Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

Подождите немного. Документ загружается.

Dynamical Groups 4.1 Noncompact Dynamical Groups 89

The so(4) LA can be realized either as a direct sum

so(4) = so(3) ⊕ so(3), or by supplementing so(3) with an

appropriately scaled quantum mechanical analogue of

the Laplace–Runge–Lenz (LRL) vector (cf. Sect. 3.6.2).

In the ﬁrst case, we use two commuting angular momen-

tum vectors M and N (cf. Sect. 2.5),



, M



= iε

jk





, N



= iε

jk





, N



= 0,

(

j, k,= 1, 2, 3

)

(4.14)

while in the second case we use the components of the

total angular momentum vector Jand LRL-like vector V

with commutation relations



, J



= iε

jk





, V



= iσε

jk





, V



= iε

jk



(

j, k,= 1, 2, 3

)

, (4.15)

with σ = 1. For σ =−1 we obtain so(3,1) (the LA of

the homogeneous Lorentz group), which is relevant to

the scattering problem of a particle in the Coulomb (or

Kepler) potential (see below). For σ = 0wegete(3) (the

LA of the three-dimensional Euclidean group) [4.18–

20]. Note that (4.14)and(4.15) are interrelated by

M =

(

J+ V

)

, N =

(

J− V

)

(4.16)

so that J = M⊕ N and V = 2M− J. The two Casimir

operators C

and C

are

= σJ

+ V

= σJ

−

+ V

−

+ V

+ σJ

− 2),

(

V · J

)

(

J· V

)

(

−

+ V

−

)

+ V

, (4.17)

where again

= X

± iX

, X = J or V . (4.18)

For so(3,1) and e(3), only inﬁnite dimensional non-

trivial irreps are possible, while for so(4), only ﬁnite

dimensional ones arise. To get unirreps, we require

J and V to be Hermitian. Using



, J

, C



a complete set of commuting operators for so(4), we la-

bel the basis vectors by the four quantum numbers as

|γ jm≡|( j

,η)jm,sothat

|γ jm= j( j + 1)|γ jm ,

|γ jm=m|γ jm ,

|γ jm=



− η

− 1



|γ jm ,

|γ jm= j

η|γ jm , (4.19)

with 2| j

| being a nonnegative integer and

j =|j

|, | j

|+1,... ,η− 1 ;

η =|j

|+k, k = 1, 2,... (4.20)

(see, e.g., [4.17] for the action of J

, V

and V

To obtain the hydrogenic (or Kepler) realization of

so(4), we consider the quantum mechanical analog of

the classical LRL vector

V =

(

p∧ L− L∧ p

)

− Zr

−1

− p(r · p) + rH ,

L = r ∧ p ,

(4.21)

which commutes with the hydrogenic Hamiltonian

H =

− Zr

−1

. (4.22)

Note that

[

L, H

]



V, H



= 0 ,



L·





V · L



= 0 ,

= 2H



+ 1



+ Z

, (4.23)

while the components of L and

V satisfy the commuta-

tion relations



, L



= i

jk







= i

jk







= (−2H)i

jk



. (4.24)

Thus, restricting ourselves to a speciﬁc bound state

energy level E

, we can replace H by E

and deﬁne

= (−2E

)

−1/2

( j = 1, 2, 3), (4.25)

obtaining the so(4) commutation relations (4.15) (with

J replaced by L). This is Pauli’s hydrogenic realization

of so(4) [4.21–23]. [In a similar way we can consider

continuum states E > 0 and deﬁne V = (2E)

−1/2

V, ob-

taining an so(3,1) realization.] The last identity of (4.23)

now becomes

=−



+ 1



− Z

/2E

, (4.26)

which immediately implies Bohr’s formula, since V

= 4M

=−1 − Z

/2E

,sothat

=−Z

/2(2 j

+ 1)

=−Z

/2n

, (4.27)

where n = 2 j

+ 1and j

is the angular momentum

quantum number for M,(4.16). In terms of the ir-

Part A 4.1

90 Part A Mathematical Methods

rep labels (4.20), we have that j

= 0, η = n,sothat

|γm=|(0, n)m≡|nm,  = 0, 1,... ,n − 1.

Using the stepwise merging of so(4) and so(2,1)

[adding ﬁrst T

which leads to so(4,1) and subsequently

and T

], we arrive at the hydrogenic realization of

so(4,2) having ﬁfteen generators L, A, B, Γ , T

, T

namely (cf. [4.2, 14, 15, 17])

L = R∧ P ,



− P(R· P) ∓ R,

Γ = RP ,





∓ R





+ L

−1

∓ R



= R· P− iI = RP

. (4.28)

Relabeling these generators by the elements of an anti-

symmetric 6 × 6 matrix according to the scheme

↔







0 L

−L

0 L

0 A

0 T







(4.29)

we can write the commutation relations in the following

standard form



, L

m





j

+ g

j

− g

k

− g

k



(4.30)

with the diagonal metric tensor g

deﬁned by the matrix

G = diag[1, 1, 1, 1, −1, −1]. The matrix form (4.29)

also implies the subalgebra structure

so(4, 2) ⊃ so(4, 1) ⊃ so(4) ⊃ so(3),

(4.31)

with so(4,1) generated by L, A, B and T

,andso(4)by

L, A.[L, B also generate so(3,1).]

The three independent Casimir operators (quadratic,

cubic and quartic) are [4.24], (summation over all in-

dices is implied)

= L

+ A

− B

− Γ

+ T

− T

ijkmn

k

= T

(B· L) + T

(Γ · L) + T

(A· L)

+ A· (B∧ Γ ),

= L

k

m

. (4.32)

For our hydrogenic realization Q

=−3, Q

= Q

= 0.

Thus, our hydrogenic realization implies a single unirrep

of so(4,2) adapted to the chain (4.31).

4.2 Hamiltonian Transformation and Simple Applications

The basic idea is to transform the relevant Schrödinger

equation into an eigenvalue problem for one of the

operators from the complete set of commuting opera-

tors in our realizations, e.g., T

for so(2,1). Instead of

using a rather involved “tilting” transformation ([4.1,

p. 20], and [4.2,14,15]), we can rely on a simple scaling

transformation [4.16, 25]

r = λR, p = λ

−1

P , r = λR , p

= λ

−1

(4.33)

where

r =







j=1





1/2

, (4.34)

and

= p

−2



(N − 1)(N − 3) + L



(4.35)

with p

deﬁned analogously to P

in (4.7); r and p are

the physical operators in terms of which the Hamiltonian

of the studied system is expressed. Recall that L

has

eigenvalues [4.26]

( + N − 2), = 0, 1, 2,... for N ≥ 2 ,

(4.36)

and we can set  = 0forN = 1 (angular momentum term

vanishes in one-dimensional case). The units in which

m = e =

= 1, c ≈ 137 are used throughout.

4.2.1 N-Dimensional Isotropic Harmonic

Oscillator

Considering the Hamiltonian

H =

, (4.37)

Part A 4.2

Dynamical Groups 4.2 Hamiltonian Transformation and Simple Applications 91

with p

in the form (4.35), transforming the cor-

responding Schrödinger equation using the scaling

transformation (4.33) and multiplying by

,weget

for the radial component



−2

ξ +

−



(λR) = 0 ,

(4.38)

with

ξ =

(N − 1)(N − 3) +

( + N − 2).

(4.39)

Choosing λ such that



ω/2



= 1, we can

rewrite (4.38) using the so(2,1) realization (4.9) with

ν = 2as



−



(λR) = 0 . (4.40)

Thus, using the second equation of (4.6) we can interre-

late ψ

(λR) with |kq and set

E = q,sothat

E = 4q/λ

= 2qω, (4.41)

with q given by (4.12)and(4.13), i. e.,

q = q

+ µ, µ= 0, 1, 2,... (4.42)

and

= k+ 1 =



1 ±



 +

N − 1



(4.43)

Now, for N = 1weset = 0sothatq

and

, yielding for E = 2qω the values



+ 2µ



ω and



+ 2µ + 1



ω, µ = 0, 1, 2,.... Combining both sets

we thus get for N = 1 the well known result

E ≡ E



n +



ω, n = 0, 1, 2,... .

(4.44)

Similarly, for the general case N ≥ 2 we choose the

upper sign in (4.43)[sothatk > −1] and get

E ≡ E



n +



ω, n = 0, 1, 2,...

(4.45)

whereweidentiﬁed(+ 2µ) with the principal quantum

number n.

4.2.2 N-Dimensional Hydrogenic Atom

Applying the scaling transformation (4.33) to the hydro-

genic Hamiltonian (4.22)inN-dimensions, we get for

the radial component (after multiplying from the left by





+ R

−1

ξ − 2λ



− λZ



(λR) = 0 ,

(4.46)

where now

ξ =

(N − 1)(N − 3) + ( + N − 2).

(4.47)

In this case we must set 2λ

E =−1 and use realiza-

tion (4.9) with ν = 1 to obtain

− λZ)ψ

(λR) = 0 . (4.48)

This immediately implies that

λZ = q

(4.49)

and

= k+ 1 =

[

1 ± (2 + N − 2)

]

(4.50)

Choosing the upper sign [since  ≥ 0andk > −1],

so that q

=  +

(N − 1), and identifying q with the

principal quantum number n, we have ﬁnally that

E ≡ E

=−

2λ

=−

. (4.51)

The N-dimensional relativistic hydrogenic atom can

be treated in the same way, using either the Klein–

Gordon or Dirac–Coulomb equations [4.2, 14–17].

4.2.3 Perturbed Hydrogenic Systems

The so(4,2) based Lie algebraic formalism can be con-

veniently exploited to carry out large order perturbation

theory (see [4.27–29]andChapt.5) for hydrogenic sys-

tems described by the Schrödinger equation

[

+ εV(r)

]

ψ(r) = (E

+ ∆E)ψ(r), (4.52)

with H

given by (4.22)andE

by E of (4.51). Ap-

plying transformation (4.33), using (4.49), (4.51)and

multiplying on the left by λ

R,weget



− q +

R+ ελ

RV(λR) − λ

R∆E



Ψ(R)

= 0 ,

(4.53)

wherewesetψ(λR) ≡ Ψ(R). For the important case

of a 3-dimensional hydrogenic atom [N = 3,ξ = ( +

1), q ≡ n] we get using the so(4,2) realization (4.28)[or

so(2,1) realization (4.9) with ν = 1]

(K + εW − S∆E)Ψ(R) = 0 ,

(4.54)

Part A 4.2

92 Part A Mathematical Methods

with

K = T

− n ,

W = λ

RV(λR),

S = λ

R . (4.55)

We also have that λ = n/Z and for the ground state case

n = q = 1. Although (4.54) has the form of a general-

ized eigenvalue problem requiring perturbation theory

formalism with a nonorthogonal basis (where S rep-

resents an overlap), T

is Hermitian with respect to

a (1/R) scalar product, and the required matrix ele-

ments can therefore be easily evaluated [4.2, 14, 15, 17,

27–29].

For central ﬁeld perturbations, V(r) = V(r), the prob-

lem reduces to one dimension and since R = T

− T

the so(2,1) hydrogenic realization (ν = 1) can be em-

ployed. For problems of a hydrogenic atom in a magnetic

ﬁeld (Zeeman effect) [4.27–30]oraone-electrondi-

atomic ion [4.31], the so(4,2) formalism is required

(note, however, that the LoSurdo–Stark effect can also

be treated as a one-dimensional problem using parabolic

coordinates [4.32]).

The main advantage of the LA approach stems from

the fact that the spectrum of T

is discrete, so that no

integration over continuum states is required. Moreover,

the relevant perturbations are closely packed around the

diagonal in this representation, so that inﬁnite sums are

replaced by small ﬁnite sums.

For example, for the LoSurdo–Stark problem when

V(r) = F z,whereF designates electric ﬁeld strength

in the z-direction, we get (4.54) with ε = F and

W = (n/Z)

RZ, S = (n/Z)

R. Since both R and

Z are easily expressed in terms of so(4,2) genera-

tors,

Z = B

− A

, R = T

− T

, (4.56)

we can easily compute all the required matrix elem-

ents [4.2, 14, 15, 17].

Similarly, considering the Zeeman effect with

V(r) =

B L



− z



(4.57)

where B designates magnetic ﬁeld strength in the

z-direction, we have for the ground state when n = 1,

 = m = 0thatε =

, K = T

− 1, W = Z

−4

R(R

−

) and S = Z

−2

R. Again, the matrix elements of

W and S are obtained from those of Z and R,(4.56)

by matrix multiplication (for tables and programs,

see [4.17]).

One can treat one-electron diatomic ions [4.2,14,15,

31] and screened Coulomb potentials, including charmo-

nium and harmonium [4.10–12, 17, 33, 34], in a similar

way.

Note, ﬁnally, that we can also formulate the

perturbed problem (4.54) in a standard form not in-

volving the “overlap” by deﬁning the scaling factor

as λ = (−2E)

−1/2

,whereE is now the exact energy

E = E

+ ∆E. Equation (4.54) then becomes

+ εW −λZ)Ψ(R) = 0 , (4.58)

with the eigenvalue λZ. In this case any conventional

perturbation formalism applies, but the desired energy

has to be found from λZ [4.35].

4.3 Compact Dynamical Groups

Unitary groups U(n)andtheirLAs often play the role

of (compact) dynamical groups since

1. quantum mechanical observables are Hermitian and

the LA of U(n) is comprised of Hermitian operators

[under the exp(iA) mapping],

2. any compact Lie group is isomorphic to a subgroup

of some U(n),

3. “nothing of algebraic import is lost by the unitary

restriction” [4.36].

All U(n) irreps have ﬁnite dimension and are thus

relevant to problems involving a ﬁnite number of bound

states [4.3–6, 10–12, 36–43].

4.3.1 Unitary Group

and Its Representations

The unitary group U(n)hasn

generators E

spanning

its LA and satisfying the commutation relations



, E

k



= δ

i

− δ

i

(4.59)

and the Hermitian property

†

= E

. (4.60)

They are classiﬁed as raising (i < j), lowering (i > j)

and weight (i = j) generators according to whether they

Part A 4.3

Dynamical Groups 4.3 Compact Dynamical Groups 93

raise, lower and preserve the weight, respectively. The

weight vector is a vector of the carrier space of an irrep

which is a simultaneous eigenvector of all weight gener-

ators E

of U(n) (comprising its Cartan subalgebra), and

the vector m = (m

, m

,... ,m

) with integer compo-

nents, consisting of corresponding eigenvalues, is called

a weight.Thehighest weight m

(in lexical ordering),

= (m

, m

,... ,m

), (4.61)

with

≥ m

≥···≥m

, (4.62)

uniquely labels U(n) irreps, Γ(m

),andmayberepre-

sented by a Young pattern. Subducing Γ(m

) of U(r)to

U(r− 1), embedded as U(r − 1) ⊕ 1inU(r), gives [4.41]

Γ(m

) ↓ U(r − 1) =



Γ(m

r−1

), (4.63)

where the sum extends over all U(r−1) weights m

r−1

1,r−1

, m

2,r−1

,...m

r−1,r−1

) satisfying the so-called

“betweenness conditions” [4.38]

≥ m

i,r−1

≥ m

i+1,r

(i = 1,... ,r − 1).

(4.64)

Two irreps Γ(m

) and Γ(m



) of U(n) yield the

same irrep when restricted to SU(n)ifm

= m



+ h, i =

1,... ,n.TheSU(n) irreps are thus labeled with highest

weights with m

= 0. The dimension of Γ(m

) of U(n)

is given by the Weyl dimension formula [4.36]

dim Γ(m

) =



i< j



− m

+ j −i





1!2!···(n− 1)! .

(4.65)

The U(n) Casimir operators have the form

U(n)



,... ,i

···E

k−1

(4.66)

The ﬁrst order Casimir operator is given by the sum

of weight generators and equals the sum of the highest

weight components.

Since U(1) is Abelian, the Gel’fand-Tsetlin [4.42]

canonical chain (Sect. 3.4.3)

U(n) ⊃ U(n − 1) ⊃···⊃U(1)

(4.67)

can be used to label uniquely the basis vectors of the

carrier space of Γ(m

) by triangular Gel’fand tableaux

[m] deﬁned by

[

]







n−1

···













······ m

1,n−1

······m

n−1,n−1

···







(4.68)

with entries satisfying betweenness conditions (4.64).

Matrix representatives of weight generators are diagonal

[m



]|E

|[m] = δ

[m],[m



]







j=1

−

i−1



j=1

j,i−1





(4.69)

while those for other generators are rather in-

volved [4.42, 43]. Note that only elementary (E

i,i+1

)

raising generators are required since

[m



]|E

|[m] = [m]|E

|[m



] (4.70)

and



i,i+1

, E

i+1, j



(4.71)

In special cases required in applications ([4.39, 40]

and Sect. 4.3.4) efﬁcient algorithms exist for the com-

putation of explicit representations.

4.3.2 Orthogonal Group O(n)

and Its Representations

Since O(n) is a proper subgroup of U(n), its repre-

sentation theory has a similar structure. The suitable

generators are

= E

− E

, F

=−F

= 0 , F

†

=−F

(4.72)

and satisfy the commutation relations



, F

k



= δ

i

+ δ

i

− δ

j

− δ

j

(4.73)

The canonical chain has the form

O(n) ⊃ O(n − 1) ⊃···⊃O(2).

(4.74)

The components of the highest weight m

= (m

, m

,... ,m

), (4.75)

Part A 4.3

94 Part A Mathematical Methods

satisfy the conditions

≥ m

≥···≥m

≥ 0forn = 2k + 1 ,

(4.76)

and

≥ m

≥···≥|m

| for n = 2k , (4.77)

where m

are simultaneously integers or half-odd

integers. The former are referred to as tensor rep-

resentations (since they arise as tensor products of

fundamental irreps), while those with half-odd integer

components are called spinor representations. Note that

for n = 2k, we have two lowest (mirror-conjugated)

spinor representations, namely m

(+)

= (

,... ,

)

and m

(−)

= (

,... ,

, −

). Only tensor representa-

tions can be labeled by Young tableaux.

Subducing O(n)toO(n − 1), the betweenness con-

ditions (branching rules) have the form

≥ m

i,n−1

≥ m

i+1,n

(i = 1,... ,k − 1) (4.78)

together with

k,2k+1

≥|m

k,2k

| (4.79)

when n = 2k + 1. The m

i,n−1

components are integral

(half-odd integral) if the m

are integral (half-odd inte-

gral). The U(n) ⊃ O(n) [or SU(n) ⊃ SO(n)] subduction

rules are more involved [4.44].

4.3.3 Clifford Algebras and Spinor

Representations

While all reps of U(n)orSL(n) arise as tensor powers of

the standard rep, only half of the reps of SO(m)orO(m)

arise this way, since SO(m) is not simply connected

when m > 2. A double covering of SO(m) leads to spin

groups Spin(m). The best way to proceed is, however,

to construct the so-called Clifford algebras C

, whose

multiplicative group (consisting of invertible elements)

contains a subgroup which provides a double cover of

SO(m). The key fact is that C

is isomorphic with gl(2

)

and C

2k+1

with gl(2

) ⊕ gl(2

). The reps of C

thus

provide the required spinor reps.

A Clifford algebra C

is an associative algebra

generated by Clifford numbers α

satisfying the anti-

commutation relations

{α

,α

}=2δ

(i, j = 1,... ,m). (4.80)

Since α

= 1, dim C

= 2

and a general element of

is a product of Clifford numbers α

···α

with

= 0or1.

To see the relation with so(m + 1), note that

=−

iα

, F



,α



( j = k)

(4.81)

satisfy the commutation relations (4.73).

As an example, C

can be realized by Pauli matrices

by setting

= σ

,α

= σ

−i0

(4.82)

Clearly, the four matrices 1

,α

and α

are lin-

early independent (note that σ

= iσ

),sothatC

isomorphic to gl(2,

Similarly, considering Dirac–Pauli matrices

−i1

0 −i1

= iγ

0 iσ

−iσ

,(k = 1, 2, 3)

(4.83)

we have that

{γ

, γ

}=2δ

,(i, j = 1,... ,4) (4.84)

so that γ

(i = 1,... ,4) or (i = 0,... ,3) represent

Clifford numbers for C

and 1

, γ

(i < j), γ

(i < j < k) and γ

≡ γ

= iγ

form an

additive basis for gl(4,

) (the γ

themselves are said to

form a multiplicative basis). For general construction of

Clifford numbers in terms of direct products of Pauli

matrices see [4.45, 46].

4.3.4 Bosonic and Fermionic Realizations

of U(n)

Designating by b

†

) the boson creation (annihilation)

operators (Sect. 6.1.1) satisfying the commutation rela-

tions



, b





†

, b

†



= 0,



, b

†



= δ

, we obtain

a possible U(n) realization by deﬁning its n

generators

as follows

= b

†

. (4.85)

The ﬁrst order Casimir operator, (4.66) with k = 1, then

represents the total number operator

N ≡ C

U(n)



i=1



i=1

†

, (4.86)

Part A 4.3

Dynamical Groups 4.3 Compact Dynamical Groups 95

and the physically relevant states, being totally symmet-

ric, carry single row irreps Γ(N

0) ≡ Γ(N0 ···0).

Similarly for fermion creation (annihilation) opera-

tors X

†

) that are associated with some orthonormal

spin orbital set {|I}, I = 1, 2,... ,2n, and satisfy the

anticommutation relations



, X





†

, X

†



= 0,



, X

†



= δ

, the operators

= X

†

(4.87)

again represent the U(2n) generators satisfying (4.59)

and (4.60). The ﬁrst-order Casimir then represents the

total number operator

N =

†

, while the possible

physical states are characterized by totally antisymmet-

ric single column irreps Γ(1

0) ≡ Γ(11 ···10···0).

4.3.5 Vibron Model

Similar to the uniﬁed description of nuclear collec-

tive rovibrational states using the interacting boson

model [4.47–49], one can build an analogous model

for molecular rotation-vibration spectra [4.8]. For di-

atomics, an appropriate dynamical group is U(4) [4.8,

50–52] and, generally, for rotation-vibration spectra in

r-dimensions one requires U(r + 1). For triatomics, the

U(4) generating algebra is generalized to U(4) ⊗ U(4),

and for the (k + 1) atomic molecule to U

(1)

(4) ⊗···⊗

(k)

(4) [4.8, 50–52].

For the bosonic realization of U(4), we need four

creation (b

†

, i = 1,... ,4) and four annihilation (b

)

operators (Sect. 4.3.4). The Hamiltonian may be gener-

ally expressed as a multilinear form in terms of boson

number preserving products



†



, so that using (4.85)

we can write

H = h

(0)



(1)



ijk

(2)

ijk

k

+··· .

(4.88)

The energy le vels (as a function of 0, 1, 2,...-body

matrix elements h

, h

(1)

, h

(2)

ijk

, etc.) are then determined

by diagonalizing H in an appropriate space, which is

conveniently provided by the carrier space of the totally

symmetric irrep Γ(N 000) ≡ Γ(N

0) of U(4).

The requirement that the resulting states be charac-

terized by angular momentum J and parity P quantum

numbers necessitates that the boson operators in-

volved have deﬁnite transformation properties under

rotations and reﬂections [4.8]. The boson operators

are thus subdivided into the scalar operators



†

,σ



J = 0, and vector operators



†

,π

; µ = 0, ±1



, J = 1

with parity P = (−)

. All commutators vanish except

for



σ, σ

†



= 1,



,π

†





= δ

µµ



. (4.89)

Since H preserves the total number of vibrons N = n

, the second order Hamiltonian (4.88) within the irrep

Γ(N

0) can be expressed in terms of four independent

parameters (apart from an overall constant) as

H = e

(0)

+ e

(1)



†



(0)

+ e

(2)





†

× π

†



(0)



π ×



(0)



(0)

+ e

(2)





†

× π

†



(2)



π ×



(2)



(0)

+ e

(2)





†

× π

†



(0)



σ ×



(0)



†

× σ

†



(0)



π ×



(0)



(0)

+··· , (4.90)

where

σ = σ,

= (−)

1−µ

−µ

and square brackets in-

dicate the SU(2) couplings.

In special cases the eigenvalue problem for H can be

solved analytically, assuming that H can be expressed in

terms of Casimir operators of a complete chain of sub-

groups of U(4) [referred to as dynamical symmetries].

Requiring that the chain contain the physical rotation

group O(3), one has two possibilities

(I) U(4) ⊃ O(4) ⊃ O(3) ⊃ O(2),

(II) U(4) ⊃ U(3) ⊃ O(3) ⊃ O(2).

(4.91)

These imply labels (quantum numbers): N [total vibron

number deﬁning a totally symmetric irrep of U(4)],

ω = N, N − 2, N − 4,... ,1 or 0 [deﬁning a totally

symmetric irrep of O(4)]andn

= N, N − 1,... ,0

[deﬁning the U(3) irrep], in addition to the O(3) ⊃ O(2)

labels J, M;|M|≤J. In terms of these labels one ﬁnds

for the respective Hamiltonians

(I)

= F + AC

O(4)

+ BC

O(3)

(II)

= F + εC

U(3)

+ αC

U(3)

+ βC

O(3)

, (4.92)

where F, A, B, ε, α, β are free parameters and C

U(k)

O(k)

are relevant Casimir operators, the following ex-

pressions [4.8, 50–52] for their eigenvalues

(I)

(N,ω, J, M) = F + Aω(ω + 2) + BJ(J + 1),

(II)

(N, n

, J, M) = F + n

+ αn

+ 3)

+ βJ(J +1).

(4.93)

Part A 4.3

96 Part A Mathematical Methods

The limit (I) is appropriate for rigid diatomics and limit

(II) for nonrigid ones [4.8, 50–52].

In addition to handling di- and tri-atomic systems,

the vibron model was also applied to the overtone spec-

trum of acetylene [4.53], intramolecular relaxation in

benzene and its dimers [4.54, 55], octahedral molecules

of the XF

type (X = S, W, and U) [4.56], and to lin-

ear polyatomics [4.57]. Most recently, the experimental

(dispersed ﬂuorescence and stimulated emission pump-

ing) vibrational spectra of H

OandSO

in their ground

states, representing typical local-mode and normal-

mode molecules, respectively, have been analyzed,

including highly excited levels, by relying on the U(2) al-

gebraic effective Hamiltonian approach [4.58–60]. The

U(2) algebraic scheme [4.61] also enabled the treat-

ment of Franck–Condon transition intensities [4.62,63]

in rovibronic spectra. The attempts at a similar heuris-

tic phenomenological description of electronic spectra

have met so-far with only a limited success [4.64].

4.3.6 Many-Electron Correlation Problem

In atomic and molecular electronic structure calculations

one employs a spin-independent model Hamiltonian

H =



i, j



σ=1

†

iσ

jσ



i, j,k,

ij,k



σ,τ=1

†

iσ

†

jτ

τ

kσ

, (4.94)

where X

†

≡ X

†

iσ

) designate the creation (annihi-

lation) operators associated with the orthonormal spin

orbitals |I≡|iσ=|i⊗|σ; i = 1,... ,n; σ = 1, 2



σ = 1, 2 labeling the spin-up and spin-down eigenstates

of S



,andh

=i|

h| j,v

ij,k

=i(1) j(2)|

v|k(1)(2)

are the one- and two-electron integrals in the orbital ba-

sis {|i}. As stated in Sect. 4.3.4, e

≡ e

iσ, jτ

= X

†

iσ

jτ

may then be regarded as U(2n) generators, and the

appropriate U(2n) irrep for N-electron states is Γ





Similar to the nuclear many-body problem [4.65],

one deﬁnes mutually commuting partial traces of spin

orbital generators e

,(4.87),



σ=1

iσ, jσ



σ=1

†

iσ

jσ

στ



i=1

iσ,iτ



i=1

†

iσ

iτ

, (4.95)

which again satisfy the unitary group commutation re-

lations (4.59) and property (4.60), and may thus be

considered as the generators of the orbital group U(n)

and the spin group U(2). The Hamiltonian (4.94) is thus

expressible in terms of orbital U(n) generators

H =



i, j



i, j,k,

ij,k

j

− δ

i

(4.96)

We can thus achieve an automatic spin adaptation by

exploiting the chain

U(2n) ⊃ U(n) ⊗ U(2)

(4.97)

and diagonalize H within the carrier space of two-

column U(n) irreps Γ(2

) ≡ Γ(a, b, c) with [4.39,

66]

a =

N − S , b = 2S ,

c = n− a− b = n−

N − S ,

(4.98)

considering the states of multiplicity (2S+ 1) involving

n orbitals and N electrons. The dimension of each spin-

adapted subproblem equals [4.39, 66]

dim Γ(2

) =

b+ 1

n + 1

(4.99)

where





designate binomial coefﬁcients.

Exploiting simpliﬁed irrep labeling by triples

of integers (a, b, c),(4.98), at each level of the

canonical chain (4.67), one achieves more efﬁcient

state labeling by replacing Gel’fand tableaux (4.68)

by n ×3 ABC [4.66]orPaldus or Gel’fand–Paldus

tableaux [4.40,67–75]

[

]

[

]

(4.100)

where a

+ b

+ c

= i. Another convenient labeling uses

the ternary step numbers d

, 0 ≤ d

≤ 3[4.66–68,76,77]

= 1 + 2(a

− a

i−1

) − (c

− c

i−1

). (4.101)

An efﬁcient and transparent representation of this

basis can be achieved in terms of Shavitt graphsand dis-

tinct row tables ([4.67, 68], cf. also [4.10–12, 39, 69]).

An efﬁcient evaluation of generator matrix representa-

tives, as well as of their products, is formulated in terms

of products of segment values, whose explicit form has

been derived in several different ways [4.10–12, 66–69,

73–75, 77, 78]. Since the dimension (4.99) rapidly in-

creases with n and N, various truncated schemes (limited

CI) are often employed. The unitary group formalism

Part A 4.3

Dynamical Groups 4.3 Compact Dynamical Groups 97

that is based either on U(n) or on the universal envelop-

ing algebra of U(n) proved to be of great usefulness

in various post-Hartree–Fock approaches to molecu-

lar electronic structure [4.79], especially in large-scale

CI calculations (in particular in the columbus Pro-

gram System [4.80]; see also [24–31] in [4.12]) and

in the spin-adapted UGA version of the coupled cluster

(CC) method [4.81–83] (cf. Chapter 5; for applications,

see [4.84–86]), as well as in various other investigations

(e.g. quantum dots [4.87], charge migration in fragmen-

tation of peptide ions [4.88, 89]; see also [4.10–12]for

other references).

4.3.7 Clifford Algebra Unitary Group

Approach

The Clifford algebra unitary group approach (CAUGA)

exploits a realization of the spinor algebra of the rotation

group SO(2n + 1) in the covering algebra of U(2

)to

obtain explicit representation matrices for the U(n)[or

SO(2n + 1) or SO(2n)] generators in the basis adapted

to the chain [4.90–94]

U(2

) ⊃ Spin(m) ⊃ SO(m) ⊃ U(n),

(m = 2n + 1orm = 2n)

(4.102)

supplemented, if desired, by the canonical chain (4.67)

for U(n).

To realize the connection with the fermionic Grass-

mann algebra generated by the creation



†



and

annihilation (X

) operators, I = 1,... ,2n, note that

it is isomorphic with the Clifford algebra C

when we

deﬁne [4.12,25]

= X

+ X

†

,α

I+2n

= i



− X

†



(I = 1,... ,2n).

(4.103)

For practical applications, the most important is the

ﬁnal imbedding U(2

) ⊃ U(n), (for the role of interme-

diate groups, see [4.90–92]). All states of an n-orbital

model, regardless the electron number N and the total

spin S, are contained in a single two-box totally sym-

metric irrep 2

0 of U

(

)

[4.93, 94]. To simplify the

notation, one employs the one-to-one correspondence

between the Clifford algebra monomials, labeled by the

occupation numbers m

= 0orm

= 1 (i = 1,... ,n),

and “multiparticle” single-column U(n) states labeled

p ≡ p{m

}=2

− (m

···m

)

, (4.104)

where the occupation number array (m

···m

) is in-

terpreted as a binary integer, which we then regard as

one-box states | p) of U(2

). The orbital U(n) generators

may then be expressed as simple linear combinations

of U(2

) generators E

=|p)(q| with coefﬁcients equal

to ±1[4.93, 94].

Generally, any p-column U(n) irrep is contained

at least once in the totally symmetric p-box irrep of

U(2

). For many-electron problems, one thus requires

a two-box irrep 2

0. Any state arising in the U(n) irrep

Γ(a, b, c) can then be represented as a linear combi-

nation of two-box states, labeled by the Weyl tableaux

[i| j]≡

i j . In particular, the highest weight state of

Γ(a, b, c) is represented by



b+c



. Once this repre-

sentation is available, it is straightforward to compute

explicit representations of U(n) generators, since E

act trivially on [i| j] [4.94]. Deﬁning unnormalized states

(i| j) as

(i| j) =



1 + δ

[i| j] , (4.105)

we have

(i| j) = δ

( p| j) + δ

(i|p). (4.106)

The main features of CAUGA may thus be summa-

rized as follows: CAUGA

1. effectively reduces an N-electron problem to a num-

ber of two-boson problems;

2. enables an exploitation of an arbitrary coupling

scheme (being particularly suited for the valence

bond method);

3. can be applied to particle-number nonconserving

operators;

4. easily extends to fermions with an arbitrary spin;

5. drastically simpliﬁes evaluation of explicit represen-

tations of U(n) generators and of their products;

6. can be exploited in other than shell-model ap-

proaches [4.95–101].

4.3.8 Spin-Dependent Operators

The spin-adapted U(n)-based UGA is entirely satis-

factory in most investigations of molecular electronic

structure. However, when exploring the ﬁne structure

in high-resolution spectra, the intersystem crossings,

phosphorescent lifetimes, molecular predissociation,

spin–orbit interactions in transition metals, and like phe-

nomena, the explicitly spin-dependent terms must be

included in the Hamiltonian. Since in most cases the total

spin S represents a good approximate quantum number,

so that the spin-adapted N-electron states render an ex-

cellent point of departure, it is necessary to consider the

Part A 4.3

98 Part A Mathematical Methods

corresponding matrix elements of general spin-orbital

U(2n) generators in terms of which the relevant spin-

dependent terms may be expressed. This was ﬁrst done

in the context of the symmetric group and Racah algebra

by Dr ake and Schlesinger [4.78] and later on in terms of

the Gel’fand–Paldus tableaux [4.102–106].

In general, the U(2n) generators e

iσ, jτ

≡ e

may be

resolved into the spin-shift components e(±)

that in-

crease (+) or decrease (−) the total spin S by one unit

and the zero-spin component e(0)

that preserves S.

The relevant matrix elements can then be expressed in

terms of the matrix elements of a single U(n) adjoint ten-

sor operator ∆, which is given by the following second

degree polynomial in U(n) generators,

∆ = E(E+ N/2 − n− 2), E =E

 (4.107)

and by the well-known matrix elements of U(2) or SU(2)

generators in terms of the pure spin states [4.102, 103]

(see also [4.107, 108]). The operator (4.107), referred

to as the Gould–Paldus operator [4.109], also plays

a key role in the determination of reduced density ma-

trices [4.110, 111], and has been recently exploited in

the multireference spin-adapted variant of the density

functional theory [4.109].

References

4.1 A. O. Barut: Dynamical Groups and Generalized Sym-

metries in Quantum Theory, Vols. 1 and 2 (Univ.

Canterbury, Christchurch 1971)

4.2 A. Bohm, Y. Ne’eman, A. O. Barut: Dynamical Groups

and Spectrum Generating Algebras (World Scientiﬁc,

Singapore 1988)

4.3 R. Gilmore: Lie Groups, Lie Algebras, and Some of

Their Applications (Wiley, New York 1974)

4.4 B. G. Wybourne: Classical Groups for Physicists (Wi-

ley, New York 1974)

4.5 J.-Q. Chen: Group Representation Theory for Physi-

cists (World Scientiﬁc, Singapore 1989)

4.6 J.-Q. Chen, J. Ping, F. Wang: Group Representa-

tion Theory for Physicists, 2nd edn. (World Scientiﬁc,

Singapore 2002)

4.7 O. Castaños, A. Frank, R. Lopez-Peña: J. Phys. A:

Math. Gen. 23, 5141 (1990)

4.8 F. Iachello, R. D. Levine: Algebraic Theory of

Molecules (Oxford Univ. Press, Oxford 1995)

4.9 A. Frank, P. Van Isacker: Algebraic Methods in Mo-

lecular and Nuclear Structure (Wiley, New York 1994)

4.10 J. Paldus: Mathematical Frontiers in Computational

Chemical Physics, ed. by D. G. Truhlar (Springer,

Berlin, Heidelberg 1988) pp. 262–299

4.11 I. Shavitt: Mathematical Frontiers in Computational

Chemical Physics, ed. by D. G. Truhlar (Springer,

Berlin, Heidelberg 1988) pp. 300–349

4.12 J. Paldus: Contemporary Mathematics,Vol.160(AMS,

Providence 1994) pp. 209–236

4.13 J. C. Parikh: Group Symmetries in Nuclear Structure

(Plenum, New York 1978)

4.14 B. G. Adams, J.

Cí

zek, J. Paldus: Int. J. Quantum

Chem. 21, 153 (1982)

4.15 B. G. Adams, J.

Cí

zek, J. Paldus: Adv. Quantum Chem.

19, 1 (1988)

4.16 J.

Cí

zek, J. Paldus: Int. J. Quantum Chem. 12,875

(1977)

4.17 B. G. Adams: Algebraic Approach to Simple Quantum

Systems (Springer, Berlin, Heidelberg 1994)

4.18 A. Bohm: Nuovo Cimento A 43, 665 (1966)

4.19 A. Bohm: Quantum Mechanics (Springer, New York

1979)

4.20 M. A. Naimark: Linear Representations of the Lorentz

Group (Pergamon, New York 1964)

4.21 W. Pauli: Z. Phys. 36, 336 (1926)

4.22 V. Fock: Z. Phys. 98,145(1935)

4.23 A. O. Barut: Phys. Rev. B 135,839(1964)

4.24 A. O. Barut, G. L. Bornzin: J. Math. Phys. 12,841

(1971)

4.25 G. A. Zaicev: Algebraic Problems of Mathemati-

cal and Theoretical Physics (Science Publ. House,

Moscow 1974) (in Russian)

4.26 A. Joseph: Rev. Mod. Phys. 39, 829 (1967)

4.27 M. Bednar: Ann. Phys. (N. Y.) 75, 305 (1973)

4.28 J.

Cí

zek, E. R. Vrscay: Group Theoretical Methods in

Physics, ed. by R. Sharp, B. Coleman (Academic, New

York 1977) pp. 155–160

4.29 J.

Cí

zek, E. R. Vrscay: Int. J. Quantum Chem. 21,27

(1982)

4.30 J. E. Avron, B. G. Adams, J.

Cí

zek, M. Clay, L. Glasser,

P. Otto, J. Paldus, E. R. Vrscay: Phys. Rev. Lett. 43,

691 (1979)

4.31 J.

Cí

zek, M. Clay, J. Paldus: Phys. Rev. A 22, 793 (1980)

4.32 H. J. Silverstone, B. G. Adams, J.

Cí

zek, P. Otto: Phys.

Rev. Lett. 43, 1498 (1979)

4.33 E. R. Vrscay: Phys. Rev. A 31, 2054 (1985)

4.34 E. R. Vrscay: Phys. Rev. A 33, 1433 (1986)

4.35 H. J. Silverstone, R. K. Moats: Phys. Rev. A 23,1645

(1981)

4.36 H. Weyl: Classical Groups (Princeton Univ. Press,

Princeton 1939)

4.37 A. O. Barut, R. Raczka: Theory of Group Repre-

sentations and Applications (Polish Science Publ.,

Warszawa 1977)

4.38 J. D. Louck: Am. J. Phys. 38, 3 (1970)

4.39 J. Paldus: Theoretical Chemistry: Advances and Per-

spectives, Vol. 2, ed. by H. Eyring, D. J. Henderson

(Academic, New York 1976) pp. 131–290

Part A 4