low-spin excitations of quasi-one-dimensional polyacetylene chains in hartree-fock models

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Page 1: Low-spin excitations of quasi-one-dimensional polyacetylene chains in Hartree-Fock models

LOW-SPIN EXCITATIONS OF QUASI-ONE-DIMENSIONAL POLYACETYLENE CHAINS

IN HARTREE-FOCK MODELS

G. E. Vaiman, A. V. Luzanov, M. M. Mestechkin, Yu. F. Pedash, and S. I. Smirnov

UDC 539.192

In the q-electron approximation taking into account Coulomb repulsion and electron correlation, we have calculated the excitation energy %s of~ the state with spin s = i, 2, 3 in long polyacetylene chains. From numerical calculations, we have established the dependence %s = (2s - i)~I, whkch is satisfactorily satisfied for the different wave functions used (from the unrestricted and two variants of the extended Hartree-Fock method). For some of the solutions, localization of the excitation in 2s - 1 fragments of the chain is characteristic, which indicates the possibility of inter- preting it in terms of a soliton.

Determination of the structure of the energy spectrum of quasi-one-dimensional objects is an unsolved problem in the theory of low-dimensional systems. Only some general results are known, obtained within the Hubbard model based on the Bethe formulation (see, for example, [i, 2]). The Hubbard model especially corresponds to polyacetylene [the polyene (CH=CH)nH2] [3], where a node with a single electron (a ~ electron) correlates to each carbon atom, and the entire treatment is restricted to the q-electron system as N ~ ~ (N = 2n). Various aspects of the electronic structure of the polyacetylene modeled in this way are considered in many works [i, 3-7]. In most investigations (with rare exceptions of the type in [8-10]), the Hubbard Hamiltonian is used, by definition ignoring the long- range part of the electronic interaction. Such a model masks important physical effects, imposing, for example, nonadditivity onto the dipole polarizability (characteristically an additive parameter).

The goal of our work involves a numerical study for the example of rather long chains (40-60 ~ centers), taking into account long-range effects in the spectrum of low-spin exci- tations in polyacetylene. We note that the length of the polyene considered in this work is close to that observed in an average continuous n-conjugation chain [2].

As an unavoidable approximation for such large systems, we assumed a quasi-one-dimen- sional model called the generalized Hartree-Fock method [i, 4] or the spin-extended Hartree- Fock method (EHF) [ii]. Its simplified variant, the unrestricted Hartree-Fock method (UHF), is used most often. In contrast to EHF, in the UHF method we do not perform an exact spin projection of the Slater determinant with different orbitals for opposite spins onto the pure spin state. In turn, in the rigorous realization of the extended Hartree-Fock method for even-electron systems, there are s + 1 possibilities for obtaining the specified state with spin s (from the number of values of the modulus of the spin projection, Sz, in the determinant function to be projected) [ii, 12]. In the standard version of EHF, which we will symbolize as (s, s) EHF, it is obtained by projection of the determinant with maxi- mum projection s z = s. In the more important other version, symbolized as (s, O) EHF, the state with the same spin s is separated by the corresponding projection of the "quasising- let" Slater determinant with projection s z = 0. It is precisely the application of the (s, 0) EHF that most often gives an underestimated value of the energy [12]. Therefore, in order to obtain more reliable physical results, we should compare data from all the variants of the generalized Hartree-Fock method.

Let us first consider the results of the calculation for a 60- and 50-atom polyene chain in the UHF method, the subject of preliminary study in [13]. We used the system of

Institute of Physical Organic Chemistry and Carbon Chemistry, Academy of Sciences of the Ukrainian SSR, Donetsk. Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 26, No. 6, pp. 716-719, November-December, 1990. Original article submitted May 5, 1989.

0040-5760/90/2606-0669512.50 �9 1991 Plenum Publishing Corporation 669

Page 2: Low-spin excitations of quasi-one-dimensional polyacetylene chains in Hartree-Fock models

TABLE i. Energies (eV) of Low-Spin Excitations of a Polyene in Different Models

6 0 ( o r 59) c e n t e r s

URF formula ( i )

40 centers

(s, O) 1/~r (s, s)

1 0,6134 0,6134 0,5141 0,88t9 3'2 1,2262 1 . 2 2 6 8 - -

2 1..8407 1,8402 1,5107 2.4-~56 5/2 2,4699 2,4536 - - - - 3 3,1234 3,0670 2,9634 42302

parameters which was used previously for short polyenes in [14,. 15] (resonance integral B = -2.274 eV, electron repulsion integrals on different nodes were calculated by inter- polation of the Ono formula for repulsion parameter Y0 = 11.16 eV per node). The values of the excitation energy X s obtained in such calculations for excitation from the singlet state to the state with spin s (Table i) are described remarkably accurately by a linear dependence on the spin s:

~.~= ( 2 s - - 1 ) k l , ( i )

where X i is the energy of the singlet-triplet excitation of the polyene chain. The fact that such a dependence is not entirely obvious for small values of s ~ n follows from analysis of the limiting cases (everywhere in the following, the number of electrons N is even, while s is a fortiori an integer).

The simplest method for the given problem is the H{ickel method: the strong binding approximation with complete neglect of the Coulomb interaction (u ~ 0). Then from the known values of the one-particle energies for a linear N-atom chain, we find that for s spin flips we need the energy

."is \ , . x k s : ~ I ~ COS ~ uSlFl ~ S~.l ,

, V + 1 ] ~ I) ( 2 )

where for s << n and sufficiently long chain length X I -- -2B~/(N + i) ~ is the only nonzero matrix element of the Huckel Hamiltonian between the ~ functions on adjacent units). The spin dependence in Eq. (2) is quadratic, contradicting (i).

Let us now consider the case 70 + ~- Due to the energetic unfavorability of states of the -C + - C- type in this case, all the lower states prove to be purely homeopolar, and for such states the polyacetylene chain is equivalent to a Heisenberg spin chain. In the free-magnon approximation, we can expect a simple proportionality %s = sX~, nevertheless different from (i). Therefore the next step is to determine the appearance of the depend- ence of % s on s in models of the generalized Hartree-Fock method, more rigorous than the unrestricted Hartree-Fock method.

From general considerations concerning the unimportance of the spin projection in large systems (see, for example, [I, 4]), the (s, s) EHF and (s, 0) EHF methods should preserve the result (I). In fact, for N = 40, EHF and UHF give comparable correlation energies c. in the ground-state singlet (in the indicated parametrization, ekopp UHF =

gopp 0.063 eV per electron and Ekopp EHF = 0.095 eV per electron). The energies of the spin

excitations prove to be close only in the case of the UHF and (s, 0) EHF, but each version of EHF approximately confirms the "2s - i rule" (I) (Table i). In this case, the symmetry of the lowest spin states is the same in all three approaches, such as for the triplet SB u in UHF, (i, i) EHF, and (i, 0) EHF. In the (i, 0) EHF case, we could obtain a level of another space symmetry 3A , but it proved to be 0.68 eV higher than the BB u level. Thus, dependence (I) is reproducedgeven in more correct models which take into account electron correlation.

The origin of rule (i), however, remains unclear. In the UHF and (s, s) EHF case, the coefficient 2s - i in (I) can be connected with localization of the spin excitation. The latter can be determined using bond orders: the one-particle averages

6 7 0

Page 3: Low-spin excitations of quasi-one-dimensional polyacetylene chains in Hartree-Fock models

~ ~ : f

zO 2O 30 #0 50 /~

Fig. i. Distribution along the chain of the absolute values of the changes in bond orders (3) upon excitation.

in the state IS(s)> with spin s, where u;i: is the creation operator on the B-th node of an electron with spin "up". The change

L. : I P2,, I (3) in adjacent bond orders P~,~+z upon going from the singlet to the specified multiplet char- acterizes the localization of the spin excitation on the ~-th node. Qualitatively, the course of L~ along the chain for different s in UHF and (s, s) EHF is reflected by Fig. i, and the absolute values of the spin densities behave similarly. As is evident, the number of regions of localization of excitation exactly coincides with 2s - i.

At the same time, in (s, 0) EHF there are generally no appreciable excitation peaks. This fact can be interpreted in terms of soliton excitations. We will assume that the (s, 0) EHF method, as giving the lowest value of the energy, approximates the spectroscopic state with spin s, while (s, s) EHF approximates soliton excitation of the corresponding spin term. In this hypothesis, the number of localization regions 2s - 1 for the soliton excitation is taken as a topological characteristic of the soliton solution. The differ- ence in the energies of the indicated two states (for example, for s = i, it is 0.37 eV) in order of magnitude gives an acceptable estimate for the energy of the soliton itself. The results of the calculations are qualitatively preserved even for slight alternation of the resonance integrals, which is important in the given context. However, further studies are needed for a deeper understanding of the observed "ls - i rule".

LITERATURE CITED

i. A. A. Ovchinnikov, I. I. Ukrainskii, and G. F. Kventsel', "Theory of one-dimensional Mott semiconductors and electronic structure of long molecules with conjugated bonds," Usp. Fiz. Nauk, 108, No. i, 81-96 (1972).

2. Yu. A. Izyumov and Yu. N. Skryabin, Statistical Mechanics of Magnetically Ordered Systems [in Russian], Nauka, Moscow (1987).

3. J. Simon and J.-J. Andre, Molecular Semiconductors [Russian translation], Mir, Moscow (1988).

4. I. A. Misurkin and A. A. Ovchinnikov, "Electronic structure and properties of poly- meric molecules with conjugated bonds," Usp. Khim., 46, No. i0, 1835-1870 (1977).

5. Io I. Ukrainskii, "Effective electron-electron interaction in conjugated polymers," Phys. Status Solidi B, 106, No. i, 55-62 (1981).

6. S. Suhai, "Perturbation investigation of electron correlation effects in infinite metallic and semiconducting polymers," Int. J. Quantum Chem., 23, No. 6, 1239-1256 (1983).

7. K. Tanaka and T. Yamabe, "Electronic structure of conductive conjugated systems and their physicochemical properties," Adv. Quantum Chem., 17, 251-284 (1985).

8. J. Cizek, G. Biczo, and J. Ladic, "Some comments on the bond structure calculations of linear chains in the semiempirical SCF LCAO crystal approximation," Theor. Chim. Acta., 8, No. 2, 175-177 (1967).

9. I. I. Ukrainskii, "Energy bond structure of polymer chains with screw axes of sym- metry," Theor. Chim. Acta, 38, No. 2, 139-148 (1975).

671

Page 4: Low-spin excitations of quasi-one-dimensional polyacetylene chains in Hartree-Fock models

i0. M. M. Mestechkin, G. T. Klimko, and L. S. Gutyrya, "Model treatment of quasi-one- dimensional systems," Ukr. Fiz. Zh., 32, No. 8, 1154-1162 (1987).

ii. M. M. Mestechkin, G. E. Vaiman, V. Klimo, and J. Tino, Extended Hartree-Fock Method and Its Application to Molecules [in Russian], Naukova Dumka, Kiev (1983).

12. M. M. Mestechkin, A. G. Gershikov, and G. E. Whyman, "On the efficiency of different EHF wavefunctions," Chem. Phys. Lett., 91, No. 6, 443-446 (1982).

13. A. V. Luzanov and Yu. F. Pedash, "Electronic properties of conjugated radicals in the restricted and unrestricted Hartree-Fock methods," in: Abstracts, 9th All-Union Conference on Quantum Chemistry (Chernogolovka, June 10-12, 1985), Pt. i, p. 150.

14. V. A. Kuprievich, V. E. Klimenko, and O. V. Shramko, "Dependence of the correlation energy of even polyenes on chain length," Teor. ~ksp. Khim., 12, No. 6, 732-738 (1976).

15. Yu. I. Gorlov and I. I. Ukrainskii, "Application of the generalized Hartree-Fock method for investigation of triplet excitations of short polyenes," Preprint 73-138 R., Inst. Teor. Fiz., Akad. Nauk Ukr. SSR, Kiev (1973).

RADIATIONLESS ELECTRON TRANSFER IN MOLTEN-SALT SYSTEMS

S. V. Volkov and V. A. Zasukha UDC 541.128

Theoretical concepts are applied to the effects of cations outside the coor- dination sphere on electron transfer between complexes in solution and in melts; the activation energy and electron-transfer matrix element are altered.

In [i-6], there are discussions on processes in condensed coordinated systems in the cluster approximation: radiative electron transitions in single celters [i, 2], vibra- tional transitions [2-4], and electron-spin [2] and nuclear ones [2, 5, 6]. The main at- tention was given to coordination compounds in molten salt systems [7], where cluster con- ditions occur most clearly [2]: there is direct contact between external cations and the internal anion coordination polyhedra [8]. Here similar concepts are applied to a kinetic aspect: radiationless electron transfer between cluster centers in molten salts.

Background ions influence electron transfer rates between complexes in aqueous and nonaqueous media, which is illustrated by Table 1 for a similar species set. The electron- transfer rates in aqueous solution increase from Li + to Cs + [9]. The main reason is re- duced electrostatic repulsion between the coordinated ions of the same sign (because of the bridging external A+), so one explains the observed relationship from the location of the A + between the coordinated anions in conjunction with the hydrate and in general solvate shells in such bridges [9]. Then Cs +, which has less hydration than Li + and Na + ( and even somewhat than K + and Rb+), which provides shorter distances. These transfer rates are correlated with thermodynamic parameters: the stabilities of the associated forms of alkali cations A + formed with the anionic Fe(CN)6 ~-/~- complexes increase in the series Li + < ... < Cs + for ~ = 0.i, which confirms that the hydrated shells are preserved [9].

However, that trend for low concentrations (about 0.i-i M) does not occur at higher ionic strengths (3-4 M), and that stability series for contact pairs and the electro- potential series will be reversed, e.g., for Fe(CN)63-/4- one gets K + < Na + < Li + [9].

That trend occurs also in the kinetic parameters, particularly the electron transfer rates between clusters, and this we consider to be expected in molten salts because there

Institute for General and Inorganic Chemistry, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 26, No. 6, pp. 719-723, November-December, 1990. Original article submitted May 16, 1989.

672 0040-5760/90/2606-0672512"50 �9 1991 Plenum Publishing Corporation