ≤ (‖u‖+ ‖v‖)2;- Cauchy-Schwarz inequality:

|〈u | v 〉|2 = 〈u | v 〉〈u | v 〉∗ = 〈u | v 〉〈 v |u 〉 ≤ 〈u |u 〉〈 v | v 〉;- parallelogram equality:

‖u+ v‖2 + ‖u− v‖2 = 2(‖u‖2 + ‖v‖2)

A vector | v 〉 with ‖v‖ = 1 is called ‘normalized to unity’ (or just

‘normalized’).

Two vectors |u 〉, | v 〉 with 〈u | v 〉 = 0 are called ‘orthogonal to each

other’ (or just ‘orthogonal’).

A set of vectors {| vk 〉} (k = 1, . . . ,m) with

〈 vk | vl 〉 = δkl =

{1 for k = l0 for k 6= l

(96)

is called an ‘orthonormal’ set.


With any ordered pair of vectors, (|u 〉, | v 〉), may be associated a

scalar product 〈u | v 〉 ∈ K¶. A scalar product has, in general, the

following properties:

- 〈 v | v 〉 ∈ R; 〈 v | v 〉 ≥ 0; 〈 v | v 〉 = 0 ⇒ | v 〉 = | o 〉;- 〈u | v 〉 = 〈 v |u 〉∗;- 〈α1 u1 + α2 u2 | v 〉 = α1

∗〈u1 | v 〉+ α2∗〈u2 | v 〉 and

〈u |β1 v1 + β2 v2 〉 = β1〈u | v1 〉+ β2〈u | v2 〉.A linear vector space with scalar product is also known as ‘inner

product space’ or ‘pre-Hilbert‖ space’.

A scalar product can be used to define the length (or norm) of a

vector:

‖v‖=√〈 v | v 〉 ≥ 0 (95)

in which case the linear vector space turns into a unitary space, where

the following relations hold:

¶ This is called a ‘bra-c-ket’ in the Dirac notation used here.

‖ D. Hilbert (1862-1943)


and — with restriction to cases where 〈u | v 〉 ∈ R — the angle φ =

](|u 〉, | v 〉) between them can be defined as

cos (φ) =〈u | v 〉‖u‖ ‖v‖ (0 ≤ φ ≤ π) (101)

Hilbert space:

A complete unitary linear vector space (a rigorous definition is not

attempted here).

Some examples for linear spaces:

• | v 〉 → v (associate ‘ket’ vectors with ‘ordinary’ vectors):

This yields the n-dimensional Euclidean space Rn with a basis set

{bk} (k = 1, . . . , n), so that the expansions

c =n∑

k=1

bk γk = (b1, . . . , bn)

γ1...γn

, d =

n∑

k=1

bk δk

exist and a scalar product can be defined as


An orthonormal set of basis vectors, {| bk 〉} (k = 1, . . ., n) with

〈 bk | bl 〉 = δkl, makes the evaluation of the expansion coefficients βkin a linear combination particularly simple:

〈 bk | bl 〉 = δkl ⇒ | v 〉 =n∑

k=1

| bk 〉 βk , βk = 〈 bk | v 〉 . (97)

Substitution of this expression for βk gives an expression for the iden-

tity (or unit) operator 1 which is known as ‘resolution of the identity’:

| v 〉 =n∑

k=1

| bk 〉〈 bk | v 〉 =

n∑

k=1

| bk 〉〈 bk | | v 〉 = 1 | v 〉 (98)

⇒ 1 =n∑

k=1

| bk 〉〈 bk | . (99)

The distance d between two vectors |u 〉, | v 〉 can be obtained from

d = ‖u− v‖= ‖v − u‖ , (100)


based on the Lebesgue∗∗ integral definition. This leads to nor-

malization integrals

‖f‖ =

(∫

Gf∗(r) f(r) dr

)1/2=

(∫

G|f(r)|2 dr

)1/2

Special cases included herein are

- L2([−p/2, p/2]) used in the harmonic analysis of periodic func-

tions f(x) = f(x+ p) ∈ C; for period p = 2π:

Orthonormal basis (dimension d =∞, denumerably infinite):{

1√2π

eikx

}(k ∈ Z) ,

1

2π

∫ π−π

e−i(k−l)x dx = δkl

Fourier†† series expansion:

f(x) =∞∑

k=−∞

ck√2π

eikx , ck =1√2π

∫ π−π

e−ikx f(x) dx

∗∗ H. Lebesgue (1875-1941)

†† J. Fourier (1768-1830)


c · d = cT

Md , with cT = (γ1, . . . , γn) and d =

δ1...δn

,

and the metric matrix M = (mij) where mij = bi · bj = mji. Whenthe basis is chosen to be orthonormal (bi · bj = δij), the metricmatrix reduces to the n × n unit matrix, and the scalar productcollapses to the simple familiar form where only the coefficients(coordinates) γk and δk are involved:

bi · bj = δij ⇒ c · d = cTd = (γ1, . . . , γn)

δ1...δn

=

n∑

k=1

γk δk

• | f 〉 → f (associate ‘ket’ vectors with ‘ordinary’ functions):This leads, e.g., to the infinite-dimensional spaces L2(G) of com-plex-valued functions of n variables, f(r) = f(x1, x2, . . . , xn) ∈ C

(r ∈ Rn), that are square-integrable over a range (or region) G ⊆Rn in the sense of the scalar product

〈 f | g 〉 =∫

Gf∗(r) g(r) dr


when the MO coefficients cik are collected into MO (column)

vectors ck, which form the matrix C. Completeness of the

basis set is a delicate issue, which is usually left untouched in

practical work.

- L2((R3 × S)n) [ where S denotes a ‘spin space’ ] and the con-

struction of (spin-adapted) n-electron state functions Ψk by

linear combination of Slater determinants Φi (called configu-

ration interaction or ‘CI expansion’, since every Slater deter-

minant can be associated with an ‘electron configuration’):

Ψk(x1, . . . ,xn) =m∑

i=1

Φi(x1, . . . ,xn)Cik

or equivalently

(Ψ1,Ψ2, . . . ,Ψm) = (Φ1,Φ2, . . . ,Φm)C

when the CI coefficients Cik are collected into CI (column)

vectors Ck, which form the matrix C. In the limit m→∞, the

CI expansion is exact, if the spin orbitals — from which the

Slater determinants are formed — constitute a complete basis

(of single-particle functions).


- L2(R) and Fourier transformation:

Orthonormal basis (dimension d =∞, continuum):{

1√2π

eikx

}(k ∈ R) ,

1

2π

∫ ∞−∞

e−i(k−k′)x dx = δ(k − k′)

Fourier transform pairs (symmetric form):

f(x) =1√2π

∫ ∞−∞

f(k) eikx dk , f(k) =1√2π

∫ ∞−∞

f(x) e−ikx dx

- L2(R3) and the construction of molecular orbitals ψk(r) by lin-

ear combination of basis functions χi(r) (for historical reasons,

this is called ‘molecular orbitals by linear combination of atomic

orbitals’, or ‘MO-LCAO approach’):

ψk(r) =m∑

i=1

χi(r) cik

or equivalently

(ψ1, ψ2, . . . , ψm) = (χ1, χ2, . . . , χm)C


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