McWeeny-Volume

Eigenvalues of Model Hamiltonian Matrices

from Spectral Density Distribution Moments.

The Heisenberg Spin Hamiltonian

JACEK KARWOWSKI, DOROTA BIELINSKA–WAZ

AND JACEK JURKOWSKI

Instytut Fizyki, Uniwersytet Miko laja Kopernika, Grudziadzka 5,87-100 Torun, Poland

Abstract

An approach aimed at approximating the extreme (the lowest and/or the high-est) eigenvalues of matrices representing many-electron model Hamiltonians from aknowledge of several spectral density distribution moments is proposed. A detaileddiscussion of the Heisenberg spin Hamiltonian spectrum is presented as an exampleof an application. PACS:

31.15-p — calculations and mathematical techniques in atomic and molecular physics

31.15.ct — semi-empirical and empirical calculations

75.10.Dg — crystal-field theory and spin Hamiltonians

1

1. Introduction

In his “Overview of Molecular Quantum Mechanics” presented at 1991 NATO ASIin Bad Windsheim, Professor Roy McWeeny says: “The fundamentals of molecu-lar quantum mechanics have changed remarkably little since the 1930’s: the mostcommonly used methods of constructing molecular wavefunctions are still dominatedby expansions in terms of Slater determinants in which individual electrons are de-scribed by spin-orbitals (...). It is true that the techniques have changed. (...) Butbeautiful as some of these techniques may be, they are simply sophisticated and cleverways of doing the same things.” [1]. This deep and fundamentally true statementstimulates thinking about molecular quantum mechanics in very general terms, sothat details of this or another approach do not obscure the general structure of thetheory. Then, in the vast majority of methods of molecular quantum mechanicswe simply attempt to solve the eigenvalue problem of a matrix which representsthe N -electron Hamiltonian in an N -electron (i.e. antisymmetric and usually spin-adapted) finite-dimensional Hilbert space being an N -fold Cartesian product of aone-electron 2K-dimensional spin-orbital space being itself a Cartesian product ofa K-dimensional orbital space and a 2-dimensional spin space. Specific methodsdiffer by the way the one-electron space is selected and by the approximations usedto solve the eigenvalue problem in the N -electron model space. Their overview isgiven, e.g., in an excellent monograph by McWeeny [2].

Also in this paper we aim at determining eigenvalues of matrices representingHamiltonians of many-electron systems in spin-adapted antisymmetric model spaces.However our approach is based on ideas developed mostly in nuclear physics [3, 4] andis rather exotic in molecular quantum mechanics. We try to deduce the eigenvaluesof a Hamiltonian matrix from a knowledge of several moments of its spectral densitydistribution. This attempt, if successful, may open a way to developing methodsof approximating the eigenvalues by quantities whose evaluation is not limited bydimensions of the matrices. Spectral density distribution moments characterize theHamiltonian and the model space. They are independent of a specific representationand can be determined a priori, without any N -electron matrix element evaluation[5]. The results always correspond to the complete N -electron model space. Since,in this language, no bound property for the approximations to individual eigenvaluesexists, the accuracy of the results may be estimated, in principle, by an extrapo-lation of the cases in which a direct comparison with the exact values is possible.This weakness of the approach is compensated by its generality and insensitivityto the dimension of the problem. In particular, it is expected that it may be veryuseful in establishing different types of extrapolation and asymptotic formulas forthe eigenvalues.

In the present paper the Heisenberg spin Hamiltonian has been used to illustratethe way in which the spectral density distribution moments may be applied to gen-erate specific eigenvalues. The moment-generated eigenvalues are compared withthe exact ones for two simple chains of spins. However, without any modification ofthe method, an arbitrary lattice of spins may be studied in exactly the same way.Our formalism is based on a valence-bond type representation of the HeisenbergHamiltonian. The paper is dedicated to Professor Roy McWeeny, whose works onmolecular quantum mechanics, in particular on the valence-bond method [1, 2, 6],gave to us much inspiration.

2

2. Moment-Generated Spectra

The density ρ(E) of a discrete spectrum E1, E2, . . . , ED may be expressed as

ρ(E) =1

D

D∑

i=1

δ(E − Ei) , (1)

with∞∫

−∞

ρ(E) dE = 1 . (2)

The corresponding moments Mn, n = 1, 2, . . . , D are

Mn =

∞∫

−∞

Enρ(E) dE =1

D

D∑

i=1

Eni =

1

DTrHn , (3)

where H is the Hamiltonian matrix. The average energy E is equal to M1, the width(dispersion) of the spectrum is defined as σ = (M2 − M2

1 )1/2 and the asymmetry

parameter (skewness) is given by γ = (M3 − 3M2M1 + 2M31 )/σ

3.

The distribution function F (E) is defined by

F (E) =

E∫

−∞

ρ(E) dE . (4)

The discrete spectral density function may be approximated by a continuous densityfunction ρ(E) chosen so, that a given number of moments calculated with both thefunctions are the same. The continuous m−moment equivalent of ρ(E) is thendetermined by the following set of conditions:

∞∫

−∞

(E − E)n[ρ(E)− ρ(E)] dE = 0 , n = 0, 1, . . . ,m . (5)

It is convenient to introduce the dimensionless variable

x = (E − E)/σ . (6)

With a continuous distribution function F (x) one may associate a discrete (smoothed)spectrum. The smoothed spectrum may be defined as the one which corresponds tothe set of values xi which satisfies the condition [3, 4]

F (xi) = (i− 1/2)/D, i = 1, 2, . . . , D . (7)

The difference between the real spectrum and the smoothed one, i.e the qualityof the moment-generated spectrum, depends on the kind of the continuous spectraldensity function selected for the fitting procedure and on the number of momentstaken into account. Most frequently Gaussian-like distributions based on the Gram-Charlier expansion [4, 7, 8] are used. Then [9]

ρ(x) =1√(2π)

e−x2/2m∑

n=1

cnHn(x) , (8)

3

where Hn(x) is the n-th order Hermite polynomial. Multiplying by Hj(x) andintegrating from −∞ to ∞ we have

cj =1

j!

∞∫

−∞

ρ(x)Hj(x) dx . (9)

Assuming that the spectrum is normalized so that E = 0 and σ = 1, in virtue ofEqs. (3) and (5),

∞∫

−∞

ρ(x)xn dx = Mn

and cj may be expressed as a linear combination ofMk, k = 1, 2, . . . , j. In particular,for the central (i.e. calculated about the average) moments, c0 = 1, c1 = 0, c2 =(M2 − 1)/2, c3 = M3/6, c4 = (M4 − 6M2 + 3)/24, . . . [9].

Using Gaussian-like approximations of ρ is justified by a theorem which says thatasymptotically, when the dimension, K, of the orbital space approaches infinity,N ≫ 1, N/K → 0 and the one-particle interactions dominate, the spectral densitydistribution becomes Gaussian, independent of the specific form of the one-particlepotential [10]. However, this theorem is not valid for finite systems and for modelsin which the values of K and N are of comparable magnitude. In such cases thequestion which form of the continuous density function is most appropriate remainsopen. In particular, a polynomial expansion

ρ(E) = (Emax − E)(E − Emin)m−2∑

n=0

cnEn (10)

with ρ(E) ≥ 0 for Emin < E < Emax and ρ(Emin) = ρ(Emax) = 0 is a reasonableoption. The parameters Emin, Emax and cn, n = 0, 1, . . . ,m−2, are determined from

Emax∫

Emin

ρ(E) dE = 1 , (11)

Emax∫

Emin

ρ(E)En dE = Mn , for n = 1, 2, 3, . . . ,m . (12)

One of the advantages of using the polynomial approximation of ρ(E) is that therange of the spectrum (Emin and Emax) is explicitly determined by the moments. Adisadvantage, compared to the Gram-Charlier expansion, is the occurence of morecomplicated expressions for cn (for m > 3 they have to be evaluated numerically).

The formalism outlined, has successfully been applied in nuclear [3, 4] and inatomic [7, 8] spectroscopy. It belongs to a field referred to as statistical theory ofspectra. A review of applications of this theory in atomic physics has recently beenpublished by Bauche and Bauche-Arnoult [11]. An appealing area of its applicationin molecular physics and quantum chemistry is studying behaviour of eigenvalues ofH as functions of N and of quantum numbers, as e.g. of the total spin S. Amongmodels which are most suitable for this kind of study, the Heisenberg spin Hamilto-nian belongs to the most attractive ones.

4

3. The Heisenberg Hamiltonian

The Heisenberg Hamiltonian for an N -electron system or rather for a system ofN spins, may be written as [12–14]

H =N∑

i<j

Jij (1

2+ 2~si~sj) (13)

=1

2

N∑

i 6=j

Jij(ij) , (14)

where Jij are the exchange parameters, ~si is the one-electron spin operator of the i-th

electron and (ij) ∈ SN is a transposition operator. The Hamiltonian is defined in acomplete antisymmetric and spin-adapted space of covalent structures correspondingto a single electronic configuration in which all the orbitals are singly occupied.Then, the number of spinorbitals is twice the number of electrons, i.e. K = N inthis case. The dimension of the model space is equal to [12]

f(S,N) =2S + 1

N + 1

(N + 1

N/2− S

), (15)

where S is the quantum number corresponding to the total spin operator S2. TheHamiltonian matrix elements in this space are [5]

Hmn =1

2

N∑

i 6=j

JijVNS ((ij))mn , (16)

whereV NS (P )mn = 〈SM,m|P |SM, n〉 , (17)

with |SM, n〉 being an eigenvector of S2 and Sz is a matrix element of an irreduciblerepresentation of the permutation group of N objects, SN . The representation isassociated with a two-row Young shape containing

p = N/2 + S (18)

cells in the first row andr = N/2− S (19)

cells in the second row.An arbitrary power of the Heisenberg Hamiltonian may be expressed in terms

of permutation operators (c.f. Eq. 14). Therefore its spectral density distributionmoments may easily be related to the characters of the pertinent representations ofthe symmetric group. They were studied in this context already by Heisenberg inhis original paper [12] and then by many other authors [15–18]. Since

Hn =1

2n

N∑

i1 6=j1

N∑

i2 6=j2

· · ·N∑

in 6=jn

Ji1j1Ji2j2 · · · Jinjn (i1j1)(i2j2) · · · (injn) , (20)

we have

Mn =1

2n∑

C

χ[pr][C]

f(S,N)

N∑

{i}

N∑

{j}

Ji1j1Ji2j2 · · · Jinjn , (21)

5

where the first sum is extended over those classes [C] of S2n ⊂ SN which can be

generated by products of n transpositions, χ[pr][C] denote characters of the pertinent

representation of SN and the sums over {i} and {j} extend to those combinationsof the indices i1, i2, . . . , in and j1, j2, . . . , jn, which result in permutations belongingto the appropriate class. In particular,

M1 = X[pr][2]

1

2

N∑

ij

′ Jij , (22)

M2 = X[pr][22]

1

4

N∑

ijkl

′JijJkl +X[pr][31]

N∑

ijk

′ JijJjk +X[pr][14]

1

2

N∑

ij

′J2ij , (23)

M3 = X[pr][23]

1

8

N∑

ijklmn

′JijJklJmn +X[pr][321]

3

2

N∑

ijklm

′JijJjkJlm

+ X[pr][214]

12

N∑

ij

′J3ij +

N∑

ijk

′(3J2ijJjk + JijJjkJki) +

3

4

N∑

ijkl

′J2ijJkl

+ X[pr][412]

N∑

ijkl

′ (JijJikJil + 3JijJjkJkl) , (24)

where primes mean that all the summation indices are different and

X[pr][C] =

χ[pr][C]

f(S,N)(25)

are the normalized irreducible characters, introduced and studied by Klein et al.[18].

A class [C] of S2n is defined by a partition t1, t2, . . . , tq. In the simplest case,

χ[pr][1q ] = f(S,N). In a general case, the characters χ

[pr][C] may be expressed, using the

Murnaghan-Nakayama rule [19]. As one can demonstrate [20], in particular, if q = 1then

χ[pr][t1]

= f(S0, N1) + ǫ1f(S1, N1) , (26)

whereN1 = N − t1 ,

ǫ1 = sgn(2S − t1 + 1)

with sgn(x) = −1, 0, 1 if, respectively, x < 0, x = 0, x > 0;

S0 = S +t12

(27)

and

S1 =

S − t12, if t1 ≤ 2S

t12− S − 1 , if 2S + 1 < t1 ≤ N

2+ S + 1 .

(28)

For t1 outside the ranges specified in Eq. (28). If q = 2, then f(S1, N1) = 0;

χ[pr][t1t2]

= f(S00, N2) + f(S01, N2) + ǫ10f(S10, N2) + ǫ11f(S11, N2) , (29)

whereN2 = N1 − t2 ,

6

ǫ10 = ǫ1 ,

ǫ11 = ǫ1sgn(2S1 − t2 + 1) ,

S0i2 = S0 + (−1)i2t22, (30)

S10 = S1 + t2/2 , (31)

and

S11 =

S1 − t22, if t2 ≤ 2S1

t22− S1 − 1 , if 2S1 + 1 < t2 ≤ N1

2+ S1 + 1 .

(32)

Finally,

χ[pr][t1t2t3···tq ]

=1∑

i1=0

1∑

i2=0

· · ·1∑

iq=0

ǫi1i2···iqf(Si1i2···iq , Nq)′ , (33)

whereNq = Nq−1 − tq ,

ǫi1i2···iq−10 = ǫi1i2···iq−1 ,

ǫi1i2···iq−11 = ǫi1i2···iq−1sgn(2Si1i2···iq−1 − tq + 1) ,

Si1i2···iq−10 = Si1i2···iq−1 + tq/2 (34)

and

Si1i2···iq−11 =

Si1i2···iq−1 −tq2, if tq ≤ 2Si1i2···iq−1

tq2− Si1i2···iq−1 − 1 , if 2Si1i2···iq−1 + 1 < tq ≤ Nq−1

2+ Si1i2···iq−1 + 1 .

(35)Prime in Eq. (33) means that f(Si1i2···iq , Nq) 6= 0 only if the inequalities:

0 ≤ Si1 ≤ N1/2

0 ≤ Si1i2 ≤ N2/2

· · · · · · · · · (36)

0 ≤ Si1i2···iq ≤ Nq/2

are simultaneously fulfilled.The general character expressions given by Eqs. (26) and (29) may be transformed

to more explicit forms:

χ[pr][t1]

=2S + t1 + 1

N − t1 + 1

(N − t1 + 1

N/2− S − t1

)+

2S − t1 + 1

N − t1 + 1

(N − t1 + 1

N/2− S

), (37)

and

χ[pr][t1t2]

=2S + t1 + t2 + 1

N − t1 − t2 + 1

(N − t1 − t2 + 1

N/2− S − t1 − t2

)∆0

+2S + t1 − t2 + 1

N − t1 − t2 + 1

(N − t1 − t2 + 1

N/2− S − t1

)∆0

+2S − t1 + t2 + 1

N − t1 − t2 + 1

(N − t1 − t2 + 1

N/2− S − t2

)∆1

+2S − t1 − t2 + 1

N − t1 − t2 + 1

(N − t1 − t2 + 1

N/2− S

)∆1 , (38)

7

where∆0 = Θ(N/2− S − t1)

and∆1 = Θ(N/2 + S − t1 + 1)(1− δ(2S + 1, t1)) .

The symbol Θ(x) stands for the Heaviside function:

Θ(x) =

{0, if x < 01, if x ≥ 0

(39)

and δ(x, y) is the Kronecker delta. Combining Eqs. (25) and (33) one can see that

the normalized irreducible characters X[pr][C] are polynomials of the 2n-th order in S

(in fact they are n-th order polynomials in S(S + 1)) with the coefficients beingrational functions of N in which the numerators and denominators are polynomialsof at most 2n-th order [18].

It is convenient to transform the incomplete sums of products of the exchangeparameters in Eqs. (22)–(24) to complete sums which may be related to topologicalinvariants of the corresponding molecular graphs [5]. Then, after some algebra, weget:

Mn =l(n)∑

i

Tn(i)Rn(i) , (40)

where l(1) = 1, l(2) = 3, l(3) = 8, . . .. The elements Tn(i) expressed in terms ofcharacters of SN and their asymptotic forms corresponding to N ≫ 1 are as follows:

T1(1) = X[pr][2] ∼ z + 1

4,

T2(1) = 14X

[pr][22] ∼ z2 + 1

2z + 1

16,

T2(2) = −X[pr][22] +X

[pr][31] ∼ −4z2 + z − 3

4N,

T2(3) = 12X

[pr][22] −X

[pr][31] +

12X

[pr][14] ∼ 2z2 − 2z + 3

8,

T3(1) = 18X

[pr][23] ∼ z3 + 3

4z2 + 3

16z + 1

64,

T3(2) = −6X[pr][23] + 15X

[pr][321] − 9X

[pr][412] ∼ −48z3 + 36z2 − 6z + 9

2N,

T3(3) = 34X

[pr][23] − 3

2X

[pr][321] +

34X

[pr][214] ∼ 6z3 − 9

2z2 − 3

8z + 9

32,

T3(4) = −X[pr][23] + 3X

[pr][321] +X

[pr][214] − 3X

[pr][412] ∼ −8z3 + 6z2 − 5

2z + 3

8,

T3(5) = 2X[pr][23] − 3X

[pr][321] +X

[pr][412] ∼ 16z3 − 4z2 + 10

Nz − 15

4N2 ,

T3(6) = 3X[pr][23] − 6X

[pr][321] + 3X

[pr][412] ∼ 24z3 − 12z2 + 3

2z − 9

8N,

T3(7) = −32X

[pr][23] +

32X

[pr][321] ∼ −12z3 + 63

Nz2 + 3

4z − 9

16N,

T3(8) = 2X[pr][23] − 6X

[pr][321] −X

[pr][214] + 5X

[pr][412] ∼ 16z3 − 14z2 + 4z − 3

8,

(41)

where z = S(S + 1)/N2. The combinations of products of the exchange parametersare given by the following set of equations:

R1(1) =∑ijJij , R2(1) =

∑ijkl

JijJkl , R2(2) =∑ijk

JijJjk ,

R2(3) =∑ijJ2ij , R3(1) =

∑ijklmn

JijJklJmn , R3(2) =∑ijk

J2ijJik ,

R3(3) =∑ijkl

J2ijJkl , R3(4) =

∑ijk

JijJjkJki , R3(5) =∑ijkl

JijJikJil ,

R3(6) =∑ijkl

JijJjkJkl , R3(7) =∑

ijklmJijJjkJlm , R3(8) =

∑ijJ3ij .

(42)

8

Table 1: W[C] coefficients for a linear and for a comb-like chain of atoms.

Coefficient linear chain comb-like chainW[2] 2(N − 1) 2(N − 1)W[22] (N − 2)(N − 3) (N − 5

3)(N − 4)

W[31] 2(N − 2) 83(N − 7

4)

W[14] N − 1 N − 1W[23] (N − 3)(N − 4)(N − 5) (N − 3)(N − 4)(N − 7)W[321] 6(N − 3)(N − 4) 8(N − 4)(N − 7

2)

W[412] 6(N − 3) 10(N − 175)

W[214] 3(N − 1)(N − 53) 3(N − 1)(N − 5

3)

The coefficients Rn(i) carry all information about topology of the molecule. Theirmeaning becomes particularly transparent in the tight binding approximation. If allnearest neighbour exchange parameters are equal to J and all the others vanish, thenJij/J is the topological matrix of the molecule - its elements are 1 for neighboringatoms and 0 otherwise. Let a denotes the number of bonds in the molecule, bi —

the number of bonds connected to the i-th atom, βp =N∑i=1

bpi , α =N∑i,jbibjJij/J ,

n(△) — the number of three-member cycles formed by the bonds and Rn(i)′ =

Rn(i)/J n. Then R1(1)′ = 2a, R2(1)

′ = 4a2, R2(2)′ = β2, R2(3)

′ = 2a, R3(1)′ = 8a3,

R3(2)′ = β2, R3(3)

′ = 4a2, R3(4)′ = n(△), R3(5)

′ = β3, R3(6)′ = α, R3(7)

′ = 2aβ2,R3(8)

′ = 2a. Eqs. (22)–(24) may be rewritten in the form

Mn = J n∑

C

X[pr][C] W[C] , (43)

where

W[2] = 2a ,

W[22] = a(a+ 1)− β2 ,

W[31] = −2a+ β2 ,

W[14] = a ,

W[23] = a(a2 + 3a+ 4)− 3(a+ 2)β2 + 3α + 2β3 − n(△) ,

W[321] = −6a(a+ 2) + 3(a+ 5)β2 − 6α− 3β3 + 3n(△) ,

W[412] = 10a− 9β2 + 3α + β3 − 3n(△) ,

W[214] = a(3a− 2) + n(△) .

Asymptotically, for N → ∞, Rn(i)′ behave like N ℓ, with ℓ ≤ n. For each n, the

term (∑

ij Jij)n = (2a)n becomes dominant for N sufficiently large. Combining this

result with the asymptotic form of Eqs. (41) we get

Mn ∼ J n X[pr][2n]a

n . (44)

This result is quite general and may be proved by induction. Then, for N → ∞non-centered moments of the spectral density distribution become

Mn ∼ [2JN(z +1

4)]n = Mn

1 . (45)

9

Table 2: Exact ground (first entry) and first excited (second entry) state energies (Eexact) of a linearchain with J = ±1, and errors of the energies derived from two- and three-moment polynomials(δP2 and δP3) and from two-, three-, four-, and n-moment Gram-Charlier (δG2, δG3, δG4, δGn)spectral densities. The values of n are displayed in the last column. The errors are defined asdifferences between the exact and the moment generated energies.

J N 2S + 1 Eexact δP2 δP3 δG2 δG3 δG4 δGn n1 10 1 –8.39 0.12 –0.36 1.00 0.50 0.36 0.11 9

–7.70 0.19 –0.17 0.40 0.14 0.14 0.01 93 –8.55 0.34 –0.13 1.67 1.08 0.76 0.46 9

–8.33 0.16 –0.23 0.79 0.43 0.33 0.11 95 –8.68 0.38 0.05 1.46 1.09 0.74 0.45 9

–8.43 0.23 –0.04 0.70 0.47 0.37 0.16 111 11 2 –9.51 0.18 –0.37 1.90 1.11 0.75 0.36 11

–9.34 –0.01 –0.49 0.94 0.43 0.28 0.01 114 –9.63 0.27 –0.20 2.01 1.33 0.85 0.44 11

–9.44 0.16 –0.26 1.19 0.73 0.51 0.20 116 –9.74 0.34 0.24 1.66 1.26 0.80 0.45 11

–9.53 0.22 –0.05 0.90 0.65 0.47 0.21 11–1 10 1 –4.02 –1.51 –0.72 –0.62 –0.13 –0.38 –0.13 3

–2.29 –0.40 0.05 –0.19 0.11 0.05 0.05 43 –3.36 –1.28 –0.49 0.06 0.56 0.12 0.00 10

–2.55 –0.86 –0.29 –0.24 0.14 –0.07 0.03 115 –1.40 –0.75 –0.28 0.34 0.68 0.20 0.00 7

–0.73 –0.46 –0.11 0.00 0.24 0.05 0.00 2–1 11 2 –4.26 –1.85 –0.83 –0.13 0.42 0.04 0.03 9

–3.50 –1.44 –0.66 –0.50 0.00 –0.25 0.00 34 –3.02 –1.49 –0.68 0.25 0.68 0.22 –0.01 11

–2.32 –1.08 –0.42 –0.05 0.39 0.05 0.01 96 –0.69 –0.80 0.34 0.52 0.85 0.29 0.06 10

–0.12 –0.55 –0.19 0.13 0.38 0.11 0.00 9

The coefficients Rn(i) describe specific features of the molecule. They may bedetermined from a knowledge of the molecular graph and are referred to as thetopological invariants of the molecule. The coefficients Tn(i) express the way inwhich these specific features propagate when N and S change. Therefore they arecalled propagation coefficients [3].

4. Some Numerical Results

Numerical calculations have been performed for two simple chains of atoms: alinear chain

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Table 3: The same as Table I, but for the comb-like chains.

J N 2S + 1 Eexact δP2 δP3 δG2 δG3 δG4 δGn n1 7 2 –5.17 0.15 0.27 0.30 0.38 0.33 0.09 11

–4.65 –0.17 –0.13 –0.34 –0.32 –0.16 –0.09 74 –5.45 0.43 — 0.57 0.22 0.22 –0.06 11

–4.48 0.63 — 0.47 0.38 0.40 0.33 56 –5.73 0.47 — 0.38 0.27 0.38 0.02 10

–5.00 0.19 — 0.03 0.10 0.23 0.03 21 10 1 –8.11 0.10 –0.08 0.93 0.77 0.44 0.04 8

–7.08 0.54 0.41 0.74 0.65 0.63 0.42 73 –8.36 0.32 –0.05 1.60 1.65 0.77 0.33 11

–8.15 0.15 –0.16 0.75 0.48 0.35 0.07 115 –8.54 0.51 — 1.59 0.89 0.76 0.29 11

–8.31 0.35 — 0.82 0.41 0.38 0.10 11–1 7 2 –1.55 0.34 0.23 0.50 0.42 0.41 0.09 11

–0.71 0.34 0.29 0.16 0.15 0.31 0.15 34 –1.91 –1.17 — –1.02 –0.55 –0.57 –0.25 11

0.83 0.80 — 0.64 0.77 0.79 0.64 26 1.59 –0.21 — –0.30 –0.14 –0.05 0.05 10

2.27 –0.54 — –0.70 –0.79 –0.61 –0.08 9–1 10 1 –2.33 –0.12 0.09 0.71 0.87 0.46 0.06 11

–1.54 0.08 0.22 0.28 0.37 0.32 0.15 113 –2.59 –0.71 –0.18 0.57 0.95 0.43 0.26 6

–2.18 –0.68 –0.27 –0.08 0.20 –0.03 0.00 55 –2.81 –2.16 — –1.07 –0.47 –0.68 –0.43 11

–0.29 –0.03 — 0.44 0.88 0.77 0.44 2

In both cases a = N − 1 and n(△) = 0. The other quantities needed to determinethe topological invariants for the first three moments are:

• Linear chain: β2 = 4N − 6, β3 = 8N − 14, α = 8N − 16;

• Comb: β2 = (14N − 20)/3, β3 = 12N − 18, α = 10N − 22.

Coefficients W[C] for both cases are collected in Table I. The asymptotic (N ≫ 1)expressions for the dispersion, σ2, and for the skewness, γ, of the spectral densitydistributions are:

• Linear chain:

σℓ =1

2|J |

√3N(1− y)(1− y/3) ,

γℓ = −Jσℓ

1− 2y/3

1− y/3(1− y) ,

• Comb:

σc =1

2|J |

√3N(1− y)(1− y/9) ,

γc = −Jσc

1− y2/3

1− y/9,

where y = 4S2/N2. Since 0 ≤ y ≤ 1, we have

0 ≤ σℓ ≤ σc ≤√3N/2 . (46)

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In each case the width of the spectrum monotonically increases with decreasing spinand in the low-spin limit (S ≪ N), σℓ = σc =

12|J |

√3N . Behaviour of the skewness

parameters is more interesting. If S ≪ N then γℓ ∼ γc ∼ −2sgn(J )/√3N , i.e. the

skewness approaches 0 for large N . However for large S, i.e. for y → 1, in the caseof a linear chain γℓ → 0, while in the case of a comb γc → −∞.

Calculations have been performed for J = −1 (ferromagnetic case) and forJ = 1 (antiferromagnetic case), assuming nearest neighbour approximation. Theexact energies (obtained by diagonalization of the Hamiltonian matrix) of the groundand first excited states are compared with the moment-generated ones in Tables II(linear chain) and III (comb). As one can see, adding more moments in generalimproves quality of the results. In particular, the multi-moment Gram-Charlierdensity produces in nearly all cases reasonable results. In the case of the multi-moment Gram-Charlier density, the value of n has been selected in such a way thatadding more terms in the corresponding expansion, i.e. taking larger n, does notconsiderably improve the results. The value of the “optimum” n is displayed inthe last columns. In some cases surprisingly good energies are obtained from thepolynomial density. However if the polynomial is not positive in the range of thespectrum then the corresponding polynomial density does not exist. In such casesthe corresponding entry in the Table is empty. As it is seen, in most cases themoment-generated energies give reasonably good approximation to the exact ones.We conclude that the results encourage further studies on moment generated spectra,in particular on different forms of approximations of the low-moment spectral densitydistributions.

Acknowledgements

The authors are indebted to B. G. Wybourne for numerous discussions and toD. J. Klein for his very helpful remarks. Continuous support from the Polish KBNis gratefully acknowledged.

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