3. Mathematical tools for quantum chemistryMathematical tools for quantum chemistry 4. The...
Transcript of 3. Mathematical tools for quantum chemistryMathematical tools for quantum chemistry 4. The...
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 1/ 42-3
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
h 2 hc5 1 hcc5 4 dih5
h 1 hc6 2 hcc6 3 dih6
h 2 hc7 1 hcc7 4 dih7
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 4/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 3/ 42-3
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 6/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 5/ 42-3
• ‘+’ 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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3. Ternary operators require an ordered triple of objects:
O(a, b, c) = z
Examples:
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 8/ 42-3
- 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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=
(b
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)
xT(1×m) B(m×k) = c
T(1×k)
(xT
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)
2. Binary operators require an ordered pair of objects:
O(a, b) = z
Examples:
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 7/ 42-3
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).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 10/ 42-3
• 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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 9/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 12/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 11/ 42-3
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!
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 14/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 13/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 16/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 15/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 18/ 42-3
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’.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-02 17/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 20/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 19/ 42-3
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’’
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 22/ 42-3
The symmetry operation C2 and the water molecule
21
1
2
12φ = π/2φ = π/2
φ = π. yx
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 21/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 24/ 42-3
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?
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 23/ 42-3
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 26/ 42-3
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?
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 25/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 28/ 42-3
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 27/ 42-3
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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FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 30/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 29/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 32/ 42-3
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,+, ·)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 31/ 42-3
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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 34/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 33/ 42-3
- 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.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 36/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 35/ 42-3
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 38/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 37/ 42-3
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)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 40/ 42-3
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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 39/ 42-3
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).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 42/ 42-3
- 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
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-09 41/ 42-3