Coupled-perturbed Hartree–Fock theory for quasi–one-dimensional periodic systems: Calculation of static and dynamic nonlinear optical properties of model systems

Coupled-Perturbed Hartree–FockTheory for Quasi–One-DimensionalPeriodic Systems: Calculation of Staticand Dynamic Nonlinear OpticalProperties of Model Systems

A. MARTINEZ,1 P. OTTO,1 J. LADIK1,2

1Lehrstuhl fur Theoretische Chemie, Friedrich-Alexander-Universitat, Erlangen-Nurnberg,Egerlandstr. 3, 91058, Erlangen, Germany2Laboratory of the National Foundation for Cancer Research, Egerlandstr. 3, 91058, Erlangen, Germany

Received 22 January 2002; accepted 26 April 2002

DOI 10.1002/qua.10750

ABSTRACT: An alternative method to solve the coupled-perturbed Hartree–Fock(CPHF) equations for infinite quasi–one-dimensional systems is presented. The newprocedure follows a proposal made by Langhoff, Epstein, and Karplus to obtain perturbedwavefunctions free from arbitrary phase factors in each order of perturbation. It is based onthe intermediate orthonormalization of the perturbed wavefunctions (which is differentfrom the usual one) and a corresponding selection of the Lagrangian multipliers. In thisway it is possible to incorporate the orthonormalization conditions into the set of CPHFequations. Moreover, a new, advantageous procedure to determine the derivatives of thewavefunction with respect to the quasimomentum k is presented. We report calculations ofthe dipole moment, the polarizability �, and the first hyperpolarizability � for differentpolymers (poly-HF, poly-H2O, trans-polyacetylene, polyyne, and polycarbonitrile) fordifferent frequencies. These results are extensively compared with oligomer calculations.© 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 251–268, 2003

Key words: coupled-perturbed Hartree–Fock; hyperpolarizability; nonlinear optic;polarizability; polymer

Introduction

T he interest in the prediction of nonlinear op-tical (NLO) properties of chemical systems has

grown in the last years [1–3]. Special attention is

Correspondence to: A. Martinez; e-mail: [email protected]

Contract grant sponsor: DFG.Contract grant number: 51/9-4.

International Journal of Quantum Chemistry, Vol 94, 251–268 (2003)© 2003 Wiley Periodicals, Inc.

given to polymers due to the potential applicationin electro-optical communication and computa-tional devices. Quantum chemical calculations canbe a powerful alternative to experimental work forthe explanation of structure–property relationships,which can be exploited for the design of materialswith technological applications.

Although for molecules and clusters well-estab-lished methods for the calculation of dynamic NLOproperties are known [4–6], methods for infiniteperiodic polymers (which take advantage of theperiodic symmetry) are still under development.Though some efforts have been made in the past tosolve this problem [7–10], it was only in the lastyear that a systematic solid theoretical development[11] based on the time-dependent coupled-per-turbed Hartree–Fock (CPHF) approach togetherwith numerical applications for static [12] and dy-namic [13] (hyper)polarizabilities was published byKirtman, Gu, and Bishop.

Unfortunately, the authors do not make a de-tailed comparison for the dynamic case [13], withresults obtained by means alternative to those usedin the static case [12]. Such comparisons could bevery illustrative with respect to the validity of thetheory and the correctness of the computer pro-grams because essential differences exist betweenimplementations of the CPHF theory for polymersfor the cases of static and dynamic fields. In thedynamic case the symmetry properties of the ma-trices of the unperturbed case, which hold also forthe static case, are broken and new symmetry rela-tions have to be taken into account.

Parallel to Kirtman and coworkers [11–13], ourgroup is also engaged in the establishment of amethod for the solution of the CPHF equations forperiodic infinite systems employing solid-statephysical concepts [7–10]. In this article we presentan alternative way to solve the CPHF equations thatwas already successfully applied to molecules [14].We also describe a new, advantageous method forthe calculation of the derivatives of the wavefunc-tion with respect to the quasimomentum k. Wepresent calculations of static and dynamic polariz-abilities and first hyperpolarizabilities of five differ-ent polymers employing a minimal basis set andcomparisons with oligomer calculations. Finally,we report static and dynamic NLO properties ofpolycarbonitrile using a double-zeta basis set.These results are compared with corresponding oli-gomer and other polymer calculations.

Theory

Detailed description of the CPHF equations forpolymers have been previously published [10, 11].Therefore, we give only a brief outline of themethod up to second order and then present ourway of solving them. Some interesting aspects inwhich our approach differs from the previous ones[9–13] are discussed in detail.

PERTURBATION OPERATOR OF THEEXTERNAL ELECTRIC FIELD FORPOLYMERS

The one-electron Hamiltonian describing a peri-odic polymer interacting with a homogeneous elec-tric field E� (t) is given by:

HS�r�, t� � �er�E� �t� (1)

Applying this operator to the delocalized crystalorbitals �n

k� (r�, t)

�nk� �r�, t� � eik�r�un

k� �r�, t� (2)

one can show that HS(r�, t)�nk� (r�, t) can be rewritten as

[15–17].

�er�E� �t��nk��r�, t� � �ieE� �t�eik�r��k�e�ik�r��n

k��r�, t�

� ieE� �t��k��nk��r�, t� (3)

The first part does not cause a change in k�; there-fore, it is periodic in direct space (q-space). Theoperator is responsible for the polarization due tothe mixing of bands with the same value of k�. Thesecond term is unbounded in q-space, causing achange of the quasimomentum k� within a band andconsequently a polarization current. For insulatingsystems with completely filled bands and a largegap, such a current cannot occur and therefore onecan omit this term in the operator [10, 15–17].

Finally, the periodic perturbation operator HE(k�,r�, t), describing the interaction of a polymer with anexternal electric field, has the form:

HE�k� , r�, t� � HS�k� , r�, t� � ieE� �t��k�

� �ieE� �t�eik�r��k�e�ik�r� (4)

The same operator can be obtained taking the vec-tor potential (A� ) as the perturbation [11].

MARTINEZ, OTTO, AND LADIK

252 VOL. 94, NO. 5

The quasimomentum vector k� becomes for one-dimensional systems (0, 0, kz) assuming that thepolymer axis coincides with the z-axis and will bewritten simply as k (�kz). Because we are interestedin the longitudinal components of the dynamicNLO properties, we take the external field only in zdirection:

E� �t� � �0, 0, Ez�e�i�t � e�i�t�� (5)

CPHF EQUATIONS FOR POLYMERS

The starting point for the CPHF method in theCrystal Orbital Approach is the time-dependent HFequation:

� F�k, r�, t� � i�

�t��nk�r�, t� � �

l

nl�k, t��lk�r�, t�;

(6)

�nk(r�, t) is the time-dependent Bloch function. In this

equation and in the following ones, the indices nand l run over the doubly occupied bands. Theoperator F(k, r�, t) contains the time-independentunperturbed Fock operator, the perturbation oper-ator HE(k, r�, t), and the time-dependent contributionto the Fock operator due to the time dependence ofthe crystal orbitals. Using perturbation theory F(k, r�,t), �n

k(r�, t), and nl(k, t) are expanded in a powerseries of the perturbation parameter :

F�k, r�, t� � F�0��k, r�� F�1��k, r�, t�

� 2F�2��k, r�, t� � . . . (7a)

�nk�r�, t� � �n

k�0��r�� nk�1��r�, t� � 2�n

k�2��r�, t� . . .

(7b)

nl�k, t� � nl�0��k� � nl

�1��k, t� � 2nl�2��k, t� � . . .

(7c)

The perturbed wavefunction satisfies the “interme-diate” orthonormalization condition, as suggestedby Langhoff et al. [18]

��nk�0��r��l

k�r�, t�� nk�0��r��l

k�0��r�� nl (8)

instead of the usual orthonormalization condition[6, 12],

��nk�r�, t��l

k�r�, t�� nk�0��r��l

k�0��r�� nl (8a)

We denote by �nk(r�, t) the perturbed wavefunction,

which satisfies Eq. (8a) (as obtained in Refs. 11, 12,and 13, to distinguish it from our perturbed wave-function �n

k(r�, t). The choice of this orthonormaliza-tion condition, (Eq. [8]) instead of the usual one (Eq.[8a]), ensures that the wavefunction of each pertur-bation order will be free of additional phase factors.

Collecting terms of the same order of (for sim-plicity we take nl

(0)(k) � 0 for n l, in the solutionof the unperturbed case, which can be achieved bymeans of an unitary transformation of the equation)we obtain the following integro-differential equa-tions:

Zeroth order:

F�0��k, r�� n�0��k��n

k�0��r�� 0 (9a)

First order:

F�0��k, r�� n�0��k� � ��n

k�1��r��

� hE�1��k, r�� 1��k, r��n

k�0��r��

� �l

nl�1��k��l

k�0��r�� 0 (9b)

Second order:

F�0��k, r�� n�0��k� � 2��n

k�2��r�� hE�1��k, r��

� �1��k, r��nk�1��r��

l


k�1��r��

� �2��k, r��nk�0��r��

l


k�0��r�� 0

(9c)

F�0��k, r�� n�0��k��n

k�2�0�r�� hE�1��k, r��

� �1��k, r��nk�1��r��

l


k�1��r��

� hE�1��k, r�� 1��k, r��n

k�1��r��

� �l


k�1��r�� 2�0�k, r��nk�0��r��

� �l

nl�2�0�k��l

k�0��r�� 0 (9d)

For the time-dependent wavefunctions of higherorders the usual ansatz [18] is used:

QUASI–ONE-DIMENSIONAL PERIODIC SYSTEMS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 253

�nk�1��r�, t� � �n

k�1��r��ei�t � �nk�1��r��e�i�t (10a)

�nk�2��r�, t� � �n

k�2��r��ei2�t � �nk�2��r��e�i2�t � �n

k�2�0�r��

(10b)

and the operator F(k, r�, t) is partitioned in the fol-lowing form:

F�0��k, r�� 12 �1

2 � �A

ZA

r1A

� �l

��lk�0��r��1 �

12 Pn7l

rnl��l

k�0��r�� (11a)

F�1��k, r�, t� � HE�1��k, r�, t� � �1��k, r��ei�t

� �1��k, r��e�i�t (11b)

F�2��k, r�, t� � �2��k, r��ei2�t � �2��k, r��e�i2�t

� �2�0�k, r�� (11c)

Here F(0)(k, r�) is the unperturbed Fock operator andPn7l is the permutation operator for the coordinatesof electrons n and l, respectively. The operators are the Fock operators of higher order:

�1��k, r�� l

��lk�0��r��1 �

12 Pn7l

rnl��l

k�1��r�� l

k�1� �r��1 �12 Pn7l

rnl��l

k�0��r�� (12a)

�2��k, r�� l

��lk�0��r��1 �

12 Pn7l

rnl��l

k�2��r�� l

k�2� �r��1 �12 Pn7l

rnl��l

k�0��r�� l

k�1� �r��1 �12 Pn7l

rnl��l

k�1��r��(12b)

�2�0�k, r�� l

��lk�0��r��1 �

12 Pn7l

rnl��l

k�2�0�r�� l

k�2�0�r��1 �12 Pn7l

rnl��l

k�0��r�� l

k�1��r��1 �12 Pn7l

rnl��l

k�1��r�� l

k�1��r��1 �12 Pn7l

rnl��l

k�1��r��(12c)

The time-independent operator hE(1)(k, r�) is defined

by combining Eqs. (4) and (5):

HE�1��k, r�, t� � hE

�1��k, r��e�i�t � e�i�t� (13)

The index (1) of hE(1)(k, r�) is the perturbation order.

The definition of the matrix elements of this oper-ator will be discussed in details under The MatrixElements of hE

(1)(k, r�).The set of Eqs. (9) are the CPHF equations in the

crystal orbital approach.We use the intermediate orthonormalization

condition [Eq. (8)] instead of the usual one [Eq. (8a);see also Eq. (14a–c) of [12]]. Expanding Eq. (8) weobtain for all n and l:

Zeroth order:

��nk�0��l

k�0�� nl (14a)

First order:

��nk�0��l

k�1�� 0 (14b)

Second order:

��nk�0��l

k�2�� 0 (14c)

��nk�0��l

k�2�0� � 0 (14d)

Notice that in the orthonormalization equation of agiven order no perturbation contribution of lowerorder of the wavefunction occurs. If the usual or-thonormalization conditions are used (see Ref. 12)


254 VOL. 94, NO. 5

extra terms appear on the left side of Eqs. (14b),(14c), and (14d) of the form

��nk�1��l

k�0��, ��nk�2��l

k�0�� nk�1��l

k�1� �,

and

��nk�2�0��l

k�0�� nk�1��l

k�1�� nk�1� ��l

k�1� �,

respectively.The orthonormalization conditions (Eq. 14) can

be incorporated into the CPHF equations by meansof the Lagrangian multipliers [14, 18]. The solutionof the CPHF equations found with this orthonor-malization conditions are free of undesirable phasefactors. This point will be clarified later.

SOLUTION OF CPHF EQUATIONS

The Lagrangian multipliers are obtained by multi-plying Eq. (9a–d) on the left by �l

k(0)(r�) and integrat-ing over the whole space, using Eqs. (14a–f):

n0�k� � ��n

k�0��r��F�0��k, r��nk�0��r�� (15a)

nl�1��k� � ��l

k�0��r��hE�1��k, r�� 1��k, r��n

k�0��r��

(15b)

nl�2��k� � ��l

k�0��r��hE�1��k, r�� 1��k, r��n

k�1��r��

� ��lk�0��r�� 2��k, r��n

k�0��r�� (15c)

nl�2�0�k� � ��l

k�0��r��hE�1��k, r�� 1��k, r��n

k�1��r��

� ��lk�0��r��hE

�1��k, r�� 1��k, r��nk�1��r��

� ��lk�0��r�� 2�0�k, r��n

k�0��r�� (15d)

For the solution of the CPHF equations the crystalorbitals are expressed as linear combinations ofatomic orbitals (LCAO) in each perturbation order �(� � (0), (1)�, (2)�, and (2)0):

�nk��r�� 2N � 1��1/ 2 �

q��N

N

eikaq � �1

m

C nk� �

q�r�� (16)

Here, N is the number of neighboring cells whoseinteractions with the reference cell are taken explic-itly into account, �

q(r�) denotes the atomic orbital�

q(r� � aqe�z � r�p) located in cell q at position r�p, m isthe number of atomic orbitals in the elementarycell, and a the translation length.

Equation (16) is substituted into the CPHF equa-tions and the resulting equations are multipliedfrom the left with ��

0 (r�) and integrated over thespace, to obtain the LCAO form of the CPHF equa-tions.

Equation (9a), the zeroth order one, is solved inthe standard way, with the help of the self-consis-tent–field method. The equation of order � is solvedusing the results of previous orders to calculate theinitial values of � and �, finding the coefficientsC n

k� iteratively until self-consistency in the determi-nation of �, �, and C n

k� is reached. The coefficientsare the solution of a system of equations of the formA n(k)Cn(k) � �Bn(k) for each occupied band n andeach k point in each iteration. The matrix An(k)contains the factor that appears in the first term ofequations (9b–d) multiplying �n

k�. The vector Bn(k)contains all the other terms of Eqs. (9b–d) begin-ning with the second one.

The solution of the CPHF equations fulfills onlythe intermediate orthonormalization conditions.We have to transform them in each iteration step tothe exact solutions that fulfill the general orthonor-malization equations [Eq. (8a)]. The transforma-tions are [14, 18]:

�nk�0��r�� n

k�0��r�� (17a)

�nk�1��r�� n

k�1��r�� (17b)

�nk�2� �r�� n

k�2� �r��

�12 �

l

��lk�1��r��n

k�1� �r��lk�0��r�� (17c)

�nk�2�0�r�� n

k�2�0�r�� 12 �

l

��lk�1��r��n

k�1��r��

� ��lk�1��r��n

k�1��r��lk�0��r�� (17d)

The corrected-perturbed wavefunctions of Eq. (17)are to be used for the determination of � and �

[Eqs. (12) and Eq. (15)] in each iteration step. To beexact, one has to rewrite Eq. (12) and Eq. (15) put-ting �n

k�(r�) in place of �nk�(r�).

One should emphasize that in the general case,the �n

k(r�), obtained using the orthonormalizationconditions of Eq. (14) in the perturbation procedurefollowed by the transformations of Eq. (17), are notnecessarily identical to the �n

k(r�), which are obtainedusing the orthonormalization conditions of Eq. (8a)[Eq. (14a–c) in Ref. 12] directly in the perturbation



procedure. Contrary to the second one, our �nk(r�) are

free from arbitrary phase factors, which can appearif one uses the other method of solution. This pointis discussed in detail by Langhoff et al. in a veryillustrative review [18]. The authors showed thatthe solutions of higher order obtained using di-rectly Eq. (8a) are arbitrary with respect to theaddition of imaginary time-independent multiplesof the unperturbed wavefunction and, conse-quently, these solutions differ from the solutionsobtained with Eq. (8) by a time-independent phasefactor.

If one is interested in a quantity of the perturbedwavefunction of any order, for example, the deriv-atives of the wavefunction with respect to thequasimomentum k, it is very advantageous to ob-

tain a wavefunction without arbitrary phase fac-tors. Depending on the method used, the presenceof arbitrary phase factors could induce errors in thecalculation of the derivatives. This is very impor-tant because these derivatives appear both in theperturbation equations and in the calculation of theexpectation values of the NLO properties.

MATRIX ELEMENTS OF HE(1)(K, R� ): NEW

WAY TO CALCULATE THE DERIVATIVES OFTHE WAVEFUNCTION WITH RESPECT TO K

The matrix elements of hE(1)(k, r�) are obtained by

acting on the crystal orbital of each perturbationorder �n

k�(r�) in LCAO form and multiplying fromthe left with the atomic orbital ��

0 (r�) and integratingover the whole space:

hE� �1� �k�C n

k� � ��0 �r��hE

�1��k, r��nk��r��

0 �r�� ieEzeikz�

�k e�ikz��nk��r��

� ��0 �r�� ieEzeikz

�

�k e�ikz� �q��N

N

eikaq � �1

m

C nk� �

q�r�� 0 �r�� eEzz� �

q��N

N

eikaq � �1

m

C nk� �

q�r��

� ��0 �r��eEzaq� �

q��N

N

eikaq � �1

m

C nk� �

q�r�� 0 �r�� ieEz� �

q��N

N

eikaq � �1

m�

�k C nk� �

q�r�� eEz��D�

z �k�C nk� � iS�� k�C n

k� � iS� �k�C� n��k��; C� n

��k� ��

�k C nk� (18)

D� z (k) and S�� (k) are matrix elements of the dipole

matrix in z direction in k space and of the matrix ofthe derivatives of the overlap matrix S(k) with re-spect to k, respectively:

D� z �k� � �

q��N

N

eikaqD� z0q

� �q��N

N

eikaq��0 �r��z��

q�r��

(19)

S�� k� ��

�k S� �k� ��

�k �q��N

N

eikaqS� 0q

� �q��N

N

iaqeikaqS� 0q (20)

To find the derivatives of the wavefunction withrespect to k usually one uses Pople’s analyticalmethod (calculation of the derivatives of the unper-turbed CPHF equations and the orthonormalizationcondition with respect to k, followed by complexarithmetic operations). For more details of thismethod see Refs. [10] and [11]. Alternatively, onecan use numerical derivatives, if one takes a suffi-cient large number k points in the Brillouin zone forthe calculation.

We propose a simple and accurate method todetermine the derivatives with respect to k of thewavefunctions of any perturbation order.

In the development of the crystal orbital methodthe matrix of coefficients in k space C(k) can berelated with a correspondent matrix C(q) in q-space(see, e.g., Chapter 1 of Ladik [19]). For these matri-


256 VOL. 94, NO. 5

ces the same relations hold as for the pairs of ma-trices F(k)-F(q) and S(k)-S(q), respectively:

C nk� � �

q��N

N

eikaqC nq� ; C n

q � �k��Nkp

Nkp

e�ikaqC nk� (21)

Here Nkp is the number o k points in the half of theBrillouin zone. From these relations it is very easyto determine the derivatives of C(k) in the samemanner as for the overlap matrix S(k) [Eq. (20)]:

C� nk� �

�

�k �q��N

N

eikaqC nq� � �

q��N

N

iaqeikaqC nq�

� �q��N

N

iaqeikaq �k��Nkp

Nkp

e�ik�aqC nk�� (22)

This way is very advantageous in comparison withthe methods available until now. It is simple and ofgeneral validity for each perturbation order. Onecan obtain all the elements of the derivative matrix(some difficulties exist to obtain the diagonal ele-ments of the first derivative matrix employing Po-ple’s method [12]). It is also very accurate and su-perior to the numerical derivative method.

The only difficulty with respect to its applicabil-ity is the necessity of “well-behaved” wavefunc-tions in k space (this is also true for the numericalderivatives). This means that the phase factors ofthe wavefunction for each k point may not be arbi-trary to prevent a discontinuity on the wavefunc-tion. This would affect the derivatives because inthese methods the values of the coefficients fordifferent k points are used for their determination.The phase angle �n

�(k) for each band n must be acontinuous function of k, and in particular it mustbe valid that �n

�(��/a) � �n�(�/a). This additional

condition for a correct transformation of the matrixin k space into q space and reverse are not requiredin the case of Fock, overlap, and density matrices,because they have “mean value” character (theycan be considered as expectation values of physicalproperties and phase factors cancel in their calcu-lation, at least in the zeroth-order equations).

In the case of the determination of the deriva-tives using Pople’s method the phase factors areincorporated as a factor in the derivatives and theyresult as an overall multiplicative term in the CPHFequations. In this sense Pople’s derivatives are de-pendent on the phase factors, but this dependence

ensures that the derivatives correspond to thewavefunctions that determine them.

To ensure the continuity of �n(0)(k), we enforce the

phase factors of �nk(0)(r�) such that the sum of the

squares of the imaginary parts of the coefficients isminimal for each k point. The wavefunctions ofhigher perturbation orders cannot be corrected aposteriori because to introduce new phase factorswould change the calculated mean values of phys-ical properties. Langhoff et al. [18] have shown thatthe application of the ordinary orthonormalizationdoes not guarantee that the perturbed wavefunc-tion would be free from arbitrary phase factors.However, the use of the intermediate orthonormal-ization conditions [Eq. (14)], which we are employ-ing, prevents this problem. Therefore our perturbedwavefunctions of each order are free from arbitraryphase factors and we can apply Eq. (22) to find thederivatives of the wavefunction.

FREQUENCY-DEPENDENT(HYPER)POLARIZABILITIES

We can obtain the (hyper)polarizabilities fromthe perturbed wavefunction. We use a dipole mo-ment operator for the polymer cell in k space, de-fined in analogy with the molecular case:

D�z�k� �

hE�1��k�

eEz(23)

With this operator we obtain for the dipole mo-ment:

�z �a

2� ��/a

�/a �n,l

��nk�0�� D�

z�k��lk�0��dk (24a)

For the dynamic polarizability:

�zz�� a

2� ��/a

�/a �n,l

��nk�0�� D�

z�k��lk�1��

� ��lk�1� � D�

z�k��nk�0��dk (24b)

For the dynamic first hyperpolarizabilities:

�zzz��2�; ��, �� a

2� ��/a

�/a �n,l

��nk�0�� D�

z�k��lk�2��



� ��lk�2� � D�

z�k��nk�0��

� ��nk�1� � D�

z�k��lk�1��dk

(24c)

�zzz�0; �, �� a

2� ��/a

�/a �n,l

��nk�0�� D�

z�k��lk�2�0�

� ��lk�2�0� D�

z�k��nk�0��

� ��nk�1�� D�

z�k��lk�1��

� ��nk�1�� D�

z�k��lk�1��dk

(24d)

Expanding the crystal orbital in a linear combina-tion of atomic orbitals (LCAO) the quantities takethe form:

Dipole moment:

�z � �q��N

N �,�

��D�z0q

� iS��0q�P�

0q�0� � iS�0q P�

0q�0��

(25a)

Dynamic polarizability:

�zz�� q��N

N �,�

��D�z0q

� iS��0q�P�

0q�1�� iS�

0q P�0q�1��

�

(25b)

Dynamic first hyperpolarizabilities:

�zzz��2�; ��, ��

� �q��N

N �,�

��D�z0q

� iS��0q�P�

0q�2�� iS�

0q P�0q�2��

� (25c)

�zzz�0; �, ��

� �q��N

N �,�

��D�z0q

� iS��0q�P�

0q�2�0� iS�

0q P�0q�2�0

� (25d)

The density matrices are defined as:

P�0q�0�

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�o��k�C�i

�0�*�k��e�ikqadk

(26a)

P�0q�1��

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�1��k�C�i

�0�*�k�

� Ci�o��k�C�i

�1� *�k��e�ikqadk (26b)

P�0q�2��

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�2��k�C�i

�0�*�k�


�2� *�k� � Ci�1��k�C�i

�1� *�k��e�ikqadk

(26c)

P�0q�2�0

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�2�0�k�C�i

�0�*�k�


�2�0*�k� � Ci�1��k�C�i

�1��*�k�

� Ci�1��k�C�i

�1��*�k��e�ikqadk (26d)

C nk� are the coefficients of the corrected perturbed

wavefunctions of Eq. (17).The matrices P0q�, are similar to the density ma-

trices but combine products of the coefficients andthe derivatives of the coefficients with respect to k.

P�0q�0�

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�o��k�C��i

�0�*�k��e�ikqadk

(27a)

P�0q�1��

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�1��k�C��i

�0�*�k�

� Ci�o��k�C��i

�1� *�k��e�ikqadk (27b)


258 VOL. 94, NO. 5

P�0q�2��

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�2��k�C��i

�0�*�k�


�2� *�k� � Ci�1��k�C��i

�1� *�k��e�ikqadk

(27c)

P�0q�2�0

�a

2� ��/a

�/a �2 �i�1

n/ 2

Ci�2�0�k�C��i

�0�*�k�


�2�0*�k� � Ci�1��k�C��i

�1��*�k�

� Ci�1��k�C��i

�1��*�k��e�ikqadk (27d)

Computational Details and Results

We have calculated the longitudinal dipole mo-ment � per unit cell, and static and dynamic longi-tudinal � and � per unit cell, for poly-HF, poly-H2O, trans-polyacetylene (PA), polyyne (PY), andtwo configurations of polycarbonitrile (PCN-A andPCN-B). The atomic coordinates of the elementarycell and the translation vector a of the systems aregiven in Table I. Some geometries are taken fromthe literature [12] to enable the comparison. Cle-menti’s minimal basis (MB) set [20] was used in allcalculations except for PCN-B, where, additionally,calculations with Clementi’s double-zeta basis set(DZ) [21] were done.

The crystal orbital calculations have been per-formed in the ninth (N � 9) and 15th (N � 15)

TABLE I ______________________________________________________________________________________________Atomic coordinates and translation vector a (in Å) of poly-HF, poly-H2O, PA, PY, and two configuration ofpolycarbonitrile (PCN-A and PCN-B).

X Y Z

Poly-HFa � 2.82F 0.0 0.0 0.0H 0.0 0.0 �0.90poly-H2Oa � 2.72O 0.0 0.0 0.0H 0.0 0.0 �0.96H �0.9294217 0.0 0.2403648PAa � 2.46C 0.0 0.0 0.0H �1.076 0.0 0.0C 0.6533929 0.0 1.1595856H 1.7293929 0.0 1.1595856PYa � 2.5685C 0.0 0.0 0.0C 0.0 0.0 1.2015PCN-Aa � 2.3065C 0.0 0.0 0.0N 0.0 0.6627 1.096H 0.0 �1.0880 0.0PCN-Ba � 2.306484092C 0.0 0.0 0.0N 0.0 0.661897045 1.095578522H 0.0 �1.087999755 0.000729158



neighbors interaction approximation in the strictsense (all the integrals are cut off at N cells). Weused 121 k points (Nkp � 121) in the half of theBrillouin zone (Bishop et al. [12] have shown thatfor those systems Nkp � 91 is sufficient to reachconvergence according to the number of k points).No long-range Coulomb interactions (LRCI) wereincluded.

For the calculation of the dipole moments nospecial procedure is necessary for the determina-tion of the diagonal elements of the derivative ma-trix (in contrast to Ref. [12]), because we obtain allelements with the procedure explained under Ma-trix Elements of hE

(1)(k, r�). In this way our newapproach to calculate the derivatives of the wave-function of any order with respect to k is more

TABLE II ______________________________________________________________________________________________Comparison of oligomer and polymer calculations for poly-HF with Clementi’s minimal basis set.a

Oligomern �� (0; 0)

� � 0.0656��(��; �) ��(0; 0, 0)

� � 0.0656��(0; ��, �)

� � 0.0656��(�2�; �, �)

26 �1.00418 4.412 4.454 3.670 3.728 3.85027 �1.00418 4.412 4.454 3.669 3.727 3.84928 �1.00419 4.412 4.454 3.668 3.727 3.84829 �1.00419 4.412 4.454 3.668 3.726 3.84830 �1.00420 4.412 4.454 3.667 3.726 3.847� (polyn) �1.00427b 4.412 4.454 3.660 3.718 3.843� (Pade) 4.412 4.454 3.660 3.719 3.838

PolymerN � �(0; 0) �(��; �) �(0; 0, 0) �(0; ��, �) �(�2�; �, �)

9 �1.00408 4.414 4.456 3.626 3.683 3.803Ref. 12c 9 �1.004 4.414 3.62615 �1.00429 4.413 4.455 3.649 3.707 3.828

a All results are in a.u.b The dipole moment was fitted to a convergent geometrical series.c Ninety-one k points were used in this calculation.

TABLE III _____________________________________________________________________________________________Comparison of oligomer and polymer calculations for poly-H2O with Clementi’s MB set.a


� � 0.0656��(��; �) ��(0; 0, 0)

� � 0.0656��(0; ��, �)

� � 0.0656��(�2�; �, �)

26 �0.87349 6.946 7.027 2.934 3.063 3.34327 �0.87350 6.946 7.027 2.932 3.061 3.34128 �0.87351 6.946 7.027 2.930 3.059 3.33929 �0.87339 6.947 7.028 2.928 3.057 3.33730 �0.87353 6.947 7.027 2.927 3.056 3.336� (polyn) �0.87355b 6.949 7.028 2.887 3.010 3.303� (Pade) 6.949 7.029 2.905 3.003 3.312

PolymerN � �(0; 0) �(��; �) �(0; 0, 0) �(0; ��, �) �(�2�; �, �)

9 �0.87310 6.951 7.031 2.794 2.920 3.193Ref. 12c 9 �0.873 6.951 2.79415 �0.87345 6.949 7.029 2.867 2.995 3.271

a All results are in a.u.b The dipole moment was fitted to a convergent geometrical series.c Ninety-one k points were used in this calculation.


260 VOL. 94, NO. 5

consistent and compact. The dipole moments canbe determined only up to modulus a, as shown inRef. [12]. The corrected values are reported in ourtables.

The dynamic NLO properties were calculated fora frequency of 0.0656 a.u. (wave length � 6843 Å)except for PCN-A and PCN-B where a series ofcalculations for different frequencies have been per-formed.

Additionally we have carried out static and dy-namic calculations for oligomers with a length be-tween 25 and 30 units (except for PCN-B, DZ basis,with lengths between 23 and 28 units) using theprogram GAMESS [22]. The reported values are thedifferences of �, �, or � between two oligomerswhich differ by one unit (� values). For the deter-mination of the extrapolated values of the oligomercalculations we have fitted the average values of �or � (calculated value for an oligomer divided bythe number of units of the oligomer) to an inversepolynomial of the number of cells as suggested byKirtman [23]. Dalskov et al. [24] showed that thisextrapolation method for the average values givesreliable results. We have also fitted the averagevalues to a Pade approximant following the proce-dure described in Ref. [24] to have an independentcriteria of the quality of the extrapolation. The av-erage dipole moments � are almost converged. Tofind the extrapolated values we have fitted thesmall variations of � to a convergent geometricalseries.

We take as reference for the comparison the in-verse polynomial extrapolated values [�(polyn)],although we have no criteria to affirm that this isbetter than the Pade approximant values [�(Pade)].However, in most cases, both values are very close,and we conclude that we have a good referencepoint to compare our approach. The greatest differ-ence between the two extrapolation methods(�zzz(0; ��, �), � � 0.0656, polycarbontrile-A) is0.7% of the extrapolated value. We also cannotprove the superiority of the oligomer method overthe polymer approach, but it is expected that bothwill converge to a common result. Consequently, inthis work, the more neighbors are included for thecrystal orbital calculation, the nearer are the valuesof the polymer method and the extrapolated limitof the oligomer series to each other. For simplicitywe refer to �z, �zz(��; �), �zzz(0; ��, �), and�zzz(�2�; �, �) as �, �(��; �), �(0; ��, �), and�(�2�; �, �), respectively. The most important re-sults are shown in Tables II–X.

For the model systems poly-HF and poly-H2O(Tables II and III), the calculated values for N � 15are very close to the extrapolated oligomer values.The calculated dipole moments agree with the oli-gomer values. The differences for � are 0.02% and0.01% of the extrapolated values of oligomers forpoly-HF and poly-H2O, respectively. The differ-ences for � are below 0.7% and 1.0% for poly-HFand poly-H2O, respectively. The static calculationswith N � 9 reproduce the results of Ref. [12].

TABLE IV _____________________________________Comparison of oligomer and polymer calculationsfor trans-PA with Clementi’s MB set.a

Oligomern ��(0; 0)

� � 0.0656��(��; �)

26 116.71 150.8127 116.76 150.9128 116.80 150.9929 116.84 151.0630 116.87 151.13� (polyn) 117.23 151.81� (Pade) 117.26 151.96

PolymerN �(0; 0) �(��; �)

9 118.42 152.4815 118.16 152.05

a All results are in a.u.

TABLE V ______________________________________Comparison of oligomer and polymer calculationsfor polyyne with Clementi’s MB set.a

Oligomern ��(0; 0)

� � 0.0656��(��; �)

26 119.71 135.5327 119.77 135.6028 119.82 135.6729 119.86 135.7330 119.89 135.77� (polyn) 120.30 136.25� (Pade) 120.38 136.39

PolymerN �(0; 0) �(��; �)

9 122.77 137.6815 120.64 136.77




For the �-conjugated systems PA and PY and theMB basis set (Tables IV and V) the agreement of thepolymer � values with the oligomer extrapolationsare very good. The differences are below 0.9% forPA and 0.4% for PY for N � 15. The predictedenhancement of �(��; �) for � � 0.0656 a.u. (whichis more significant as in poly-HF and poly-H2O) isconsistent with the oligomer calculations.

The results for PCN-A and PCN-B with the MBbasis set (Tables VI–IX) are qualitatively the mostinteresting ones, because they are extended for a

wide range of frequencies. The calculations withN � 9 are already close to the oligomer values. Theimprovement from 9 to 15 neighbors measuresfrom 0.4% to 0.2% for the differences of the � be-tween oligomer and polymer calculations and from3% to 1% for the �.

In Figure 1 are plotted the values of �(��; �) andin Figure 2 the values �(0; ��, �) and �(�2�; �, �)with increasing � for PCN-A. The typical disper-sion of the three magnitudes is shown. In the case of�(��; �) there is good agreement between the

TABLE VI _____________________________________________________________________________________________Comparison of oligomer and polymer calculations for � and �zz(��; �) of PCN-A with Clementi’s MB set.a


� � 0.015��(��; �)

� � 0.030��(��; �)

� � 0.045��(��; �)

� � 0.0656��(��; �)

26 0.13089 107.45 108.67 112.52 119.65 137.6227 0.13101 107.53 108.75 112.61 119.75 137.7628 0.13112 107.59 108.81 112.68 119.83 137.8829 0.13121 107.64 108.87 112.74 119.91 138.0030 0.13128 107.69 108.92 112.79 119.97 138.09� (polyn) 0.13182b 108.16 109.39 113.31 120.55 138.89� (Pade) 108.15 109.40 113.28 120.53 138.86

PolymerN � �(0; 0) �(��; �) �(��; �) �(��; �) �(��; �)

9 0.13273c 108.56 109.82 113.79 121.09 139.6715 0.13260c 108.36 109.60 113.54 120.82 139.28

a All results are in a.u.b The dipole moment was fitted to a convergent geometrical series.c Five hundred one k points were used in this calculation.

TABLE VII ____________________________________________________________________________________________Comparison of oligomer and polymer calculations for �zzz(0; ��, �) and �zzz(�2�; �, �) of PCN-A withClementi’s MB set.a

Oligomern ��(0; 0, 0)

� � 0.015��(0; ��, �)

� � 0.015��(�2�; �, �) � � 0.030 � � 0.030 � � 0.045 � � 0.045

� � 0.0656��(0; ��, �)

26 1393.80 1424.63 1489.74 2252.5827 1403.15 1434.31 1500.11 2272.5028 1411.11 1442.55 1508.96 2289.5329 1417.92 1449.60 1516.52 2304.1430 1423.78 1455.66 1523.02 2316.71� (polyn) 1468.63 1501.12 1571.93 2411.51� (Pade) 1450.89 1492.16 1562.41 2394.45

PolymerN �(0; 0, 0) �(0; ��, �) �(�2�; �, �) �(0; ��, �) �(�2�; �, �) �(0; ��, �) �(�2�; �, �) �(0; ��, �)

9 1505.61 1540.16 1611.85 1651.96 2025.01 1869.81 3257.45 2483.3015 1486.86 1520.87 1592.82 1630.94 1997.95 1845.30 3208.37 2448.55



262 VOL. 94, NO. 5

polymer and oligomer values. In the case of �(0;��, �) and �(�2�; �, �) we have calculated fewerpoints with the oligomer method, but they are alsoin good agreement (see Table VII). Calculations of�(��; �) for � � 0.1312 have not achieved theconvergence limit after 200 iterations for the oli-gomer or for the polymer methods. For this fre-

quency the system is near the resonance region,which can be estimated on the basis of the gap (inreality the resonance occurs with the frequency ofthe lowest excitonic level, which is in most casessubstantially smaller than the gap). The calculatedgap between the valence and conduction band ofPCN-A is of 0.377 a.u. �(�2�; �, �) has a similar

TABLE VIII ____________________________________________________________________________________________Comparison of oligomer and polymer calculations for � and �zz(��; �) of PCN-B with Clementi’s MB set.a


� � 0.015��(��; �)

� � 0.030��(��; �)

� � 0.045��(��; �) � � 0.0656

26 0.12936 106.62 107.82 111.60 118.5927 0.12948 106.69 107.89 111.68 118.6928 0.12958 106.75 107.96 111.75 118.7729 0.12967 106.81 108.01 111.81 118.8430 0.12975 106.85 108.06 111.86 118.90� (polyn) 0.13026b 107.37 108.55 112.36 119.50� (Pade) 107.50 108.62 112.53 119.52

PolymerN � �(0; 0) �(��; �) �(��; �) �(��; �) �(��; �)

9 0.13118c 107.73 108.96 112.84 120.02 138.18Ref. 12d 9 0.134 107.715 0.13105c 107.51 108.73 112.59 119.74 137.80

a All results are in a.u.b The dipole moment was fitted to a convergent geometrical series.c Five hundred one k points were used in this calculation.d Ninety-one k points were used in this calculation.

TABLE IX _____________________________________________________________________________________________Comparison of oligomer and polymer calculations for �zzz(0; ��, �) and �zzz(�2�; �, �) of PCN-B withClementi’s MB set.a

Oligomern ��(0; 0, 0)

� � 0.015��(0; ��, �)

� � 0.015��(�2�; �, �) � � 0.030 � � 0.030 � � 0.045 � � 0.045 � � 0.0656

26 1364.58 1394.47 1457.5427 1373.58 1403.77 1467.5128 1381.24 1411.70 1476.0129 1387.79 1418.48 1483.2730 1393.41 1424.30 1489.52� (polyn) 1435.95 1468.31 1536.78� (Pade) 1433.81 1464.73 1533.03

PolymerN �(0; 0, 0) �(0; ��, �) �(�2�; �, �) �(0; ��, �) �(�2�; �, �) �(0; ��, �) �(�2�; �, �) �(0; ��, �)

9 1473.33 1506.77 1577.47 1614.85 1974.79 1825.20 3155.24 2415.49Ref. 12b 9 147415 1455.04 1487.97 1557.57 1594.44 1948.58 1801.44 3108.07 2381.83

a All results are in a.u.b Ninety-one k points were used in this calculation.



behavior for a frequency of 0.0656 a.u., which isunderstandable, because this magnitude dependson 2�.

To find an exact value of the dipole moment �we have calculated this quantity with 501 k pointsfor these systems, because the convergence of �with increasing Nkp is significantly slower thanthe convergence of � and �. This effect was al-ready present in other calculations (see Table II ofRef. 12) and we have investigated it in detail forPCN-B. Figure 3 shows the convergence of � independence of N and Nkp. We see that the dipolemoment is very sensitive to the number of kpoints in the Brillouin zone and somewhat less soto the number of interacting neighbor cells. Thecalculated values with Nkp � 121 and Nkp � 191are not very satisfactory for any N, whereas forNkp � 299, the value for the seventh neighborinteraction calculation is already very close to theoligomer limit value.

Comparing the results for PCN-A and PCN-B wesee that a small increase of the bond length alter-nation (the difference of lengths between the CONbond in a cell and the bond of the N atom of the cellwith the C atom of the next cell) is sufficient forappreciable changes of the values of � and �. Ourpolymer calculations also reproduce the oligomer

results in this point. The values for the static � and� obtained in Ref. [12] for PCN-B are exactly repro-duced in our calculation with N � 9.

Finally, we have calculated NLO properties forPCN-B with DZ basis set. It is known that theaddition of polarization functions to the basiscauses only small changes in the NLO properties incase of polymers. In contrast to the previous dis-cussed calculations (used to verify the method andthe computational program) the results of Table Xhave predictive value in the limits of the HF ap-proximation.

For the more extended basis set the results forN � 9 are not as exact as for the MB basis set. Thedifferences of � and � between oligomer andpolymer calculations measure 1% and 4%, respec-tively. Going up to N � 15 the results for � aresignificantly improved. The differences betweenoligomer and polymer calculations decrease to0.2%, and for � the differences are about 1.3%.The results for both magnitudes for N � 15 arevery satisfactory and the calculation time is sub-stantially shorter in comparison with the oli-gomer procedure. With DZ basis no convergenceis achieved for � with 501 k points, but it can beseen that the values come closer to the oligomervalues with increasing N. For a better result prob-

TABLE X ______________________________________________________________________________________________Comparison of oligomer and polymer calculations for PCN-B with Clementi’s double-zeta basis set.a


� � 0.015��(��; �)

� � 0.030��(��; �) ��(0; 0, 0)

� � 0.015��(0; ��, �)

� � 0.015��(�2�; �, �) � � 0.030 � � 0.030

24 0.5391 119.40 120.84 125.39 2528.69 2592.69 2728.5525 0.5397 119.55 120.99 125.55 2562.68 2627.92 2766.4826 0.5400 119.68 121.12 125.70 2589.62 2655.91 2796.7527 0.5405 119.78 121.23 125.82 2614.83 2682.06 2824.9128 0.5408 119.88 121.33 125.93 2635.91 2703.93 2848.51� (polyn) 0.5416b 121.11 122.63 127.35 2909.91 2988.29 3155.35� (Pade) 121.17 122.67 127.41 2907.95 2982.90 3142.02

PolymerN � �(0; 0) �(��; �) �(��; �) �(0; 0, 0) �(0; ��, �) �(�2�; �, �) �(0; ��, �) �(�2�; �, �)

9 0.4118c 121.92 123.42 128.17 3020.87 3099.37 3267.12 3355.63 4239.48Ref. 13d 9 121.94 128.37 2915.4 3265.9 4198.515 0.4572c 121.41 122.89 127.60 2947.86 3024.11 3186.97 3272.93 4130.0830 0.4723c 121.18 122.66 127.35 2916.00 2991.26 3152.01 3236.85 4082.37

a All results are in a.u.b The dipole moment was fitted to a convergent geometrical series.c Five hundred one k points were used in this calculation.d One hundred twenty-nine k points and 6-31G basis set were used in this calculation. Long-range Coulomb interactions were alsoincluded.


264 VOL. 94, NO. 5

ably Nkp should be increased further before fur-ther increasing N.

Comparing our results for N � 15 with the re-ported values of Ref. [13] for PCN-B, we find thatthey are very close to each other for each magnitudeand frequency that can be compared, although theuse of a different basis set must be kept in mind.The authors used a split-valence basis set (6-31G),and LRCI were included in their calculations.

The rather good agreement of the oligomer andpolymer calculations for N � 15 points shows thatLRCI do not play an important role if one takesenough neighbors explicitly into account. These cal-culations are perfectly possible with the usual com-putational tools. We have made additional calcula-tions for PCN-B with the DZ basis for N � 30. Theresults in Table X demonstrate that one can make

very exact predictions with our program: the dif-ferences between oligomer and polymer calcula-tions are under 0.06% for � and 0.2% for �.

Conclusions

We have presented an alternative method tosolve the CPHF equations for quasi–one-dimen-sional systems using the crystal orbital formalism.The new algorithm makes use of the intermediateorthonormalization conditions as proposed byLanghoff et al. [18], to ensure that the perturbedwavefunctions are free from arbitrary phase factors.We have found a simple and accurate method todetermine the derivatives with respect to k of thewavefunctions of any perturbation order. This

FIGURE 1. Oligomer and polymer values for the longitudinal dynamic polarizability �(��; �) of polycarbonitrile-A(Clementi’s MB set) versus optical field frequency.



method is of general applicability and advanta-geous compared with the more complicated Pople’sderivatives [10, 11] and the inaccurate numericalderivatives.

We have implemented the new approach in acomputational program and we have made calcu-lations of dynamic NLO properties of differentpolymers applying different frequencies. The re-sults have been confirmed with independent oli-gomer calculations. The static results are exactly thesame as obtained in Ref. [12] for some of the sys-tems. This point confirms also the validity of ournew method to find the derivatives of the wave-function with respect to k.

Our program can go beyond N � 9 (the limitgiven in Refs. [12] and [13] for the number of cells

taken explicitly into account) and the results withN � 15 are very good for the employed basis sets.Furthermore, the agreement of our polymer calcu-lations with the oligomer results indicate that theinfluence of LRCIs for N � 15 are unimportant for� and they have only a minor effect for � (1% forMB and 1.3% for DZ). Going to N � 30 yields veryexact results.

The application of the method to polymers withlarge elementary cells will be published elsewhere.The extension of the method to third-order proper-ties is under development. We will use the “nonit-erative” form for calculations of the � [6, 13], be-cause it has been shown that it produces the sameresults as the iterative solution of the CPHF equa-tions. The estimation of the effects of electronic

FIGURE 2. Oligomer and polymer values for the longitudinal dynamic first hyperpolarizabilities �(0; �, �) and�(�2�; �, �) of polycarbonitrile-A (Clementi’s MB set) versus optical field frequency.


266 VOL. 94, NO. 5

correlation on the dynamic NLO properties is alsoone of next steps of our investigations.

ACKNOWLEDGMENTS

The authors are very grateful to the DFG (Deut-sche Forschungsgemeinschaft, Project To 51/9-4)for financial support.

References

1. Bishop, D. M.; Norman, P. In Nalwa, H. S., Ed. Handbook ofadvanced electronic and photonic materials, vol. 9; Aca-demic: San Diego, 2000, p. 1.

2. Champagne, B.; Kirtman, B. In Nalwa, H. S., Ed. Handbookof advanced electronic and photonic materials, vol. 9; Aca-demic: San Diego, 2000, p. 63.

3. Electronic and nonlinear optical materials: theory and mod-eling, special issue. J Phys Chem 2000, 104, 4671.

4. Sekino, H.; Bartlett, R. J. J Chem Phys 1986, 85, 976.

5. Rice, J. E.; Amos, R. D.; Colwell, S. M.; Handy, N. C.; Sanz,J. J Chem Phys 1990, 93, 8828.

6. Karna, S. P.; Dupuis, M. J Comp Chem 1991, 12, 487.

7. Ladik, J. J Mol Struct Theochem 1990, 206, 39.

8. Ladik, J.; Dalton, L. R. J Mol Struct Theochem 1991, 231, 77.

9. Ladik, J. Non-linear optical materials: theory and modeling,ACS 628, Washington, 1996, Chap. 10.

10. Otto, P.; Gu, F. L.; Ladik, J. J Chem Phys 1999, 110, 2717.

11. Kirtman, B.; Gu, F. L.; Bishop, D. M. J Chem Phys 2000, 113,1294.

12. Bishop, D. M.; Gu, F. L.; Kirtman, B. J Chem Phys 2001, 114,7633.

13. Gu, F. L.; Bishop, D. M.; Kirtman, B. J Chem Phys 2001, 115,10548.

FIGURE 3. Dependence of the dipole moment of polycarbonitrile-B (Clementi’s MB set) on the number of neighbor-ing cells (N) and the number of k points in the half of the Brillouin zone (Nkp).



14. Martinez, A.; Czaja, A.; Otto, P.; Ladik, J. Mol Struct Theo-chem 2002, 349, 589.

15. Wannier, G. Phys Rev 1960, 117, 432.16. Kittel, C. Quantum theory of solids. John Wiley & Sons: New

York, 1963.17. Otto, P. Phys Rev 1992, B45, 10786.18. Langhoff, P. W.; Epstein, S. T.; Karplus, M. Rev Mod Phys

1972, 44, 602.19. Ladik, J. In Quantum theory of polymers as solids; Plenum

Press: New York, 1988.

20. Gianolio, L.; Pavani, R.; Clementi, E. Gazz Chim Ital 1978,108, 181.

21. Gianolio, L.; Clementi, E. Gazz Chim Ital 1980, 110, 179.

22. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.;Gordon, M. S.; Jensen, J. J.; Koseki, S.; Matsunaga, N.;Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgom-ery, J. A. J Comput Chem 1993, 14, 1347.

23. Kirtman, B. Chem Phys Lett 1988, 143, 81.

24. Dalskov, E. K.; Oddershede, J.; Bishop, D. M. J Chem Phys1998, 108, 2152.


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Coupled-perturbed Hartree–Fock theory for quasi–one-dimensional periodic systems: Calculation of static and dynamic nonlinear optical properties of model systems