QUANTUM-CHEMICAL STUDIES OF QUASI-ONE-DIMENSIONAL ELECTRON SYSTEMS. PART 2. CUMULENES AND ORIGIN OF THE FORBIDDEN ZONE

This review is devoted to the basic problem in quantum theory of quasi-one-dimensional electron systems like polyenes (Part 1) and cumulenes (Part 2) – physical origin of the forbidden zone in these and analogous 1D electron systems due to two possible effects – Peierls instability (bond alternation) and Mott instability (electron correlation). Both possible contradiction and coexistence of the Mott and Peierls instabilities are summerized on the basis of the Kiev quantum chemistry team research projects.


Introduction
This review gives detailed results and thorough discussion of basic results in quantum theory of quasione-dimensional electron systems like Polyenes (Part 1 [1]) and Cumulenes (Part 2), including partly Polyacetylenes, Polydiacetylenes, and some organic crystalline conductors obtained by Kiev quantum chemistry team with my direct and consultive or conductive participation in some of the research projects below.
We began in Part 1 [1] with local electronic states in long polyene chains in the simple tight-binding approximation [2][3][4]. Then we gave condensed review of the Generalized Hartree-Fock method and its different versions with some demonstrative applications to atoms, molecules, and carbon polymers [5]. Further we turned to theory of electronic structure of long polyene neutral alternant radicals based on the different orbital for different spins SCF method [6]. Then we ended Part 1 [1] with the local electronic states in polyene chains with an impurity atom using unrestricted Hartree-Fock approach [7].
Part 2 of the review we begin with basics of the πelectronic theory of cumulenes [8,9]. Then long cumulene chains are treated by extended and unrestricted Hartree-Fock approaches [10]. Thus, we come close to the basic problem in quantum theory of quasi-onedimensional electron systemsphysical origin of their forbidden zone.
In connection with this basic and most intriguing problem two results will be described in details. In one case using unrestricted Hartree-Fock treatment of the Hubbard-type Hamiltonian for long one-dimensional chains two possible effects -Peierls instability (bond alternation) and Mott-type electron correlation spin ordering leading to energy gap formation are mutually exclusive [11]. On the other hand, it was recently shown that quite sophisticated theory based on the varying localized geminals approach predicts coexistence of the Mott and Peierls instabilities in real one-dimensional systems [12]. Moreover, it is stated that this approach permits to give the answer to the question what mechanism of the forbidden gap formation is more essentialthe electron correlation (Mott instability) or dimerization (Peierls instability). Both treatments despite their contradictions each other will be presented in details. Finally [13], the summary of the review with conclusions and perspectives is given.
The ease of cis-trans isomerization for the ECs or of stereoisomerization for the OCs is determined by the barrier height of internal rotation of the CH 2 end-groups. Rotation of one of the CH 2 groups by 180 • returns the cumulene molecule to its initial state. It is a natural suggestion that the barrier height is determined by the energy of such a molecular conformation in which one of the CH 2 groups is turned by 90 • in comparison with the most stable conformation. In the following under barrier height V we shall imply the difference between energies of the lowest singlet states of the molecular conformations with symmetry D 2h and D 2d .
The barriers V in cumulenes were considered theoretically in [8,9,19,20]. Popov [20] used a simple Huckel method which leads to the conclusion that with an increase of the number of C atoms the barrier tends to zero which is actually simply obvious from physical point of view. σ-Bonds of cumulene chains have cylindrical symmetry and their energy does not depend upon the angle of rotation of the end-groups. Therefore if direct interaction of the end-groups is neglected the barrier height is determined by the energy change of the πelectrons with the change of the molecular conformation.
Cumulenes C N H 4 have 2N-2 π-electrons. In accordance with the simple MO theory 2N-2 levels can contain either N -1 bonding levels and equally many antibonding levels in ECs or N-2 bonding and equally many antibonding levels plus 2 nonbonding levels in OCs. In the former 2N-2 π-electrons occupy all N-1 bonding levels; in the laterall N-2 bonding levels and the two remaining electrons occupy the two-fold degenerate nonbonding level. The first distribution is energetically more favorable than the second one. This is achieved for even N for planar conformations and for odd N for twisted conformations. This may be considered as a simple explanation of the known experimental fact [17] that the stable conformation of the ECs is planar, but that of the OCs is twisted with perpendicular arrangements of planes of the end-groups. This very interesting property of the cumulenes was in fact first explained by van't Hoff [21] in 1877 using the tetrahedral model of the carbon atom.
Let us choose the coordinate system in a way so that in the conformation D 2h π-AOs of the subsystem with N AOs are directed along x-axis and with N-2 AOsalong y-axis. The z-axis passes through the C atoms. Conformation D 2d is formed by a rotation of one of the end-AO's by 90 • . In this case the number of AOs which are directed along the x-and y-axis equals N-1 in both cases.
In the conformation D 2h π x -states have symmetry b 2g and b 3u , and π y -statesb 2u and b 3g . In the conformation D 2d all π-MOs transform according to the irreducible representation e. Therefore in this conformation the frontier MOs (pair of nonbonding orbitals) is degenerated by symmetry. Accidental degeneration of the frontiers MOs in the conformation D 2h remains in the Pariser -Parr -Pople (PPP) [22,23] approximation also, for in this case zero differential overlap approximation is used. It is removed by alternation of the bond lengths.
The lowest electronic configuration of the cumulene molecule in its unstable conformation has a mul-tiplet structure with states 3 A 2 , 1 B 1 , 1 A 1 , and 1 B 2 for ECs and 3 A u , 1 A u , 1 A g , and 1 A' g for OCs. We shall see later that when electronic interaction is accounted for the lowest states become 3 A 2 , 1 B 1 , resp. 3 A u , 1 A u . The states 1 A 1 , 1 B 2 , resp. 1 A g , 1 A' g correspond to electron transfer between the perpendicular x-and y-subsystems of π-AOs. The molecule in its stable conformation, which is 1 A g for ECs and 1 A 1 for OCs has a closed shell. The degeneration of the frontier π-MOs is removed for inorganic cumulenes with alternating atoms of different electronegativity. To a smaller degree the same is true if the difference in the hybridization between the parameters of inner and outer C atoms is taken into account. But even in this case the lowest singlet state may be 1 A u if the orbital energy splitting does not exceed the splitting of even and odd states.
In the following we shall neglect the difference in hybridization between outer and inner C atoms. This approximation is sufficiently good because the integrals for sp 2 and sp states are almost equal [24].
Let us the x-and y-MOs in the conformation D 2h write down as a linear combination of the π-AOs x ν and y ν with the chain of AOs y ν denoted by primed symbols ,.
The summation is extended over all AOs of the chain. In the same manner it is possible to set up the components of the degenerate pairs of the MOs in the conformation D 2d .
Let ˆ i A be the creation operator for an electron i of orbital state i  and spin state  , and ˆ i A be the same for spin state  . Degenerate orbital pairs of open shell will be denoted by the symbols k and  k , and orbitals of closed shell by j and  j . Then the wave functions of states with closed shell  c may be written as 11 1 ( , ) ,ˆˆˆ0  T  T  g   T  T  T  T  T   T  T  T  T  T   T  T  T  [26]. This is also true for the SCF method in the PPP approximation, which is assumed, if the ionization potentials and integrals are put equal for all C atoms [23,27] including the end-atoms: The energy c x E represents the π-electron energy of a hypothetical compound with the same space structure as the corresponding cumulene with closed shell but having only one system of AOs of the type x  . The same is true for the energy c y E . int E represents the energy of the static electron interaction of the two chains and does not depend upon the MO coefficients.
Analogous transformations for the states with open shell 1 1 B and 1 u A lead to the following result: 11 1 int As we see, division into two chains is possible also in this case, but now each chain is in a doublet state and has an open shell structure as in organic free radicals. However where the summation is taken over all AOs of the two chains, one gets different interaction energies for different conformations: However on physical grounds the interaction energies of positive charges in different core conformations of cumulenes can not be different. These differences are small, of course, and decrease rapidly with increasing chain length.
If one accepts the differences mentioned then the barrier height V may be found from the relation The last term will then result from compensations of charges of int E and core E . If, on the other hand, one takes the same core energies core E for both conformations then Barrier values by (4) and (5) are almost identical especially for large N.
As we see from (4) and (5) the barrier height is determined first of all by SCF energies of the π-electron subsystems which may be calculated from formulae (2) and (3) It is possible then to show analytically that barriers calculated by formula (4) tend asymptotically to zero with increasing N in accord with Huckel calculations in [20]. Now we consider very long cumulene chains using sophisticated EHF as well as UHF approaches.

Electronic Structure of Long Cumulene Chains in the Extended Hartree-Fock Method Compared with its Unrestricted Version
It is important to note once more that most properties of carbon polymeric chains like polyenes, cumulenes, polyacetylenes, polyacenes, and graphene can be explained in terms of the π-electron approximation. This fact enables methods involving electron correlation to be used for theoretical treatment of such electronic systems, which in turn gives a possibility for studying the 22 22 () We begin our consideration of long cumulenes with the UHF equations for long polyene chains.

1. The UHF equations for long polyene chains
Here will be now proved that the UHF equations for long neutral polyenes both with even and odd number of carbon atoms are the same. For this purpose let us analyze the results obtained for polyenes with even and odd [1] number of C atoms 1 N by the UHF method. If the chain boundaries are taken into account, as in the Hubbard's approximation, the Hamiltonian for a long polyene with equal bond lengths can be expressed as [29,30] ,, , , , ,ˆ( 2 cos )ˆˆˆ( , , , ), 2 where  and  are the Coulomb and the resonance integrals respectively,  is the Coulomb integral corresponding to the electron interaction with one of the nearest atoms, ˆ k A  and ˆk A  are the operators of π-electron creation and annihilation in the state where 2 1, ( ) , 24 1. ( ) is defined by the equation Let us transform the operators ˆ k A  and ˆk A  using Eqs (120) in [1] as (1) (2) 1/ 2 Substituting (10) and (11) into (07) one obtains The operators () i k A  correspond to one-electron wave functions In the ground state all levels (1) k  are filled and all levels (2) k  are empty whether N is even or odd. Thus, the relations (8)-(14) are valid in both cases. Consequently, in the UHF method the self-consistent functions (14) and the energy spectra of long polyene chains with even N coincide with those for odd N, as it should be expected so far as 1 N . By contrast, the HF solution for long poly-Фізико-математичні науки Scientific Journal «ScienceRise» №6/2(11)2015 127 ene is unstable relative to a small perturbation modeling the addition of an unpaired electron to the system. Unfortunately, the UHF wave function is not an eigenfunction of the total spin operator 2 S . To get rid of this disadvantage one has to use the EHF method. It will be shown below that both the UHF and the EHF methods being applied to large enough systems give identical results except spin density expressions. This means that the projection of the UHF wave function on the state involving the lowest multiplicity does not affect the relations (8)- (14). It should also be noted that the exact solution of the Schrodinger equation with the Hamiltonian (6) and the cyclic boundary conditions is obtained in [34]. The study of the exact solution [34, 35] has shown that there was an energy gap in the spectrum of quasi-ionic excitations active in optical spectra. Hence, it can be concluded that the UHF/EHF method treats correctly this feature of the exact solution.
Now we shall consider the values Substituting (14) into (15) one obtains The chain boundary effect is revealed in the dependence of   on  . It follows from (16)

2. Electronic Structure of Long Cumulene Chains
Now we shall turn to the treatment of long cumulenes 4 C H ( 1) N N using the results obtained just above. As we know the π-electron system of a cumulene molecule consists of the two π-subsystems which have the maxima of the electron density at two mutually perpendicular planes. From now all values corresponding to one of these subsystems will be marked with letter a/A and to anotherwith letter b/B. There are two possible conformations of a cumulene molecule which differ by mutual orientation of its end-groups CH 2 . Let us denote the conformation of symmetry D 2h in which the endgroups lie in the same plane as A and the alternative conformation of symmetry D 2das  A .
Let us consider a cumulene molecule neglecting its end-groups. In the short-range interaction approximation , kb  with σ-spin, where ˆˆˆ, , According to [33], there are two possible selfconsistent solutions: Фізико-математичні науки The expression (20) have the same form as (3). Because of this the expressions (20) are diagonalized by canonical transformation of the type (05), namely: The operators ( In the ground state all levels (1) k  of the two subsystems are filled and all levels (2) k  are empty. Thus, the UHF wave function of the cumulene ground state has the following form Using (17) and (25) one can obtain the expression for the ground state energy Atomic populations are defined as The analysis of (28) allows to reveal the dependence of Thus, as with the polyenes, the chain boundary influence on   sharply decreases when the distance from the chain boundary increases.
We shall need further the equations for the coefficients (23), which can be obtained by proper transformation of (21), namely: To obtain the equations for () it is necessary to permute markings a and b in (30). Now let us consider cumulenes taking into account the end-effects. In the conformation A with symmetry D 2h the subsystem a contains N π-electrons and the subsystem b contains N -2 π-electrons. In the conformation  A with symmetry D 2d both subsystems a and b contain the same N -1 π-electrons. In passing from the long ideal no-end-groups cumulene to a real cumulene molecule with the end-groups some alterations in the equation (30) result due to the relative shift of the cumulene π-electron subsystems a and b. Namely, the effective values of the Coulomb integrals are changed according to It follows from (31) and (29) that the effective Coulomb integrals of the end-atoms decrease by the value without regard for a change in the interaction between σand π-electrons in passing from the long ideal no-endgroups cumulene to real cumulene molecule. However, as long as the end carbon atoms of a cumulene molecule have the sp 2 hybridization, one should expect that the absence of the Coulomb interaction between π-electrons at the end-atoms is compensated by an interaction between σand π-electrons. This point of view is supported by the fact that the first ionization potential of a carbon atom in the valence sp 2 state coincides with that in the sp state within 10 -3 eV [36]. On the other hand, the exchange interaction  does not appear to be compensated Фізико-математичні науки Scientific Journal «ScienceRise» №6/2(11)2015 129 for in this case. Thus, we shall assume that the change in the Coulomb integrals (31) at the end-atoms is Let us consider the alteration of () i   at the atoms next to the end-atoms. Using (29) and (31) one can obtain for the longer subsystem and for the shorter subsystem Thus, the end-effects in cumulenes are of a local nature and can be considered by means of the local perturbation theory, which was applied to long polyenes in the framework of the UHF method in [1].
The ratio A , each πsubsystem a and b consists of an odd number of electrons, being a long polyene radical. Nevertheless, it follows from the previous section of this paragraph that the energy spectra of long even polyenes and long polyene radicals are the same in the framework of the UHF (or EHF) method. Thus, in both conformations of a cumulene, its excited states are separated from the ground state with the gap 1 2( )   in accordance with Eq. (21). Let us evaluate the difference between the ground state energy of a long cumulene chain in the conformation A and that in the conformation The value of E is usually referred to as the torsion barrier of cumulene end-groups. Using the relations (17) and (25), one can obtain (1) (1) (1) 2 1 where (1) [] In order to evaluate Eq. (32) it is important to note that if () fk is a continuous function of k then It follows from Eqs (34), (21), and (33) that (1/ )    EN , i.e., the torsion barrier tends to zero when the cumulene is lengthened. From the mathematical point of view this result is due to the fact that the intervals between the levels occupied in the ground state are of 1/ N whether the cumulene subsystems a and b consist of the even or odd number of π-electrons.
Let us evaluate the influence of the small perturbations ()  i   on the π-electronic structure of cumulenes.
As we already know from § 5 in [1], small local perturbations can give rise to local states in the forbidden zone of a system like long polyene chains. These local state energies differ from the nearest zone state energy by values 2 a , where a is the width of the forbidden zone in the ideal cumulene chain. It means that in our case the forbidden zone width is not affected practically by the end-effects. It was also shown in § 5 in [1] that local perturbations placed at the large distance from one another do not interact. Hence it follows that the endeffects in long cumulenes can not change the value of the torsion barrier. Indeed, the contributions into the ground state energy are additive relative to perturbations of atoms placed at the different ends of a long cumulene chain and, because of this, are the same whether the cumulene is in the conformation A or  A . To study spin properties of cumulenes we should pass from the UHF method to the EHF approach. As it will be shown below, the SCF equations for systems consisting of the large number of electrons are the same whether one uses the UHF or the EHF method. So, the orbitals () Let us now consider the multiplicity of the cumulene grounf state. Suppose the number of cumulene carbon atom to be even, i. e. 2  Nq . Then both cumulene subsystems a and b in the conformation A consist of the even number of π-electrons N and N-2, respectively. Hence, the total spin projection for each of the two subsystems in the ground state when all levels of both subsystems are filled is equal to zero: 0  ab MM . Therefore, the cumulene ground state in the conformation A is a singlet one (S = M = 0) and its EHF wave function, as will be shown below, has the following form where Â is the antisymmetrization operator be specified later, ,,  (36) and (37) if the following relations are taken into account, namely: Let us pass now to the calculation of the AO spin populations in long cumulene chains. Using the Eqs (23) and (24) according to the UHF method one can obtain in the two conformations of cumulenes To obtain spin populations in the framework of the EHF method one should multiply (38) by the factor / ( 1)  SS in accordance to the relation (52) below. So, the AO spin populations in long cumulene chains C N H 4 with an even number N vanish identically in the conformation A . But, they differ from zero in the conformation  A and are equal to

EHF and UHF Methods when Applied to Large Electronic Systems
Before to give the final discussion for this paragraph let us compare the UHF and EHF approaches as applied to large systems. The EHF wave function can be written as (see § where 0 12 12  is the antisymmetrization operator. In the EHF method the ground state energy of a many electron system has the form where we have used the standard notations for the electron interaction integrals, 1

( , )
S i j and 2 ( , ) S i j are the sums of some electron interaction integrals, Фізико-математичні науки Scientific Journal «ScienceRise» №6/2(11)2015 131 2 2 0 00 1/ (1 ), Since, according to (42) Taking into account that all terms in (43) are positive and using (47) and (48) one can obtain It follows from (46), (49), and (43) that the relations (45) are also valid in the framework of the multiparameter AMO method for the limit case    nm . Thus, the relation is valid in general if the system under consideration consists of a large number of electrons. Besides, it follows from equation (50) that the SCF equations are the same in the EHF and UHF methods for this case. This can be proved directly through the use of the EHF (or the GF) equations (see § 3 in [1]) obtained by Goddard.
As far as the EHF approximation, the spin density expression has the following form in the analogous spin density expressions [41, 42] there seems to be a mistake in the coefficients in the last term in (51).
Using (45) one can obtain from (51) for the case For long polyene chain from (42) and (14) one obtains Using (47) and (53) It follows from (57) that the Gaussian approximation gives the same value of S  as the approximation used for this purpose in [6]. Comparing (57) with the exact expression (56) one can see that the approximation (57) correctly reflects the asymptotic behaviour of S  when  N (except for the constant). It should be noted that the relation (50) can be obtained also by means of the rotation group theory [43]. However, using this method we lose some important details, e. g. it is impossible to obtain the asymptotic form (52) for the spin density expression (51).
Next let us discuss the excited states of long polyene chains by means of the EHF method. Let us replace an orbital (1) k  by (2) k  in (239) To summarize, the ground state energy, energies of the lowest excitations and the the SCF equations for large systems ( 1 N ) are the same in the framework of the UHF and the EHF methods. Thus, to calculate the electronic structure of the system, which consists of large number of electrons, by the EHF method one may use the simple single determinant UHF wave function rather than the much more complicated EHF wave function (39).

4. Some Conclusions
As already known the appearance of the forbidden gap of about 1 eV width in the optical spectra of long cumulene chains can be explained by means of the RHF method with the alternation of bond lengths being introduced. However, the torsion barrier of the end groups of long cumulene chain does not vanish in this model. This fact seems unnatural as far as the end-groups CH 2 of long cumulene chain C N H 4 ( 1 N ) are placed at the large distance from one another. On the other hand, the simple MO methods give 0( )     EN for cumulenes with equal bond length. But in this case the first electronic transition frequency also tends to zero which contradicts the experiment.
To put it another way, the assumption that the energy gap in the spectra of long cumulene chains is due to the bond alternation gives rise to the dependence of the gap value to the torsion barrier. The gap value is shown to be equal to the torsion barrier in this model [45]. From the mathematical point of view this correlation between the gap value and the torsion barrier results from neglecting electron correlation. Indeed, if the long cumulene chain in the conformation involving the odd number of π-electrons in each of the two subsystems a and b is treated by means of the Huckel or the RHF methods, then in the spectrum of such chain there are two levels in the ground state which correspond to the zero values of one-electron energies, whether the bond alternation is introduced or not.
It is shown in this paragraph that the appearance of the forbidden zone in spectra of long cumulene chains is not connected with the value of the torsion barrier in the framework of the EHF method in contrast to the simple models mentioned above. Furthermore, the EHF method gives zero value of the torsion barrier for long cumulenes with equal bond lengths. On the one hand, these results once more suggest the necessity for taking account of electron correlation when large conjugated systems are treated. On the other hand, we think that these results provide some further evidence for the correlation nature of the forbidden zone in spectra of long cumulene chains and, consequently, long polyene chains.

Coexistence or Contradiction of the Peierlsand Mott-type Instabilities in Quasi-One-Dimensional Systems
It has been first stated by Mott [46][47][48][49][50][51][52][53][54] that the one-dimensional array of atoms with a half-filled valence band should necessarily exhibit metaldielectric transition as a result of increasing the lattice constant. Modern developments of the Mott instability have been reviewed in [54][55][56]. In such Mott-type dielectrics the lowest quasi-ionic excitations are separated from the ground state by the energy gap of the order    IA (I and A are being the ionization potential and electron affinity correspondingly). The value of this important parameter should be ~10 eV in the case of isolated small atoms, but some factors in real systems like polarizability of the given elementary unit (CH 2 group in polyenes, TCNQ fragment in charge transfer salts) or of the neighbouring elementary units [57] reduce this gap to 34    eV for polyenes and up to 1  eV in TCNQ chains. Furthermore, electron exchange at the real interatomic distances should be taken into account which results in the broadening of previously highly degenerate ionic excited states to a conductance band of width 4 | |  , where β being the resonance integral. In the case of / | | 1   , this does not change the spectrum qualitatively and even at real distances one gets the Mott-type dielectric at zero temperature. In the opposite case / | | 1

 
, the exchange broadening is larger than the energy gap which leads to the metal type structure of the excitation spectrum of the 3d-crystal. However, in the 1d-case such a structure is unstable with respect to nuclear displacements of a special kind and the Peierls transition to the usual semiconducting state takes place [58-60]. As a result one gets an initially continuous band of allowed states split in two bands with a forbidden zone of the width 12  133 [56]. In this paragraph the 1d-problem will be treated with generalization to consider finite temperatures. It should be mentioned that for both types of instabilities the gap should be temperature dependent and should be equal to zero if the temperature raises above some critical temperature T c . This may be qualitatively understood as follows. In the case of the Peierls transition, the width of the gap is determined by two subtle effects: lowering of the total energy due to the lowering of filled energy levels and raising the energy due to lattice distortion. The energy minimum is reached at the definite distortion which determines the energy gap value. If the temperature is raised, some of the electrons pass to the band of excited states which results in two effects of the same sign: 1) the energy gain due to the energy levels lowering becomes smaller because not all those levels are now filled; 2) from the point of view of the excited electrons, decreasing the gap is preferable as it lowers their energy.
Thus, the gap width  and the lattice distortion depend on the occupation n of the one-electron states, which in its turn depends on the temperature: Thus, at a higher temperature the gap becomes smaller, which makes it easier for the electrons to occupy excited states after the temperature rise and so on. It seems likely that the process is fast enough and at some T c the gap vanishes. The quantitative treatment [59, 60] confirms this explanation.
The situation is formally similar in the case of Mott semiconductors. In this case the creation of ionic excitation makes it easier for the electron at the neighbouring atom to be excited also, i. e., the energy gap depends on the electron distribution at the levels of the ground and excited states which, in turn, is temperature dependent.
The method used below is simple and straightforward: the 1d-chain with lattice displacement x of the kind of bond alternation will be considered using the SCF calculations allowing, in principle, to get the correlation gap. The total energy or the free energy in the case of 0  T will be evaluated to investigate whether its minimum correspond to the nonzero values of both correlation gap and the lattice distortion or whether only one of them may differ from zero for the 1d-system.

1. Peierls and Mott Instabilities at T=0 • K
We start with a Hamiltonian that differs from the Hubbard Hamiltonian in two points: the lattice distortion as the bond alternation is taken into account and the repulsion of electrons when accounted for the neighbouring atoms 12  is included, namely: where  is the Coulomb integral, U is the Hubbard parameter of the electron repulsion on the same atom, 12  accounts for electron repulsion on the nearestneighbouring atoms. The second term describes the Peierls doubling of the unit cell. The first term will be omitted in the following treatment bearing in mind that it results only in a trivial equal shift of all energy levels.
The translational invariance of the Hamilton (61) may be used to reduce it to a more nearly diagonal form. Let us introduce the operators ˆ k B  and ˆk B  which create and annihilate, respectively, an electron in a state with quasi-momentum k and spin σ: ) there are N/2 equivalent pairs of sites rather than N equivalent sites. Thus, a linear combination of the operators (64) is required to diagonalize the quadratic part of the Hamiltonian. The energy spectrum breaks up into two bands, separated by a forbidden zone 4 in width. We do not follow this procedure here because it is useless in treating the last two quartic interacting terms in (65).
To treat the full Hamiltonian , we have to simplify it in an appropriate way. We wish to obtain the selfconsistent solution of our problem. Thus, we shall reolace some terms in the quartic part of the Hamiltonian by their average values. Bearing this in mind one can reduce (65) leaving only the terms we expect to have as nonzero Equation (69) evidently shows that   is proportional to the overall difference in the number of electrons with spin σ at the even and odd atoms of the chain and differs from zero only if spin alternation at the neighbouring sites of the chain take place. Retaining this term makes it possible to account for the correlation contribution to the energy gap or, in other words, to treat the Mott-type semiconductors, while, as it has been mentioned, the second term in (65) allows ua to consider the Peierls instability. In the last term of (65) we preserve the following four terms: (i) 12  kk , namely, which is merely a correction to the Hartree-type term discussed above, and in the ground state assumbed to be replacable by the C number 12 N ; (ii) 1 (76) and (75) into (77), and performing the calculations required using (78), one obtains the system of two nonlinear equations with respect to , kk UV  . Linearizing these equations, which corresponds to the SCF procedure, and using (79), one obtains the solution in the form where 12 12, , 22 .
To make the solution complete, we have to calculate   , ˆ  , and   , where the everaging is implied over the just found ground state, corresponding to all states k  occupied with the minus sign in Eq. (80), i.e. the ground state has the form where the operators ()  k b are defined by Eqs (76) We are now in a position to turn to the final step of the treatment, namely, calculation of the total energy and minimization of it with respect to x   , corresponding to the usual HF SCF procedure. We have used the non-trivial solution in the case of the regular chain structure (no alternation) because in this case it corresponds to a lower energy than the trivial solution [61]. However, for 0   this nontrivial solution does not minimize the total energy, and for some x  in the process of its increasing we get 0  ; at this point we should jump to the trivial solution because the non-trivial one ceases to exist. Hence, in this case we have an alternating-bond chain with a vanishing ( 0  ) contribution of the Mott-type correlation to the creation of the energy gap; while the gap due to the bond alternation should be calculated in a quite different way [58-60]. The results is well known: in the absence of the Mott-type contribution, the Peierls-type transition to a semiconducting state necessarily takes place, and the gap obtained can be approximately calculated as Therefore, at least at zero temperature the picture is clear: depending on the numerical values of the parameters involved the quasi-one-dimensional chain represents either a Mott-type or a Peierls-type semiconductor, but not their combination, and the choice should be done by comparison of the total energies of both states. Roughly speaking, it may be stated that the real state is the state with the larger gap calculated neglecting the possibility of the other state available. In fact, in addition to the criterion ||  U  mentioned above for the Mott metaldielectric transition, one more criterion should be formulated determining the value of the gap arised due to Peierls instability. However, it is evident that if we have || U  , then the correlation gap is large, ca. U, very likely larger than the gap due to the lattice distortion, and the situation is reversed for ||  U  .

2. Finite Temperatures
Let us now consider the same question of the possible combined nature of the energy gap in the case of a finite temperature. Only the general method of calculations and the final results will be presented below. For details of calculations see [59].
To get the temperature dependence of all the values we are interested in the following procedure may be proposed: in all equations used the average over the ground state should be replaced by statistical average calculated as Bearing in mind that the Hamiltonian expressed in terms of the operators ˆ,  kk bb at (76) Equation (99) may be considered in the same way as it has been done concerning Eq. (87). The one-electron energy levels which are now temperature dependent are also independent on x  and all the discussion following Eq. (91) may be repeated leading to the same conclusions at the finite temperature as were arrived at in the case of zero temperatute.
Let us now note that if our system is a Mott-type semiconductor with Eq. (99) becomes similar to the Peierls gap in the chain with bond alternation, namely, both of them are BCStype gaps in superconductors. Unfortunately this does not provide us much information on the nature of metal -Mott semiconductor phase transition because low-lying triplet and singlet excitations should be taken into account before one treats the quasi-ionic states with higher energies; but only these later states may be considered using the standard UHF procedure.

Coexistence of Mott and Peierls Instabilities in Quasi-One-Dimensional Systems
The quasi-one-dimensional conductors have so far being studied are of interest for both theoreticiants and experimentators. This interest, on the one hand, is due to advances in synthesis of polyacetylene (PA), polydiacetylene (PDA), organic crystalline conductors based on molecular donors and acceptors of electrons. On the other hand, 1d-conductors are nontrivial systems. Thus, 1dmetal is unstable to the transition into semiconducting state. As a result the 1d-metal with half-filled conduction band becomes the Mott semiconductor or Peierls semiconductor. The Peierls transition leads to dimerization of the uniform regular 1d lattice (bond alternation) and semiconducting energy gap is proportional to the dimerization amplitude. The Mott transition is a result of electron correlation and energy gap in the Mott semiconductor vanishes with decreasing electronelectron interaction strength. The semiconductor of the Mott and Peierls type possesses some interesting properties. For example, the Mott semiconductors are characterized by antiferromagnetic structures [80], and in the Peierls semiconductors the kink-type excitations are possible [81,82].
The influence of the Mott and Peierls instabilities on the properties of real quasi-one-dimensional systems have already long story. The main problem in theoretical studies consists in complications related to correct account of electron correlation effects. In ealier papers contradiction of the Mott and Peierls transitions was usually stated. Then it was shown that this contradiction is a result of one-electron approach in the RHF theory. The conclusion that the Mott and Peierls transitions coexist one with another was first made in [83]. This result was obtained due to more correct treatment of pair electron correlations using varying localized geminals (VLG) approach [84][85][86]. It was shown that electronelectron interaction can enhance the Peierls dimerization [83]. This somewhat surprising result initiates several theoretical studies [87-92] which conformed the conclusion that even account for a small electronelectron interaction leads to increase in dimerization. This conclusion has been received on the basis of perturbation theory for infinite chains using computations [90] and the Feynman diagram technique [91]. Numerical calculations of short polyene chains within the same geminals approach conformed this result slightly deformed by boundary conditions [83].
Thus, we can state now that the theory predicts coexistence of the Mott and Peierls instabilities in real systems. So, the experimental data on 1d-systems should not correspond to the simple picture of the Peierls or the Mott semiconductors. One must expore the more complicated theoretical model including the both phenomena. On this way only one can give correct description of real 1d materials. For example, we can now give the correct answer to the question what mechanism of the forbidden gap formation is more essentialthe electron correlation or dimerization.
In this paragraph we shall study now the simultaneous effect of the Mott and Peierls instabilities on electronic spectra and lattice distortion in real 1d conductors such as organic donoracceptor molecular crystals and conjugated polymers of PA type. These studies are based on the VLG approach [83][84][85][86].

1. The Method of Calculations and Qualitative Evaluations
Studying the electronic properties of organic 1d materials the following model of uniform chain with the adiabatic Hamiltonian is used: Thus, when 1  the adiabatic approach is good enough and the problem of 1d instabilities is reduced to studying the ground state energy dependence on the value of  (104). In other words, we need the ∆-value optimizing the expression where el  is the electronic contribution into the ground state energy per an electron pair, and is the constant of electronphonon interaction.
In order to calculate the electronic contribution into the ground state energy mentioned above the VLG approach will be used. The ground state wave function has the form 0 11ˆˆ0  (1 ) ( ) (1 ) (4 ) , can be obtained in the limit of small  . We can see from (122) that when  and, as a result, U are small the energy dependence (120) on  is nonanalytic. Thus, we can suppose strong dependence of U on 0  which minimizes the total energy. Results of numerical study of U on  will be given below. Now the evaluation of asymptotic behaviour in two limiting cases 0  U and  U will be given. When 0  U the non-interacting-electron model is valid and the energy is defined by the value of (121)  (119) and (105) and optimizing  one obtains 0 2 84 1.
We note that Eqs (121 ), (122 ), and (119) describe well the dependence of the total energy on U for any value of U>0 [84,85]. But, the correct description of the Peierls instability near the point U = 0 needs more precise relations due to the fact that the Peierls instability results from a logarithmic term. The latter just lost when passing from (112) to (125) [83]. Now we consider the effect of electronelectron interaction at neighboring sites resulting from the terms with 1  in (101). One can conclude from (114) Thus, we can see an exponential increase of In order to define the optimal value of 0  we have to look for the minimum of the energy (105) taking into account (119) in the space of 0  and λ variables, namely: where t and K are defined by (115) and (117), respectively. This task is not too complicated, but when 1 U some difficulties arise with the increase of the chain length due to the logarithmic ∆-dependence of the electronic energies in (119) and (124). As a result we cannot use the standard method of quantum-chemical optimization of the bond lengths. This method is based on the linear relations between bond length and bond order resulting from the energy expansion which depend on the model used in calculations of short or infinite chains as well as also chains with cyclic boundary conditions. Giving preference to either calculation model one must bare in mind different experimental data, not only the values of 0  .

Conclusions & Perspectives
Advances in physics and chemistry of lowdimensional electron systems have been magnificent in the last few dacades. Hundreds of quasi-1d and quasi-2d systems have been synthesized and studied experimentally and theoretically. The unusual properties of these materials attract attention of physicists, chemists, and engineers.
Quantum processes in low-dimensional systems are characterized by a number of peculiarities. Thus, special and new theoretical approaches have been developed to study low-dimensional phenomena. We will be concerned further mostly with the 1d-systems. In onedimensional physics and chemistry there is a number of difficulties and some of them are far from being overcome. On the one hand, equations of motion for 1dsystems are much simpler. This facilitates rigorous solutions of the model problems which are often impeded in case of the larger number of dimensions. On the other hand, manifestations of various interactions in 1dsystems are rather peculiar. This relates, in particular, to electronelectron and electronphonon interactions. The standard perturbation theory is inapplicable for treating both interactions. Thus, electronphonon interaction leads to the field localization of electron excitation in 1dsystems which results in soliton excitations and the Peierls deformations. Calculations of soliton excitation can not be done by decomposition in the series of electronphonon coupling constants.
Electronelectron interactions, even within the limit of a weak coupling constant, produces an energy gap in the spectrum od 1d-metal which initiate the Mott transition from metal to semiconducting state. In this case the standard perturbation theory is also not applicable.
Similar situation occurs in 1d-systems with respect to electronimpurity interactions. Started by Mott and Twose theoretical studies of this problem show that all one-electron states in 1d disordered system are localized and, as a result, cannot be calculated using the perturbation theory. State localization turns the direct current conductivity into zero.
Inapplycability of the perturbation theory is one of the main difficulties on the way to succeed in the theory of quasi-1d-systems. These difficulties were being partly surpassed in different ways.
Regardling electronphonon interaction the most fruitful method is to reduce the set of corresponding equations into a completely integrable system like the nonlinear Schrodinger equation, the sine-Gordon equation, and others.
Advances in description of electronelectron interactions turned out to be less pronounced however. The major reason for it lies in the well known complications of the many-electron theory for systems with an infinitely large number of electrons.
Traditional quantum chemistry as one of the many applications of the general theory of many-electron systems is based upon the Hartee -Fock approximation which came first as -the word came first‖. Then various many-electron theories being developed where the wave function were not represented by one Slater determinant rather then an infinite series of the determinants. If the number of particles in the system grows as  N then the number of terms in this infinite series must increase at least as aN e , where a is a constant 1  . This particular infinite complication of the theory is the main hindrance in it wide applications in calculations. It is time now to say that these difficulties are often being considerably exaggerated. As a rule, having analyzed the Hamiltonian of the system under study using the many-electron theory one can reduce the problem to a simpler Hamiltonian or without any loss in quality construct multicinfigurational wave function of the system which can be factorized into an antisymmetrized product of one-or two-electron functions. As approximations for a wave function, besides the EHF approximation described in details in § 3 in [1], the spinless fermion approximation in case of strong interactions and the VLG approximation described in previous paragraph can be mentioned.
In the EHF and spinless fermion approaches a many-electron wave function is finally factorized into the product of one-electron functions (orbitals), but in the VLG approach the factorization into the product of twoelectron functions (geminals) is performed. Now we draw attention to another aspect of the theory of quasi-1d electron systems. Real systems with one-dimensional anisotropy are, in fact, threedimensional. In case of a theoretical study it is expedient to mentally separate a 1d-system out of the real system using its specific properties. This separation of a quasi-1d-subsystem goes naturally through analysis of the total Hamiltonian represented by the sum where ˆn H is the Hamiltonian of a n-th quasi-1d subsystem (filaments, needles, chains, stacks, etc), and the operators ˆn m V describe its interactions with other quasi-1d subsystems.
Further it is usually assumed that the interaction operators do not include terms responsible for electron exchange between separate quasi-1d subsystems. Namely this predetermines the subdivision of the Hamiltonian into the sum (131). This approximation provides satisfactory description of PAs, donoracceptor molecular conducting crystals as well as many other quasi-1d electron systems.
Before we consider particular expressions for the Hamiltonians for electronphonon systems under study it is worthwhile to note the following. Most processes in quasi-1d systems are determined by the energy spectrum and the nature of elementary excitations. The low-energy region of the spectrum is mainly related to a small part of the total number of electrons in the system under study. This facilitates a rigorous enough description of electron processes occurring in these systems. As example, most interesting properties of polyenes, cumulenes, and polyacethylenes originate from the π-electron number equals or proportional to the number of carbon atoms and essentially less than the total number of all electrons in the system. Studying the most significant properties of donoracceptor molecular conducting crystals it is sufficient to consider one electron only per a donoracceptor pair. In case of TTF-TCNQ crystal it means that only one electron out of 208 is to be considered.
Despite of the simplifications mentioned above we are still have to restrict ourselves with semi-empirical models of quantum chemistry. For example, the well known Huckel -Pople (HP) Hamiltonian is very popular and useful to study many properties of molecules with conjugated bonds.
As a rule, it is sufficient in (132) to account for resonance interaction (so called electron hopping) for the adjacent atoms only, namely: As far as the electron interaction in (132) is conserned only the first several terms are usually accounted for. As an example, in the Hubbard -Anderson (HA) Hamiltonian Interaction between two neighbouring atoms is only often used:  The greatest interest with respect to newly synthesized quasi-1d and quasi-2d systems is attached to the compounds with high electric conductivity. But on the way to create good organic conductors the investigators encounter difficulties of not only technical but principal nature which relates to an electron instability of a conducting state. Their most important peculiarity lies in the fact that a metallic state of a quasi-1d crystal is unstable with respect to a transition into a dielectric or semiconductive state. The character of instability and its force strength which determines the metalinsulator transition temperature depends on structural features of the crystal.
Let us consider a system consisting of long needles packed into a 3d-crystal. The Hamiltonian of each needle is supposed to be the first term in the general expression (131) 0 1,( where the same notifications as in (132)  A 1d-metal with a half-filled conduction band is unstable with respect to the following metalinsulator transitions: 1) The Mott metalinsulator transition resulting from electron interactions. Instability of a 1d-metal with respect to this transition arises from the fact that electronelectron interactions produce the gap at T=0° K even within a weak coupling constant 0 /  U  in the Hamiltonian (132).
2) The Peierls metalinsulator transition is connected with electronphonon interactions. Alongside with the gap a periodic deformation of the crystal occurs with the period / F k  .
3) The Anderson metalinsulator transition resulting from structure disordering of the crystal. The instability of a 1d-metal in this case is stimulated by localization of electron states even by a weak random field.
When coupling constant U is large the Wigner ordering of electrons in quasi-1d conductors appears.
Early theories of quasi-1d systems came to the conclusion that various instabilities in a 1d-metal are being competive [55]. However, further analysis have shown that, in fact, a coexistence of different instabilities is possible. Thus, in [83] it was shown that the Mott and Peierls instabilities coexist both at 1   and at 1/ 2   .
In other words, a 1d Mott insulator also undergoes lattice deformation with the period / F k  . If we want to obtain a good organic conductor or even superconductoe we should stabilize the system with respect to the above transitions. All history of quasi-1d metal sysnthesis is, in fact, the history of fighting the above instabilities.
One of the effective means to fight the metalinsulator transitions is to shift electron density  from the values approaching 1, ½, ⅓ and other fractions with small denominators. This can be achieved by crystal doping with electron donors ot acceptors or by violation of a simple stoichiometric ratio. To understand why this simple and clear method is so efficient we shall discuss the instabilities and their descriptions for a system with a half-filled band with 1   in more details. The Mott metalinsulator transition.
The spectrum of a cyclic chain with Hamiltonian (143) is the spectrum of an insulator at any U > 0, that is, the excitation of states with charge transfer requires an energy E . For the first time a conclusion on the energy gap formation in such a system appeared in calculations by EHF method [55].
The Peierls metalinsulator transition. Let us consider a system with the Hamiltonian which can be represented as the sum of (139), (140), and (141) where a is a non-deformed lattice parameter, 0  is the phase of bond deformation, 2  F Qk , and The energy spectrum of conduction electrons for a half-filled band is given by where --‖ sign corresponds to a completely filled conduction subband, and -+‖ sign corresponds to the vacant subband. Thus, the gap in the one-particle spectrum is 0 8.   g ER  (148) The ground state energy correction is Some specific features of physics in one dimension remain valid also in two dimensions. Theoretical treatment of 2d-models is more complicated. For example, The Mott and Anderson metalinsulator transitions can occur also in quasi-2d systems. However, the Peierls transition in 2d case can appear only for special forms of the Fermi surface in the case of so called -nesting‖. Generally speaking, the conditions for the metalinsulator transitions in 2d-systems are stronger than those in 1d case. Passing to 2d-systems one can stabilize conducting and superconducting states.