On the Possibility of Landau-Pekar Triplet Bipolaron Existance

A theoretical analysis of a homogeneous magnetic field the action on the electronic terms of the bipolaron in the triplet 3 , 2 z x iy P ± state is made. It is shown that the action of the magnetic field, on the one hand, leads to displacement of the triplet terms to the low-energy region and thus to stabilization of the bipolaron and, on the other hand, appreciable weakens the criteria that limit the conditions of their existence. It is shown that the inclusion of electron-electron correlat ion leads to the stabilization of the triplet b ipolaron, as well as to expand the boundaries of his existence.


Introduction
It was postulated previously [1] the possibility o f the existence in polar media of a bound two polarons (bipolaron) formations in a trip let (2 3 P) state whose structure resembles the heliu m atom in the excited , 1 2 z x iy s p ± state. Such two polarons species can be formed in polar med ia where autolocalization is associated with the formation of a fairly deep and wide polarization potential well. It was shown that polarons can exist in the ground and relaxat ion-excited states. As is well known [2], optical excitation of a polaron Franck-Condon transition occurs between the electronic 1s and , 2 z x iy p ± -states, the oscillator strength is ~ 0.9. At the same t ime the possible existence of singlet axially-symmetrical b ipolarons has been widely d iscussed [3][4][5][6][7][8][9][10]. The most probable dipole-actives the optical t ransition for ground state of Landau-Pekar bipolaron will be 1 1 2 For the special case * / 1.075 ε ε ∞ = (a polaron in ammonia), we obtain the transition frequency eV , which is very close to the experimental value 0.81eV [10].
The bipo larons in triplet state will have an ano malous chemical act ivity and, in v iew of the translational invariance of the system, can transfer the trapped energy quantum in a selected direction. The spin prohibition of the emission of a photon fro m a metastable state is lifted as a result of relativistic interactions, but this deactivation mechanism is inefficient. A lso possible are T→S (triplet→singlet) transition due to the interaction of the electron spin with the transverse phonons of the polar media. They lead to a change in the spatial symmet ry of the bipolaron. Fro m a centrally symetric state it changes to a singlet quasi-mo lecular bipolaron. However, in view of the small magnitude of the magnetic interactions, the spontaneous T→S transitions are of low probability. Triplet bipolarons may be treated as active centers having an inverted population relative to the ground singlet state.
It is known [11] that a magnetic field lowers the energy of a polarons and stabilizes the relaxation-excited states relative to the processes of nonradiative deactivation. The present paper analyzes the conditions of triplet bipolarons existence and the effect of a magnetic field on the energy terms bipolaron in t rip let state.

Basic Equations and Mathematical Method
In the adiabatic approximat ion the wave function of the system, the electron and the field of longitudinal longwavelength phonons of polar mediu m can be described as of a two-electron wave function ) , ( 2 1 r r χ and a wave function Φ of normal vibrations of a d ielectric continuum.
To determine the energy of the t wo polarons system it is necessary to minimize the functional under the additional conditions | 1 to describe the behavior of electrons in a polarizing mediu m we use the generalized Hamiltonian where P z is the co mponent of the total mo mentu m operator, K z is its eigenvalue, and z v is the mean translation velocity of the polaron moving along the z axis, where r j is the radius vector of the jth electrons; b q and b q + are the Bose operators of annihilation and creation of a quantum of longitudinal phonons with quasi-mo mentum q  and longitudinal eigenfrequency ω 0 (q = 0); and the Fourier coefficients The left-hand side of Eq. (5) can be rewritten as follo ws: We introduce the unitary transformat ion which converts Eq. (6) to a diagonal form.
The unknown functions f q and f q * are determined fro m the condition of min imu m of the functional S + HS. Considering the operator identities Varying Eq. (8) with respect to f q and f q * , we find the unknown functions Substituting Eq.(9) into Eq.(8) we obtain the total energy functional, which takes into account the translational displacement of the bipolaron as a whole Considering the cylindrical symmet ry of the problem, for slow velocity z v the electronic part of the wave function will be written in the form Substituting Eq. (11) into Eq.(10), we obtain the following equation for total energy of two-polaron system Assuming that , as well as the definitions Eqs. (1) and (11), for the eigenvalue of the momentum operator, we find (13) Substituting Eq. (13) into Eq. (12), we can readily obtain a general expression for the total energy of a bipolaron in a magnetic field allo wing fo r its translational displacement: where the longitudinal translational mass of the bipolaron is Thus, for slow velocities, the energy of translational motion can be separated from the intrinsic energy of the polaron. This makes it possible to consider 0 0 [ ] ε χ separately in Eq. (15). It is convenient henceforth to change fro m the q representation in Eq. (15) to a coordinate representation, and to introduce the one-electron spinless density function, which in the general case of an N-electron system may be written Then the self-consistent total energy of the stationary bipolaron formation may be written as follo ws: where M L is the projection of the orbital angular mo mentum L on the selected axis z, the cyclotron frequency is was considered, and the operator transformations of Ref. [12] were used.
We shall consider triplet singly excited electronic states (for distances R → ∞ the energy of the two polaron system corresponds to the , 1 2 z x iy s p ± + configuration). 1 2 is the distance between centers of polarons gravity [1,9], while the vector r i is associated with oscillat ion motion of electron in polarizat ion potential well near R i . The two-electron wave function of the electronic excited state, approximating the eigenfunction 1 2 ( , ) r r χ should be antisymmetric with respect to the transposition of both electrons orthogonal to the ground state. In the approximation of quasi-independent electrons, the electronic part of the two-center wave function (the centers of the potential wells are found at points a and b, separated by distance R) will be written in the Heitler-London form where N is the normalizat ion constant. Subscripts a and b refer to polarons located at R 1 and R 2 , respectively. χ 1s and 2 p m χ are one-particle wave functions of the 1s and 2p m states, and m = z, x±iy. The coordinate one-particle wave functions are chosen in the form of orthonormalized two-parameter Gaussian-type functions 3  , m l = ± 1.
The magnetic field removes the degeneracy with respect to the magnetic quantum nu mber M L of bipolaron, and the total energy Eq. (19) for the states 2 3 P z (M L = 0) and 2 3 P x±iy (M L = ± 1) may be written in the form The variational parameters can be found only numerically, and in the selfconsistent state the virial theorem should be satisfied. We introduce the scale transformation r → tr, R → tR, χ(r) → t 3/2 χ(tr) and then obtain for the function of the total energy (without magnetic field) ) ( where K is the mean kinetic energy of electrons, U 1 is the mean energy of interaction between electrons exchanging phonons, U 2 is diamagnetic part of the energy, and the mean Coulo mb interaction energy of electrons is U 3 . M inimizing Eq. (22) with respect to t and assuming t = 1, we obtain the virial relation 0 The virial relat ion Eq.(25) should be satisfied for all interpolaron distances.
The trip let bipolaron in 3 , 2 z x iy P ± state will be stable to adiabatic dissociation by two polarons with conservation of the total spin and orbital angular mo mentu m if the binding energy → ∞ are the self-consistent total energy of the polarons in 1s and 2p m states. Figure 1 shows the total energy of the bipolaron on the distance R for dielectric parameters * / With a decrease in the value of Q ratio decreases and reaches unity at * / ε ε ∞ = 1.15 (or / s ε ε ∞ > 7.67).
Let us consider the change in the total energy of the triplet bipolaron in the limit of a weak magnetic field : 3 1 U U << . In this case, we assume that the bipolaron has central symmetry ( β α = and η ξ µ γ = = = ). Hence, to determine the energy change, use may be made of the methods of perturbation theory, which for the 2 3 P z state gives the energy correction η are the parameters for H z = 0. The corresponding inequality can also be written for 3 2 x iy P ± term: , the energy change of the 2 3 P z term becomes mo re significant than that of the 2 3 P x±iy term, and for polar med ia with a s maller ratio of * / ε ε ∞ , this shift takes place in lower fields. , a variation calcu lation of the change in total energy terms is shown in Fig.2 Here we used the follow notation for t wo-center integrals: which is always valid, since V 1 > V 2 > 0, V 3 ≈ SV 1 , V 4 ≈ SV 2 and S also are all positive. Here