All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation

Hyperfine Interact (2012) 211:99–122DOI 10.1007/s10751-011-0536-9

All 36 exactly solvable solutions of eigenvaluesfor nuclear electric quadrupole interaction Hamiltonianand equivalent rigid asymmetric rotor with expandedcharacteristic equation listing

Lorenz Harry Menke Jr.

Published online: 6 January 2012© Springer Science+Business Media B.V. 2012

Abstract This paper derives all 36 analytical solutions of the energy eigenvaluesfor nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asym-metric rotor for polynomial degrees 1 through 4 using classical algebraic theory.By the use of double-parameterization the full general solution sets are illustratedin a compact, symmetric, structural, and usable form that is valid for asymmetryparameter η ∈ (−∞, +∞). These results are useful for code developers in the areaof Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR)and rotational spectroscopy who want to offer exact solutions whenever possible,rather that resorting to numerical solutions. In addition, by using standard linearalgebra methods, the characteristic equations of all integer and half-integer spins Ifrom 0 to 15, inclusive are represented in a compact and naturally parameterizedform that illustrates structure and symmetries. This extends Nielson’s [1] listing ofcharacteristic equations for integer spins out to I = 15, inclusive.

Keywords Quadrupole · Rigid rotor · Eigenvalues · Characteristic equations ·PAC/NQR · Neumark’s method · Double-parameterization ·Microwave spectroscopy · Exact solutions

Mathematics Subject Classifications (2010) 13P15 · 08A40 · 39-02 · 15A18 · 08A45

L. H. Menke (B)University of Pittsburgh, Pittsburgh, PA, USAe-mail: [email protected]

100 L.H. Menke Jr.

1 Introduction

The solutions for quadrupole Hamiltonian characteristic polynomials are scatteredthroughout the literature. Several sources [2]1[3]2[4] – [5]3[6]4 and theses include allcases up to quadratics and some cubics. Furthermore, except for the spin 7/2 case, theother quartic solutions are absent from the literature. In general, there are no knowncomprehensive list of all known solutions available. This paper will address this prob-lem. Furthermore, Nielson’s original listing is extended and reparameterized in amore compact and illustrative form. The formal equivalence between the asymmetricrotor and the nuclear electric quadrupole Hamiltonian for integer spin is exploitedto apply these solutions and eigenpolynomial listings across PAC/NQR and micro-wave spectroscopy research.

The standard coverage of the nuclear electric quadrupole Hamiltonian derivationis given in, E. A. C. Lucken, Nuclear Quadrupole Coupling Constants [7]. TheHamiltonian operator for pure quadrupole resonance in terms of the angular mo-mentum spin operators, is

HQ = A{

3 I2z − I2 + η

2

(I2+ + I2−

)}

where I2z, I2, I2+, and I2− are the conventional Cartesian spin operators. The quadrupolecoupling constant, A, is given by:

A = − e Q Vzz

4 I (2 I − 1)

and the asymmetry parameter, η, is defined as

η = Vxx − Vyy

Vzz

where Vxx, Vyy, and Vzz are the electric field gradients in the principle coordinatesystem that diagonalized the interaction Hamiltonian. The Cartesian electric fieldgradients, Vii, satisfy Laplace’s equation, |Vxx + Vyy| = |Vzz|. The convention ischosen so that |Vxx| ≤ |Vyy| ≤ |Vzz| along with Laplace’s equation, it is concludedthat Vzz has opposite sign to Vxx and Vyy or |Vxx + Vyy| = |Vzz|. Where as η willhave the values η ∈ [0, 1]. The quadrupole coupling constant, A, has units of energyand is expressed in terms of the characteristic interaction quadrupole frequency asA = �ωQ which is given by:

ωQ = − e Q Vzz

4 I (2 I − 1) �.

The remaining constants are; Q the nuclear quadrupole moment with Q > 0 forprolate ellipsoid; Q < 0 for oblate ellipsoid; and Q = 0 for symmetric nuclear chargedistribution, I > 1/2 the spin of the intermediate state of the nuclear cascade, m is

1Covers spins 0, 1, 2, and 3.2Covers spins 1, 3/2, 2, and 5/2.3Covers spin 7/2.4Covers spin 3.

All 36 exactly solvable solutions of eigenvalues for NEQ interaction Hamiltonian 101

azimuthal magnetic projection where m ∈ [−I, +I], e the electron charge, and � isPlanck’s constant divided by 2π . The spins I = {0, 1/2} both have singular eigenstateswhich are zero and hence no interactions.

The matrix element form in the magnetic projection m-state representation isgiven in the compact form

HQ = A{[

3 m2 − I (I + 1)]δn,m + η f 1/2 (I, m + 1) δn,m+2 + η f 1/2 (I, m − 1) δn,m−2

}

where

f (I, m) = f (I,−m) =⎧⎨⎩

1

4

(I2 − m2) [(I + 1)2 − m2] , |m| < I;

0, |m| ≥ I,

and

δi, j ={

1, i = j;0, i �= j,

is the Kronecker delta function.Let the general solution be denoted by E(I)

m (η) as a function of η for a givenspin I and for magnetic quantum number m. The trace of the spin Hamiltonianvanishes, i.e.,

+I∑m=−I

E(I)m (η) = 0

with real eigenvalues due to the hermitian nature of the spin Hamiltonian.The Hamiltonian for electric field gradients with axial symmetry η = 0 is HQ =

A{3 I2z − I2

}and this is diagonal in the m-state representation. The eigenvalues of

this operator are given by E(I)m (0) = 3 m2 − I (I + 1) where the quadrupole coupling

constant has been divided into the eigenvalue, which will apply to the remainder ofthis paper.

2 Wang transformation-irreducible representation

For integer spins the quadrupole interaction Hamiltonian is mathematically equiv-alent to the quantum mechanical asymmetrical rigid rotor. Gilbert W. King et al.paper on The Asymmetric Rotor [8]5 describes a procedure based on group theo-retic analysis that factors the interaction Hamiltonian matrix into four submatrices(Hamiltonians) or step matrices

{E+,E−,O+

,O−} which are the irreducible rep-resentation of HQ. Furthermore, Bersohn’s paper on Nuclear Electric QuadrupoleSpectra in Solids [9] introduces a 4th order perturbation expansion for half-integerspins that is valid for small η and all magnetic states of a given spin. The 2 I + 1dimensional involutary unitary transformation, X, that transform from the m-staterepresentation to the Vierergruppe (Klein) four-group representation is known

5Covers spins 0, 1, 2, 3, and parts of 4 and 5.

102 L.H. Menke Jr.

Table 1 Count of exactly, partially, and non-solvable eigenpolynomials for spins I = 0 through 19/2,inclusive

Degree of polynomial Degree of polynomial

Spin I 1 2 3 4 5 Spin I 2 3 4 5 6 7 8 9 10

0 1 5 1 3

1/2 1 11/2 1

1 3 6 3 1

3/2 1 13/2 1

2 3 1 7 1 3

5/2 1 15/2 1

3 1 3 8 3 1

7/2 1 17/2 1

4 3 1 9 1 3

9/2 1 19/2 1

The shaded region are polynomial degrees that are not solvable algebraically

as a Wang transformation [10]. The transform results in X−1 HQ X = E+ ⊕ E−⊕O+ ⊕O−.

If I is half-integer, the 2 I + 1 states of orientation contain I + 1/2 doublydegenerate representations of the rhombic group so that the secular equation ofdegree 2 I + 1 factors into two identical irreducible monic equations of degreeI + 1/2. The 2 I + 1 dimensional unitary transformation, Y, obtained from the Wangtransformation by deletion of the central row and column, transforms the interactionHamiltonian from the m-state representation for half-integer spin into a two-grouprepresentation. These submatrices are identified by the symbols,M+ andM− in which(+) and (−) refer to the index sign. The transform results in Y−1 HQ Y = M+ ⊕M−.The characteristic determinant of |HQ − Em I| = 0, is the product of the determinantof either two or four submatrices.

Half-integer spins characteristic equations results in two identical polynomialswhile integer spins factor into four different (three for spin I = 1) polynomials. Thedegrees of these polynomials are listed in Table 1.6 This table shows that completesolutions are available for spins I = {0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 5, 6, 7} and thatpartial solutions (one or more solvable characteristic equations) for spins I = {8,9}. For spins I = {0, 1/2} define ω

∗Q = I (2 I − 1) ωQ to allow the interaction to be

computed.

3 Rigid asymmetric rotor

From King’s paper the energy of a rigid asymmetric rotor is expressed as

E (a, b , c) = 1

2(a − c) E(I)

τ (κ) + 1

2(a + c) I (I + 1)

6Note that there are a total of nine equations per degree.


where the moments of inertia along the principle axis are Ia, Ib , and Ic witha = �

2/2 Ia, b = �2/2 Ib , and c = �

2/2 Ic. The labeling of the axes is chosen sothat Ia ≤ Ib ≤ Ic or a ≥ b ≥ c. Ray’s [11] parameter of asymmetry κ is defined asκ = (2 b − a − c) / (a − c) so that κ ∈ [−1, +1]. The reduced energy E(I)

τ (κ) has thesymmetry property E(I)

+τ (+κ) = −E(I)−τ (−κ) where I is the spin and τ is the rank

defined below. In the limit κ → −1 or b = c is a prolate-symmetric rotor and the limitκ → +1 or b = a is a oblate-symmetric rotor while the limit κ → 0 or 2 b = a + c isan asymmetric rotor.

Using King’s papers method of labeling rigid asymmetric rotor energy levels forinteger spin by two subscripts the first |K−1|, being {0, 1, 1, 2, 2, . . . , I, I} fromlowest to highest energy levels and the second, |K+1|, being {0, 1, 1, 2, 2, . . . , I,I} from highest to lowest energy levels, gives not only the symmetry through theparity of the indices but also the rank, τ , through the relation τ = |K−1| − |K+1|which takes on the 2 I + 1 values τ ∈ [−I,+I]. The K±1 are projections of the totalangular momentum on the symmetry axis. The rank corresponds to the m-staterepresentation value, i.e., τ ≡ m.

Nielson’s [1] paper established the correlation between Wang-Klein solution ofthe rigid asymmetric top and the quadrupole interaction energies for integer spin byW = (1/3) [E + I (I + 1)] and b = η/3. The connection between King’s energy statesof rigid asymmetric rotor and quadrupole interaction for integer spin is given at theend of §5 in King’s paper, W = E(I)

τ (κ) /κ and b = 1/κ , as

E(I)τ (κ) = 1

η

[E(I)

m (η) + I (I + 1)]

the transformation between asymmetry parameters is η = 3/κ . Note that the normalrange of both asymmetry parameters do not overlap.

In the limits as κ → −1 (η → −3, type Ir right-handed permutation) and κ → +1(η → +3, type I I Ir right-handed permutation) the rigid asymmetric rotor is diagonal(Table III in King’s paper). Using Eqs. 22 and 24 from King’s paper the eigenvaluesare E(I)

K−1(+3) = 2 I (I + 1) − 6 K2

−1 where K−1 is the diagonal representation index

and E(I)K+1

(−3) = 2 I (I + 1) − 6 K2+1 where K+1 is the diagonal representation index.

The eigenvalues for the spherical-oblate limit in the m-state representation is

E(I)m (+3) = 2 I (I + 1) − 6

{(I + 1 − |m|)2, m < 0;(I − |m|)2, m ≥ 0,

∀I ∈ N0;

where N0 = {0, 1, 2, . . .} is the set of whole numbers. The eigenvalues for thespherical-prolate limit in the m-state representation is

E(I)m (−3)=2 I (I+1) − 6

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

4

⌊I + 1 − |m|

2

⌋2

, ∀[

(even I ∧ m ≥ 0)

∨ (odd I ∧ m < 0)

];

(1 + 2

⌊I − |m|

2

⌋)2

, ∀[

(even I ∧ m < 0)

∨ (odd I ∧ m ≥ 0)

].

∀I ∈ N0;

104 L.H. Menke Jr.

Table 2 Solutions of all solvable 36 characteristic equations

Spin I Solutions of characteristic equations for general η

0 E+ E ⇒ E(0)0 = 0.

1/2 M± E ⇒ E(1/2)±1/2 = 0.

1 E+ E′ + 1 ⇒ E(1)0 = −2.

1 O± E − β± ⇒ E(1)±1 = +β±.

3/2 M± E2 − 3 α ⇒⎧⎨⎩

E(3/2)±3/2 = +√

3 α,

E(3/2)±1/2 = −√

3 α.

2 E+ E′2 − 3 α ⇒⎧⎨⎩

E(2)+2 = +2

√3 α,

E(2)0 = −2

√3 α.

2 E− E′ − 3 ⇒ E(2)−2 = +6.

2 O± E + 3 β∓ ⇒ E(2)±1 = −3 β∓.

5/2 M± E′3 − 7 α E′ − 20 β. ⇒

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

E(5/2)±5/2 = +4

√7 α/3 cos (ψ) ,

E(5/2)±3/2 = −4

√7 α/3 cos

(π

3+ ψ

),

E(5/2)±1/2 = −4

√7 α/3 cos

(π

3− ψ

),

where

ψ = θ

3and θ = arccos

(10 β

(7 α/3)3/2

).

3 E+ E′2 + 6 E′ − 15 ζ ⇒⎧⎨⎩

E(3)+2 = −6 + 2

√3 (3 + 5 ζ ),

E(3)0 = −6 − 2

√3 (3 + 5 ζ ).

3 E− E ⇒ E(3)−2 = 0.

3 O± E2 − 6 β± E − 15(β2± + 8 β∓

)⇒

⎧⎪⎨⎪⎩

E(3)±3 = +3 β± + 2

√6(β2± + 5 β∓

),

E(3)±1 = +3 β± − 2

√6(β2± + 5 β∓

).

7/2 M± E4 − 126 α E2 − 1,728 β E + 945 α2 ⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(7/2)±7/2 = +A + B−

A,

E(7/2)±5/2 = +A − B−

A,

E(7/2)±3/2 = −A + B+

A,

E(7/2)±1/2 = −A − B+

A,

where

A =√

3 α[7 + 2

√21 cos (ψ)

],

B± =√

A(−A3 + 63 α A ± 432 β

),

ψ = θ

3, and θ = arccos

(49 α3 + 864 β2

(21 α2

)3/2

).


Table 2 (continued)


4 E+ E′3 − 52 α E′ − 560 β ⇒

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

E(4)+4 = +8

√13 α/3 cos (ψ) ,

E(4)+2 = −8

√13 α/3 cos

(π

3+ ψ

),

E(4)0 = −8

√13 α/3 cos

(π

3− ψ

),

where

ψ = θ

3and θ = arccos

(35 β

(13 α/3)3/2

).

4 E− E′2 − 10 E′ − 7 (8 + ζ ) ⇒

⎧⎪⎨⎪⎩

E(4)−2 = +10 − 2

√81 + 7 ζ ,

E(4)−4 = +10 + 2

√81 + 7 ζ .

4 O± E2 + 10 β∓ E − 7(

9 β2∓ + 8 β±)

⇒⎧⎪⎨⎪⎩

E(4)±3 = −5 β∓ + 2

√2(11 β2∓ + 7 β±

),

E(4)±1 = −5 β∓ − 2

√2(11 β2∓ + 7 β±

).

5 E+ E′3 + 15 E′2 − 3 (27 + 44 ζ ) E′ − 135 (9 − 4 ζ ) ⇒⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

E(5)+4 = −10 + 8

√13 + 11 ζ cos (ψ) ,

E(5)+2 = −10 − 8

√13 + 11 ζ cos

(π

3+ ψ

),

E(5)+0 = −10 − 8

√13 + 11 ζ cos

(π

3− ψ

),

where

ψ = θ

3and θ = arccos

(5 (7 − 15 ζ )

(13 + 11 ζ )3/2

).

5 E− E′2 − 27 α ⇒⎧⎨⎩

E(5)−2 = −6

√3 α,

E(5)−4 = +6

√3 α.

5 O± E3 − 15 β± E2 − 3(

71 β2± + 352 β∓)

E + 135 β±(

5 β2± − 32 β∓)

⇒⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

E(5)±5 = +5 β± + 8

√2(3 β2± + 11 β∓

)cos (ψ±) ,

E(5)±3 = +5 β± − 8

√2(3 β2± + 11 β∓

)cos

(π

3+ ψ±

),

E(5)±1 = +5 β± − 8

√2(3 β2± + 11 β∓

)cos

(π

3− ψ±

),

where

ψ± = θ±3

and θ± = arccos

(5 β± (β± + 15 β∓)[2(3 β2± + 11 β∓

)]3/2

).

106 L.H. Menke Jr.

Table 2 (continued)


6 E+ E′4 − 294 α E′2 − 7,776 β E′ + 3,465 α2 ⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(6)+6 = +2 A + 2

B−A

,

E(6)+4 = +2 A − 2

B−A

,

E(6)+2 = −2 A + 2

B+A

,

E(6)0 = −2 A − 2

B+A

,

where

A =√

α[49 + 2

√889 cos (ψ)

],

B± =√

A(−A3 + 147 α A ± 1,944 β

),

ψ = θ

3, and θ = arccos

(6,517 α3 + 472,392 β2

(889 α2

)3/2

).

6 E− E′3 − 21 E′2 − 21 (21 + 4 ζ ) E′ + 495 (3 + 4 ζ ) ⇒⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

E(6)−2 = +14 − 8

√7 (7 + ζ ) cos

(π

3− ψ

),

E(6)−4 = +14 − 8

√7 (7 + ζ ) cos

(π

3+ ψ

),

E(6)−6 = +14 + 8

√7 (7 + ζ ) cos (ψ) ,

where

ψ = θ

3and θ = arccos

(143 − 87 ζ

[7 (7 + ζ )]3/2

).

6 O± E3 + 21 β∓ E2 − 21(

25 β2∓ + 32 β±)

E − 495 β∓(

7 β2∓ + 32 β±)

⇒⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

E(6)±5 = −7 β∓ + 8

√14(β2∓ + β±

)cos (ψ±) ,

E(6)±3 = −7 β∓ − 8

√14(β2∓ + β±

)cos

(π

3+ ψ±

),

E(6)±1 = −7 β∓ − 8

√14(β2∓ + β±

)cos

(π

3− ψ±

),

where

ψ± = θ±3

and θ± = arccos

(β∓(87 β± − 7 β2∓

)[14(β2∓ + β±

)]3/2

).

It is also found that half-integer spins are also diagonal for κ → ±1 where bothspherical-oblate and spherical-prolate limits are equivalent with eigenvalues in them-state representation

E(I)m (±3) = 2 I (I + 1) − 6

(I + 1

2− |m|

)2

∀I ∈ N0 + 1

2.

Note that half-integer spins exhibit ±m degeneracy as expected.


Table 2 (continued)


7 E+

E′4 + 28 E′3 − 14 (42 + 41 ζ ) E′2

− 4 (4,688 − 1,279 ζ ) E′

− 91(

704 − 1,136 ζ − 243 ζ 2)

⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(7)+6 = −14 + 2 A + 2

B−A

,

E(7)+4 = −14 + 2 A − 2

B−A

,

E(7)+2 = −14 − 2 A + 2

B+A

,

E(7)0 = −14 − 2 A − 2

B+A

,

where

A =√

1

3

[7 (63 + 41 ζ ) + 2

√C cos (ψ)

],

B± =√

A[−A3 + 7 (63 + 41 ζ ) A ± 24 (81 − 137 ζ )

],

C = 7(

10,287 + 13,266 ζ + 5,311 ζ 2)

,

Q = 17,505,477 − 34,441,119 ζ + 46,569,519 ζ 2 + 4,184,747 ζ 3,

ψ = θ

3, and θ = arccos

(Q

C3/2

).

7 E− E′3 − 196 α E′ − 2,288 β ⇒

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

E(7)−2 = −8

√49 α/3 cos

(π

3− ψ

),

E(7)−4 = −8

√49 α/3 cos

(π

3+ ψ

),

E(7)−6 = +8

√49 α/3 cos (ψ) ,

where

ψ = θ

3and θ = arccos

(143 β

(49 α/3)3/2

).

7 O±

E4 − 28 β± E3 − 14(

83 β2± + 328 β∓)

E2

+ 4 β±(

3,409 β2± − 10,232 β∓)

E

+ 91[675 β4± + 16 β∓

(811 β2± + 972 β∓

)]⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(7)±7 = +7 β± + 2 A± + 2

B±−A±

,

E(7)±5 = +7 β± + 2 A± − 2

B±−A±

,

E(7)±3 = +7 β± − 2 A± + 2

B±+A±

,

E(7)±1 = +7 β± − 2 A± − 2

B±+A±

,

108 L.H. Menke Jr.

Table 2 (continued)


where

A± =√

1

6

[7(13 β2± + 41 β∓

)+ 2√

C± cos (ψ±)],

B+± =√

A+[−A3+ + 7

(13 β2+ + 41 β−

)A+ ± 6 β+

(7 β2+ + 137β−

)],

B−± =√

A−[−A3− + 7

(13 β2− + 41 β+

)A− ± 6 β−

(7 β2− + 137β+

)],

C± = 7[451 β4± + β∓

(2,986 β2± + 5,311 β∓

)],

Q± = 66,052 β6± + β∓

⎡⎣ 1,113,315β4± +β∓

×(

7,390,470 β2± + 4,184,747 β∓)⎤⎦ ,

ψ± = θ±3

, and θ± = arccos

(Q±

C3/2±

).

8 E+ E′5 − 1,044 α E′3 − 48,816 β E′2⇒

+ 112,320 α2 E′ + 4,665,600 α β

{E(8)

+8, E(8)+6, E(8)

+4, E(8)+2, E(8)

0

}.

8 E−

E′4 − 36 E′3 − 18 (102 + 23 ζ ) E′2

+ 540 (32 + 41 ζ ) E′

+ 675(

576 − 48 ζ + 11 ζ 2)

⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(8)−2 = +18 − 2 A − 2

B+A

,

E(8)−4 = +18 − 2 A + 2

B+A

,

E(8)−6 = +18 + 2 A − 2

B−A

,

E(8)−8 = +18 + 2 A + 2

B−A

,

where

A =√

3[129 + 23 ζ + 2

√C cos (ψ)

],

B± =√

A[−A3 + 9 (129 + 23 ζ ) A ± 216 (25 − 17 ζ )

],

C = 3(

2,547 + 906 ζ + 67 ζ 2)

,

Q = 540,189 + 31,833 ζ + 92,727 ζ 2 + 851 ζ 3,

ψ = θ

3, and θ = arccos

(Q

C3/2

).

4 Solutions

Solutions to these characteristic equations are available for spin values of I = {1,3/2, 2, 5/2, 3}. However, rarely are the solutions to higher spin values tabulated.


Table 2 (continued)


8 O±

E4 + 36 β∓ E3 − 18(

125 β2∓ + 184 β±)

E2

− 540 β∓(

73 β2∓ + 328 β±)

E

+ 675[539 β4∓ − 16 β±

(13 β2∓ − 44 β±

)]⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(8)±7 = −9 β∓ + 2 A± + 2

B±−A±

,

E(8)±5 = −9 β∓ + 2 A± − 2

B±−A±

,

E(8)±3 = −9 β∓ − 2 A± + 2

B±+A±

,

E(8)±1 = −9 β∓ − 2 A± − 2

B±+A±

,

where

A± =√

6[19 β2∓ + 23 β± + 2

√C± cos (ψ±)

],

B+± =

√√√√√√A+

⎡⎢⎣

−A3+ + 18(

19 β2− + 23 β+)

A+

± 216 β−(

17 β+ − 7 β2−)

⎤⎥⎦,

B−± =

√√√√√√A−

⎡⎢⎣

−A3− + 18(

19 β2+ + 23 β−)

A−

± 216 β+(

17 β− − 7 β2+)

⎤⎥⎦,

C± = 3[55 β4∓ + β±

(130 β2∓ + 67 β±

)],

Q± = 1,300 β6∓ + β±[3,435 β4∓ + β±

(11,910 β2∓ + 851 β±

)],

ψ± = θ±3

, and θ± = arccos

(Q±

C3/2±

).

9 E+

E′5 + 45 E′4 − 18 (129 + 98 ζ ) E′3 − 270 (467 − 94 ζ ) E′2

− 27(

22,077 − 74,652 ζ − 14,912 ζ 2)

E′

+ 144,585(

117 + 12 ζ − 64 ζ 2)

⇒

{E(9)

+8, E(9)+6, E(9)

+4, E(9)+2, E(9)

0

}.

All solutions including the two null solutions are tabulated for completeness and areexpressed as real functions. The energy labeling is from highest m-state of +I tolowest m-state of −I such that E±I > E±(I−1) > · · · > E±1/2 for half-integer spins

110 L.H. Menke Jr.

Table 2 (continued)


9 E− E′4 − 774 α E′2 − 21,600 β E′ + 41,769 α2 ⇒

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

E(9)−2 = −2 A − 2

B+A

,

E(9)−4 = −2 A + 2

B+A

,

E(9)−6 = +2 A − 2

B−A

,

E(9)−8 = +2 A + 2

B−A

,

where

A =√

3 α[43 + 2

√849 cos (ψ)

],

B± =√

A(−A3 + 387 α A ± 5,400 β

),

ψ = θ

3, and θ = arccos

(15,007 α3 + 135,000 β2

(849 α2

)3/2

).

9 O±

E5 − 45 β± E4 − 18(

227 β2± + 784 β∓)

E3

+ 270 β±(

373 β2± − 752 β∓)

E2

⇒+ 27

[67,487 β4± + 32 β∓

(26,119 β2± + 29,824 β∓

)]E

− 144,585 β±[65 β4± − 32 β∓

(29 β2± + 128 β∓

)]

{E(9)

±9, E(9)±7, E(9)

±5, E(9)±3, E(9)

±1

}.

In the following solutions the substitutions E = 2 E′, α = 3 + η2, β = 1 − η2, β± = 1 ± η and ζ =η2 are used. The angle variables θ , θ±, ψ , ψ±, etc., and quartic variables A > 0, A± > 0, B± > 0,B+± > 0, B−± > 0, C > 0, C± > 0, Q, and Q± will be local to each corresponding spin submatrix. Theequation for O+ denoted by the upper signs and O− by the lower signs. All solutions are valid forη ∈ (−∞,+∞). The irreducible quintics for spins 8 and 9 are listed for completeness

and E+I ≥ E−I > E+(I−1) ≥ E−(I−1) > · · · > E0 for integer spins for the normalquadrupole interaction domain of η ∈ [0, 1]. The sub-Hamiltonians, characteristicequations, and solutions are listed in Table 2. Solution methods include the standardquadratic formula, Viëte’s method for cubics, and Neumark’s [12] completion of thesquare for quartics. On the various solutions of the quartic, Neumark’s method isthe most compact, symmetric, and illustrative. For Neumark’s method chose theauxiliary cubic solution that does not vanish over η’s domain (quartic parametersA, A± ∝ chosen cubic root �= 0), otherwise, solutions interchanges will occur.

The Appendix lists illustrations of each solvable and partially solvable spin caseover the quadrupole domain η ∈ [0,+1]. This furnishes a quick reference of theeigenvalues of each spin case. In fact, all eigenstates of a given spin form asymptoticlines as |η| → ∞, which is evident in the lower spin state illustrations. H. Waranabe,Nuclear Level Splitting Caused By An Axially Nonsymmetric Electric QuadrupoleInteraction: 1 ≤ I ≤ 40 [13] provides numerical tables for each spin, half-integerand integer, and all the m-states as functions of η from 0 to 1 in one tenth steps.


Furthermore, graphs of each spin case up to I = 20, inclusive over the quadrupoledomain are illustrated.

5 Characteristic equations and symmetries

Denote the characteristic equations that arise from each transformation as PIM±(E,

η) for half-integer spin, PIE± (E, η), and PI

O± (E, η) for integer spin transformationsE± and O±, respectively. We find that PI

M± (E, η) → PIM±(E, η2

)and PI

E± (E, η) →PIE±(E, η2

). The symmetry relation for odd indexed integer spin is PI

O+ (E,±η) =PIO− (E, ∓η). The symmetry relation for even indexed integer and half-integer

spins are PIM± (E, ±η) = PI

M± (E, ∓η) and PIE± (E, ±η) = PI

E± (E, ∓η). Due to sym-metry the number of equations for integer spin is reduced by one. At η = ±1further symmetries become apparent. The characteristic equation for PI

M± (E, ±1)

factors as a bi-degree polynomial P(E2)

of degree (1/2) (I + 1/2) for even I +1/2 or E P

(E2) of degree (1/2) (I − 1/2) for odd I + 1/2. For even integer spin

we have PIO− (+E, ±1) = PI

E− (−E, ±1) and for odd integer spin PIO+ (+E, ±1) =

PIE+ (−E, ±1) which in each case reduces to three equations. At η ∈ {−3, 0,+3} all

equations factor as binomials. In general, all of the characteristic equations of degreen ≥ 2 are irreducible for η or η2 over rationals (Galois irreducibility) except for thenoted points and elliptic or hyperelliptic points that are are unique to each case.The following equations shows the respective degree at η = ±1 for integer spin asfunctions of P

(E2)

or E P(E2).

Characteristic degree for even I at η = ±1

E−,O± :

⎧⎪⎨⎪⎩

I4, I = 4 k;

I + 2

4

∗, I = 4 k − 2,

; k ∈ N0 and E+ :

⎧⎪⎨⎪⎩

I + 2

4, I = 4 k − 2;

I4

∗, I = 4 k,

; k ∈ N0

where (*) denotes E P(E2)

and unmarked denotes P(E2). Characteristic degree for

odd I at η = ±1

E+,O± :

⎧⎪⎨⎪⎩

I + 1

4, I = 4 k − 1;

I − 1

4

∗, I = 4 k − 3,

; k ∈ N and E− :

⎧⎪⎨⎪⎩

I − 1

4, I = 4 k − 3;

I + 1

4

∗, I = 4 k − 1,

; k ∈ N

where N ∈ {1, 2, 3, . . .} is the set of natural numbers.Using linear algebra methods we list the characteristic equations for half-integer

spin in Table 3 and those for integer spin in Table 4. Table 4 extends Nielson’s listingof characteristic equations for

{E+,E−,O+

,O−} submatrices for spins I > 10. Thenotation used in the following tables is compact and illustrates symmetries and struc-ture. Nota Bene: This extension of Nielson’s listing exceeds what most spectroscopic

112 L.H. Menke Jr.

Table 3 List of irreducible characteristic equations for half-integer spin, I, of degree 29/2 or lesswith E = 2 E′, α = 3 + η2 and β = 1 − η2

Spin I List of characteristic equations for half-integer spin, I

1/2 E

3/2 E2 − 3 α

5/2 E′3 − 7 α E′ − 20 β

7/2 E4 − 126 α E2 − 1,728 β E + 945 α2

9/2 E′5 − 99 α E′3 − 1,188 β E′2 + 1,188 α2 E′ + 11,664 α β

11/2 E6 − 1,001 α E4 − 36,608 β E3 + 172,315 α2 E2 + 7,404,800 α β E

− 9,625(

243 α3 − 4,096 β2)

13/2 E′7 − 546 α E′5 − 14,040 β E′4 + 63,297 α2 E′3 + 2,232,360 α β E′2

− 324(

3,607 α3 − 40,500 β2)

E′ − 24,766,560 α2 β

15/2 E8 − 4,284 α E6 − 293,760 β E5 + 4,488,102 α2 E4 + 460,028,160 α β E3

− 12,852(

82,651 α3 − 691,200 β2)

E2 − 94,440,668,640 α2 β E

+ 6,251,175 α(

3,575 α3 − 221,184 β2)

17/2 E′9 − 1,938 α E′7 − 85,272 β E′6 + 1,016,481 α2 E′5 + 70,612,968 α β E′4

− 1,292(

115,265 α3 − 764,988 β2)

E′3 − 10,509,076,320 α2 β E′2

+ 91,200 α(

42,633 α3 − 1,800,652 β2)

E′

+ 11,968,000 β(

12,069 α3 − 35,672 β2)

19/2 E10 − 13,167 α E8 − 1,444,608 β E7 + 50,640,282 α2 E6 + 9,109,304,064 α β E5

− 1,026(

61,173,511 α3 − 335,702,016 β2)

E4 − 12,528,768,026,880 α2 β E3

+ 290,871 α(

67,477,459 α3 − 2,152,341,504 β2)

E2

+ 5,038,848 β(

592,912,691 α3 − 1,382,400,000 β2)

E

− 1,123,052,931 α2(

520,625 α3 − 74,870,784 β2)

21/2 E′11 − 5,313 α E′9 − 355,212 β E′8 + 8,750,511 α2 E′7 + 986,003,928 α β E′6

− 621(

8,289,575 α3 − 38,712,816 β2)

E′5 − 681,057,096,300 α2 β E′4

+ 6,520,500 α(

144,865 α3 − 3,700,404 β2)

E′3

+ 167,670,000 β(

667,037 α3 − 1,280,124 β2)

E′2

− 4,694,760,000 α2(

7,007 α3 − 701,055 β2)

E′

− 85,730,400,000 α β(

22,181 α3 − 250,047 β2)

23/2 E12 − 32,890 α E10 − 5,262,400 β E9 + 351,709,215 α2 E8

+ 96,853,036,800 α β E7 − 460(

3,156,053,737 α3 − 12,812,928,000 β2)

E6

− 488,105,100,681,600 α2 β E5 + 115 α

(18,759,043,634,197 α3

− 398,760,729,600,000 β2

)E4

+ 147,200 β(

4,824,563,811,041 α3 − 7,850,794,112,000 β2)

E3

− 1,518 α2(

566,270,393,453,451 α3 − 43,269,171,738,880,000 β2)

E2

− 193,600 α β(

1,036,283,204,477,889 α3 − 9,294,823,616,512,000 β2)

E

+ 4,228,420,625

⎡⎣ 8,081,268,993 α6 − 32,768 β2

×(

66,567,177 α3 − 66,734,080 β2)⎤⎦


Table 3 (continued)


25/2 E′13 − 12,285 α E′11 − 1,158,300 β E′10 + 51,023,115 α2 E′9 + 8,417,471,400 α β E′8

− 3,645(

23,843,659 α3 − 85,542,000 β2)

E′7 − 18,013,037,636,700 α2 β E′6

+ 87,480 α(

675,594,931 α3 − 12,279,867,750 β2)

E′5

+ 15,746,400 β(

806,811,679 α3 − 1,137,886,750 β2)

E′4

− 314,928 α2(

42,275,451,869 α3 − 2,606,423,775,000 β2)

E′3

− 661,348,800 α β(

3,606,369,581 α3 − 26,789,103,000 β2)

E′2

+ 154,314,720,000[3,843,467 α6 − 27 β2

(27,143,161 α3 − 22,999,680 β2

)]E′

+ 49,374,244,613,232,000,000 α2 β(α3 − 27 β2

)

27/2 E14 − 71,253 α E12 − 15,634,944 β E11 + 1,773,253,053 α2 E10

+ 689,923,173,888 α β E9 − 21,141(

898,734,061 α3 − 2,886,623,232 β2)

E8

− 9,492,016,281,375,744 α2 β E7 + 63,423 α

(1,385,707,959,725 α3

− 21,976,705,695,744 β2

)E6

+ 584,506,368 β(

81,081,173,225 α3 − 100,793,759,744 β2)

E5

− 14,270,175 α2(

10,890,252,543,025 α3 − 561,788,847,243,264 β2)

E4

− 109,594,944,000 α β(

697,949,799,725 α3 − 4,416,443,080,704 β2)

E3

+2∑

k=0

Pk (η) Ek

where

P2 (η) = +4,637,806,875

⎡⎣ 16,522,190,140,093 α6 − 884,736 β2

×(

2,745,039,019 α3 − 2,010,659,328 β2)⎤⎦ ,

P1 (η) = +1,102,680,213,485,760,000 α2 β(

23,127,709 α3 − 500,428,800 β2)

,

P0 (η) = −78,844,209,769,040,625 α

⎡⎣49,439,995 α6 − 221,184 β2

×(

99,781 α3 − 387,072 β2)⎤⎦ .

29/2 E′15 − 25,172 α E′13 − 3,178,864 β E′12 + 227,499,142 α2 E′11

+ 51,500,775,664 α β E′10 − 3,596(

256,177,901 α3 − 744,463,592 β2)

E′9

− 272,405,811,508,944 α2 β E′8 + 899 α

(1,904,195,635,579 α3

− 26,765,565,080,640 β2

)E′7

+ 14,384 β(

39,116,392,874,255 α3 − 43,436,701,992,208 β2)

E′6

− 7,192 α2(

187,545,382,763,179 α3 − 8,306,872,117,453,236 β2)

E′5

− 1,611,008 α β(

266,273,029,251,353 α3 − 1,465,595,022,826,481 β2)

E′4

+3∑

k=0

Pk (η) E′k

where

P3 (η) = +704,816

⎡⎢⎢⎣

519,002,173,254,267 α6 − 8 β2

×(

7,743,965,695,182,661 α3

− 4,988,469,328,943,750 β2

)⎤⎥⎥⎦ ,

114 L.H. Menke Jr.

Table 3 (continued)


P2 (η) = +78,939,392 α2 β

(1,172,761,380,708,219 α3

− 21,126,318,541,272,532 β2

),

P1 (η) = −19,054,336 α

⎡⎣

1,060,247,270,525,625 α6 − 4 β2

×(

84,657,181,297,640,301 α3

− 278,808,827,495,412,500 β2

)⎤⎦ ,

P0 (η) = −45,917,261,824 β

⎡⎣

49,955,564,716,875 α6 − 1,936 β2

×(

1,339,937,264,646 α3

− 609,839,140,625 β2

)⎤⎦ .

The coefficient polynomials Pk (η) will be local to each corresponding spin

Table 4 List of irreducible characteristic equations for integer spin, I, of degree 15 or less withE = 2 E′, α = 3 + η2, β = 1 − η2, β± = 1 ± η, and ζ = η2

Spin I List of characteristic equations for integer spin, I

0 E+ E

1 E+ E′ + 1

1 O± E − β±2 E+ E′2 − 3 α

2 E− E′ − 3

2 O± E + 3 β∓3 E+ E′2 + 6 E′ − 15 ζ

3 E− E

3 O± E2 − 6 β± E − 15(β2± + 8 β∓

)

4 E+ E′3 − 52 α E′ − 560 β

4 E− E′2 − 10 E′ − 7 (8 + ζ )

4 O± E2 + 10 β∓ E − 7(

9 β2∓ + 8 β±)

5 E+ E′3 + 15 E′2 − 3 (27 + 44 ζ ) E′ − 135 (9 − 4 ζ )

5 E− E′2 − 27α

5 O± E3 − 15 β± E2 − 3(

71 β2± + 352 β∓)

E + 135 β±(

5 β2± − 32 β∓)

6 E+ E′4 − 294 α E′2 − 7,776 β E′ + 3,465 α2

6 E− E′3 − 21 E′2 − 21 (21 + 4 ζ ) E′ + 495 (3 + 4 ζ )

6 O± E3 + 21 β∓ E2 − 21(

25 β2∓ + 32 β±)

E − 495 β∓(

7 β2∓ + 32 β±)

7 E+ E′4 + 28 E′3 − 14 (42 + 41 ζ ) E′2 − 4 (4,688 − 1,279 ζ ) E′

− 91(

704 − 1,136 ζ − 243 ζ 2)

7 E− E′3 − 196 α E′ − 2,288 β

7 O± E4 − 28 β± E3 − 14(

83 β2± + 328 β∓)

E2 + 4 β±(

3,409 β2± − 10,232 β∓)

E

+ 91[675 β4± + 16 β∓

(811 β2± + 972 β∓

)]

8 E+ E′5 − 1,044 α E′3 − 48,816 β E′2 + 112,320 α2 E′ + 4,665,600 α β

8 E− E′4 − 36 E′3 − 18 (102 + 23 ζ ) E′2 + 540 (32 + 41 ζ ) E′ + 675(

576 − 48 ζ + 11 ζ 2)

8 O± E4 + 36 β∓ E3 − 18(

125 β2∓ + 184 β±)

E2 − 540 β∓(

73 β2∓ + 328 β±)

E

+ 675[539 β4∓ − 16 β±

(13 β2∓ − 44 β±

)]

9 E+ E′5 + 45 E′4 − 18 (129 + 98 ζ ) E′3 − 270 (467 − 94 ζ ) E′2

− 27(

22,077 − 74,652 ζ − 14,912 ζ 2)

E′ + 144,585(

117 + 12 ζ − 64 ζ 2)


Table 4 (continued)


9 E− E′4 − 774 α E′2 − 21,600 β E′ + 41,769 α2

9 O± E5 − 45 β± E4 − 18(

227 β2± + 784 β∓)

E3 + 270 β±(

373 β2± − 752 β∓)

E2

+ 27[67,487 β4± + 32 β∓

(26,119 β2± + 29,824 β∓

)]E

− 144,585 β±[65 β4± − 32 β∓

(29 β2± + 128 β∓

)]

10 E+ E′6 − 2,849 α E′4 − 204,160 β E′3 + 1,277,419 α2 E′2 + 112,366,720 α β E′

− 5,225(

6,075 α3 − 226,304 β2)

10 E− E′5 − 55 E′4 − 22 (251 + 62 ζ ) E′3 + 110 (905 + 1,134 ζ ) E′2

+ 11(

547,411 + 15,564 ζ + 17,984 ζ 2)

E′

+ 4,655(

1,271 − 11,884 ζ − 2,752 ζ 2)

10 O± E5 + 55 β∓ E4 − 22(

313 β2∓ + 496 β±)

E3 − 110 β∓(

2,039 β2∓ + 9,072 β±)

E2

+ 11[580,959 β4∓ + 32 β±

(12,883 β2∓ + 35,968 β±

)]E

+ 4,655 β∓[13,365 β4∓ + 32 β±

(4,347 β2∓ + 5,504 β±

)]

11 E+ E′6 + 66 E′5 − 33 (204 + 133 ζ ) E′4 − 396 (1,398 − 227 ζ ) E′3

− 99(

12,384 − 181,044 ζ − 34,865 ζ 2)

E′2

+ 1,890(

249,408 + 23,280 ζ − 93,311 ζ 2)

E′

+ 1,091,475(

4,608 − 5,568 ζ − 808 ζ 2 − 221 ζ 3)

11 E− E′5 − 2,244 α E′3 − 109,296 β E′2 + 665,280 α2 E′ + 25,401,600 α β

11 O± E6 − 66 β± E5 − 33(

337 β2± + 1,064 β∓)

E4 + 396 β±(

1,171 β2± − 1,816 β∓)

E3

+ 99[203,525 β4± + 16 β∓

(125,387 β2± + 139,460 β∓

)]E2

− 1,890 β±[179,377 β4± − 16 β∓

(81,671 β2± + 373,244 β∓

)]E

− 1,091,475{

1,989 β6± + 8 β∓[7,847 β4± + 8 β∓

(1,471 β2± + 1,768 β∓

)]}

12 E+ E′7 − 6,552 α E′5 − 662,688 β E′4 + 8,620,560 α2 E′3 + 1,228,825,728 α β E′2

+ 6,912(

224,297 α3 − 4,134,375 β2)

E′ − 144,635,092,992 α2 β

12 E− E′6 − 78 E′5 − 39 (348 + 91 ζ ) E′4 + 468 (846 + 1,045 ζ ) E′3

+ 117(

399,168 + 44,892 ζ + 17,239 ζ 2)

E′2

+ 2,106(

20,736 − 512,352 ζ − 127,763 ζ 2)

E′

− 5,589(

2,985,984 + 601,344 ζ − 782,928 ζ 2 + 11,875 ζ 3)

12 O± E6 + 78 β∓ E5 − 39(

439 β2∓ + 728 β±)

E4 − 468 β∓(

1,891 β2∓ + 8,360 β±)

E3

+ 117[461,299 β4∓‘ + 16 β±

(39,685 β2∓ + 68,956 β±

)]E2

− 2,106 β∓[619,379 β4∓ + 16 β±

(383,939 β2∓ + 511,052 β±

)]E

− 5,589{

2,816,275 β6∓ − β±[928,887 β4∓ + 8 β±

(747,303 β2∓ − 95,000 β±

)]}

13 E+ E′7 + 91 E′6 − 91 (177 + 104 ζ ) E′5 − 13 (143,341 − 19,784 ζ ) E′4

+ 91(

119,849 + 1,142,416 ζ + 214,960 ζ 2)

E′3

+ 455(

12,343,479 + 1,114,640 ζ − 3,625,968 ζ 2)

E′2

+ 25

(4,532,189,167 − 6,850,110,712 ζ

− 1,412,346,192 ζ 2 − 275,353,344 ζ 3

)E′

− 56,875(

12,432,185 + 28,672,328 ζ − 21,752,560 ζ 2 − 6,946,560 ζ 3)

116 L.H. Menke Jr.

Table 4 (continued)


13 E− E′6 − 5,369 α E′4 − 397,696 β E′3 + 5,232,955 α2 E′2 + 14,191,923,200 α β E′

− 10,625(

43,659 α3 − 447,488 β2)

13 O± E7 − 91 β± E6 − 91(

281 β2± + 832 β∓)

E5 + 13 β±(

123,557 β2± − 158,272 β∓)

E4

+ 91[1,477,225 β4± + 128 β∓

(98,271 β2± + 107,480 β∓

)]E3

− 455 β±[9,832,151 β2± − 128 β∓

(383,581 β2± + 1,812,984 β∓

)]E2

− 25

⎧⎨⎩4,005,621,081 β6± + 64 β∓

⎡⎣

1,312,607,891 β4± + 48 β∓

×(

46,633,463 β2± + 45,892,224 β∓)⎤⎦⎫⎬⎭ E

+ 56,875 β±

⎧⎨⎩12,405,393 β6± − 64 β∓

⎡⎣

4,459,059 β4± + 80 β∓

×(

532,403 β2± + 694,656 β∓)⎤⎦⎫⎬⎭

14 E+ E′8 − 13,356 α E′6 − 1,805,760 β E′5 + 41,987,862 α2 E′4 + 8,644,890,240 α β E′3

− 108(

253,479,401 α3 − 3,060,460,800 β2)

E′2 − 5,114,739,349,440 α2 β E′

+ 8,037,225 α(

142,025 α3 − 18,966,528 β2)

14 E− E′7 − 105 E′6 − 21 (1,383 + 376 ζ ) E′5 + 45 (27,639 + 33,800 ζ ) E′4

+ 63(

3,925,341 + 697,104 ζ + 200,944 ζ 2)

E′3

− 2,835(

11,097 + 3,643,056 ζ + 950,384 ζ 2)

E′2

− 243(

1,667,954,187 + 457,830,792 ζ − 339,080,976 ζ 2 + 11,653,376 ζ 3)

E′

−2,525,985(

1,375,623 − 2,615,976 ζ − 804,240 ζ 2 − 142,592 ζ 3)

14 O± E7 + 105 β∓ E6 − 21(

1,759 β2∓ + 3,008 β±)

E5

− 45 β∓(

61,439 β2∓ + 270,400 β±)

E4

+ 63[4,823,389 β4∓ + 128 β±

(68,687 β2∓ + 100,472 β±

)]E3

+ 2,835 β∓[4,604,537 β4∓ + 128 β±

(346,489 β2∓ + 475,192 β±

)]E2

− 243

⎧⎨⎩1,798,357,379 β6∓ − 64 β±

⎡⎣

23,171,379 β4∓ + 16 β±

×(

19,007,553 β2∓ − 5,826,688 β±)⎤⎦⎫⎬⎭ E

− 2,525,985 β∓

⎧⎨⎩2,187,185 β6∓ + 64 β±

⎡⎣

581,529 β4∓ + 16 β±

×(

77,001 β2∓ + 71,296 β±)⎤⎦⎫⎬⎭

15 E+ E′8 + 120 E′7 − 252 (134 + 73 ζ ) E′6 − 1,080 (4,836 − 587 ζ ) E′5

+ 54(

2,051,352 + 8,450,088 ζ + 1,564,393 ζ 2)

E′4

+ 3,240(

12,926,880 + 1,151,484 ζ − 3,199,847 ζ 2)

E′3

+ 108

(10,903,189,248 − 20,982,056,832 ζ

− 5,071,714,254 ζ 2 − 846,665,221 ζ 3

)E′2

− 3,240

(11,210,489,856 + 14,944,566,528 ζ

− 11,170,502,280 ζ 2 − 3,481,521,299 ζ 3

)E′

− 1,937,925

(537,919,488 − 460,062,720 ζ − 290,801,664 ζ 2

+ 59,672,896 ζ 3 − 5,102,125 ζ 4

)


Table 4 (continued)


15 E− E′7 − 11,256 α E′5 − 1,170,720 β E′4 + 27,917,712 α2 E′3

+ 3,910,654,080 α β E′2 − 20,736(

576,703 α3 − 4,465,125 β2)

E′

− 965,229,281,280 α2 β

15 O± E8 − 120 β± E7 − 252(

207 β2± + 584 β∓)

E6 + 1,080 β±(

4,249 β2± + 4,696 β∓)

E5

+ 54[12,065,833 β4± + 16 β∓

(5,789,437 β2± + 6,257,572 β∓

)]E4

− 3,240 β±[10,878,517 β4± − 16 β∓

(2,624,105 β2± + 12,799,388 β∓

)]E3

+2∑

k=0

Pk (η) Ek

where

P2 (η) = −108

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

15,997,247,059 β6± + 8 β∓

×⎡⎢⎣

33,665,481,003 β4± + 8 β∓

×(

7,611,709,917 β2±+ 6,773,321,768 β∓

)⎤⎥⎦

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

,

P1 (η) = +3,240 β±

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

11,503,032,805 β6± + 8 β∓

×⎡⎢⎣

17,841,001,929 β4± + 8 β∓

×(

21,615,066,177 β2±+ 27,852,170,392 β∓

)⎤⎥⎦

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

,

P0 (η) = +1,937,925

⎛⎜⎜⎜⎜⎜⎜⎝

158,374,125 β8± + 32 β∓

×

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

220,763,965 β6± − 4 β∓

×⎡⎢⎣

71,197,863 β4± − 16 β∓

×(

9,816,099 β2±− 10,204,250 β∓

)⎤⎥⎦

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

⎞⎟⎟⎟⎟⎟⎟⎠

.

The characteristic equations for integer spin are listed by the corresponding submatrix that generatedthe equation in the following order

{E+,E−,O+,O−}. The submatrix set for spin 0 is

{E+} and spin

1 is{E+,O+,O−}. O+ is denoted by the upper signs and O− by the lower signs in the combined

equations

analysts use for eigenpolynomials before resorting to numerical methods to diago-nalize the Hamiltonian and solve for the energy eigenvalues.

Denote(α3, β2

)nas a linear functional of

{(α3)n

,(α3)n−1

β2,(α3)n−2(

β2)2

, · · · ,(α3)2(

β2)n−2

, α3(β2)n−1

,(β2)n}

with(α3, β2

)0 ≡ 0. A repeating pattern of coefficients of the characteristic equationsfor M± and alternating between E+ for even spin and E− for odd spin ∀n ∈ N0 is

{(α2 β

)n (α3, β2)n

, αn+1(α3, β2)n, βn+1(α3, β2)n

,(α2)n+1(

α3, β2)n

, (α β)n+1(α3, β2

)n,(α3, β2

)n+1

}.

No general form for the numerical factors and nested(

A α3 − B β2)

pattern hasbeen identified.

118 L.H. Menke Jr.

6 Discussion

By the proper choice of double-parameterization the full general solution sets areillustrated in a compact, symmetric, structural, and usable form that is valid forη ∈ (−∞,+∞). All characteristic equations are reduced to a product of linearfactors for asymmetric symmetry η = 0 and oblate (η = +3) and the prolate (η = −3)rigid asymmetric rotor limits. In addition, at η = ±1 all characteristic equations asfunctions of E2 are reduced in degree by about 1/2. Thus, for M± the additionalhalf-integer spins {9/2, 11/2, 13/2, 15/2, 17/2} are solvable as are the additionalinteger spins for E+ we have {8, 10, 12, 14, 16}, for E− we have {11, 13, 15, 17,19}, for O+ we have {10, 12, 14, 16, 18}, and for O− we have {9, 11, 13, 15, 17}.There are also algebraically solvable cases for rational and quadratic and higherorder algebraic field values of the asymmetry parameter on a case by case analysis.Furthermore, elliptic and hyperelliptic curve analysis reveals additional particularsolutions over rationals. Finally, recent papers by King and Canfield [14] andDrociuk [15] a comprehensive solution of the general quintic is now feasible. Thisfull set of exact solutions enables accurate descriptions of quadrupole interactionsand rotation dynamics in place of numerical methods for a larger domain of spinsthen previously available.

The consistent symbolic solutions enables a clear and simple numerical solutionsto low level spins that also functions as a benchmark comparison against numericalsolutions. The exact solutions also expands the number of exactly solvable cases thatcan form the basis of illustration and education. Also since the solutions involve rootextraction, cosine, and inverse cosine evaluations the numerical solutions are simpleto code.

The characteristic equations for integer and half-integer spins out to spin 15, inclu-sive are listed. This equation listing is more organized and extensive than availablein the published literature. Structural properties are clearly illustrated along withsystematic double parametric decomposition for the coefficients of M±, E+, and O±

for half-integer and even spin and E− for odd spin. The double-parameterization ofthe characteristic equations is roughly equivalent to Horner’s method of structuringpolynomials where in each case the size of the coefficients are greatly reduced. Themultivariate coefficient factoring by α, β, β±, and ζ is not arbitrary. It is distinctivein that no other simple combination results in as compact form. Thus, this set formsa natural parameterization of the characteristic equations. All of these solutions havebeen verified using Mathematica™ Version 7.010 [16].

Acknowledgements To my loving wife Deanna Carmen Menke: 1964-2008. Short, simple, and tothe point, just the way she thought it should be.

Appendix

The solvable cases are illustrated in the following figures over the quadrupole domainof η ∈ [0,+1]. The half-integer cases spins I = 5/2 and I = 7/2 are illustrated inFigs. 1 and 2, respectively where the energy relations are functions of η. Eachsubsequent integer spin is illustrated with one spin per diagram. For spins I = 2 − 9the energy relations as functions of η are illustrated in Figs. 3, 4, 5, 6, 7, 8, 9, 10, re-spectively. For each Figure, the energy states are labeled using the m-state notation.


Fig. 1 Energy states asfunctions of η for spin I = 5/2 5/ 2

3/ 2

1/ 2

0.0 0.2 0.4 0.6 0.8 1.0

−10

− 5

0

5

10

η

E

±

±

±

Fig. 2 Energy states asfunctions of η for spin I = 7/2

7/ 2

5/ 2

±

±

±

±

3/ 2

1/ 2

0.0 0.2 0.4 0.6 0.8 1.0

− 20

− 10

0

10

20

η

E

Fig. 3 Energy states asfunctions of η for spin I = 2

+ 2−⎯ 2

+ 1

− 1

0

0.0 0.2 0.4 0.6 0.8 1.0

− 6

− 4

− 2

0

2

4

6

E

η

120 L.H. Menke Jr.


+3−3

+2−2+1

−10

0.0 0.2 0.4 0.6 0.8 1.0

−15

−10

−5

0

5

10

15

η

E


+ 4− 4

−3 3

+ 2− 2 + 1

− 10

0.0 0.2 0.4 0.6 0.8 1.0

− 30

− 20

− 10

0

10

20

30

η

E

+


+ 5− 5

+ 4− 4

+ 3− 3

+ 2 − 2+ 1

− 10

0.0 0.2 0.4 0.6 0.8 1.0

− 40

− 20

0

20

40

η

E



+ 6− 6

+ 5− 5+ 4− 4

3− 3 + 2

− 2 +1− 1

0

0.0 0.2 0.4 0.6 0.8 1.0

− 60

− 40

− 20

0

20

40

60

η

E+


+ 7− 7+ 6− 6+ 5− 5

+ 4− 4 + 3 − 3

+ 2− 2+ 1− 1

0

0.0 0.2 0.4 0.6 0.8 1.0

− 100

− 50

0

50

100

η

E


+ 8− 8+ 7− 7

+ 6− 6

+ 5− 5+ 4

− 4 + 3 − 3+ 2− 2

+1− 10

0.0 0.2 0.4 0.6 0.8 1.0

− 100

− 50

0

50

100

η

E

122 L.H. Menke Jr.


+ 9− 9+ 8− 8

+ 7− 7+ 6

− 6+ 5− 5 + 4 − 4

+ 3− 3+ 2 −2

+ 1− 10

0.0 0.2 0.4 0.6 0.8 1.0

− 150

− 100

− 50

0

50

100

150

η

E

Curves are color labeled with the repeating hue pattern of blue, red, tan, greenwith each cycle having a slightly different hue. The spin cases I = 1 and I = 3/2are not illustrated due to their simplicity. Finally, when the domain is expand tosay η ∈ [−3, +3] such as to illustrate the asymmetric rotor, the energy curves forma mirror image about the E-axis with the odd m-state curve labels switching signs.

References

1. Nielson, H.H.: Infrared bands of slightly asymmetric molecules. Phys. Rev. 38(8), 1432–1441(1931)

2. Kroto, H.W: Molecular rotation spectra. Dover Publications, Inc., Mineola, Mineola, New York(2003)

3. Butz, T.: Analytic perturbation functions for static interactions in perturbed angular correlationof γ -rays. Hyperfine Interact. 52, 189–228 (1989)

4. Ageev, S.Z., Sanctuary, B.C.: Selective excitation of single and multiple quantum transitions forspin 7/2 in NQR. Mol. Phys. 87(6), 1423–1438 (1996)

5. Creel, R.B.: Analytic solution of fourth degree secular equations: I = 3/2 zeeman-quadrupoleinteractions and I = 7/2 pure quadrupole interaction. J. Magn. Reson. 52, 515–517 (1983)

6. Creel, R.B.: Solutions of the nuclear electric quadrupole hamiltonian for spin 3. J. Magn. Reson.50, 81–85 (1982)

7. Lucken, E.A.C.: Nuclear quadrupole coupling constants. Academic Press, London and NewYork (1969)

8. King, G.W., Hainer, R.M., Cross, P.C.: The asymmetric rotor I. Calculation and symmetryclassification of energy levels. J. Chem. Phys. 2, 27–42 (1943)

9. Bersohn, R.: Nuclear electric quadrupole spectra in solids. J. Chem. Phys. 20(10), 1505–1509(1952)

10. Wang, S.C.: On the asymmetrical top in quantum mechnics. Phys. Rev. 34, 243–252 (1929)11. Ray, B.S.: Über die Eigenwerte des asymmetrischen kreisels. Zeitschrift fur Physik, 78, 74–91

(1932)12. Neumark, S.: Solution of cubic and quartic equations. Pergamon Press, 1st edn. (1965)13. Waranabe, H.: Nuclear level splitting caused by an axially nonsymmetric electric quadrupole

interactions: 1 ≤ I ≤ 40. At. Data Nucl. Data Tables 46, 285–343 (1990)14. King, R.B., Canfield, E.R.: An algorithm for calculating the roots of a general quintic equation

from its coefficients. J. Math. Phys. 32(4), 823–825 (1991)15. Drociuk, R.J.: On the complete solution to the most general fifth degree polynomial.

Physics Department, Simon Fraser University, Burnaby British Columbia, Canada (2000).arXiv:math.GM/0005026V1

16. Wolfram, S.: The mathematica book. Wolfram Media/Cambridge University Press, Mass, USA,7th edn. (2009)

http://arXiv.org/abs/math/0005026V1

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation

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