The Virial Theorem and the Ground State Problem in Polaron Theory N. I. Kashirina a, V. D. Lakhno b, and A. V. Tulub c aInstitute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine e-mail: [email protected] bInstitute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, oblast, 142292 e-mail: [email protected] cSt. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034 Russia e-mail: [email protected] Received August 15, 2011

Abstract —The virial theorem for the translation-invariant theory of a polaron [3] is discussed. It is shown that, in [3], Tulub made a nonoptimal choice of variational parameters in the strong-coupling limit, which led to the violation of the virial relations. The introduction of an additional variational parameter to the test function reduces the polaron energy and makes it possible to satisfy the relations of the virial theorem for a strong-coupling polaron (the Pekar 1 : 2 : 3 : 4 theorem). DOI: 10.1134/S1063776112030065

It is well known that one can often establish some general relations between the mean values of the = ik⋅ r + Eint 〈Ψp | ∑(Vke ak H.c. )| Ψ p 〉 , kinetic, potential, and interaction energies for classi- cal and quantum systems, which are known as the vir- k ial theorem. The virial theorem holds for both exact and Ψp is the wave function of the ground state of a and approximate wave functions, provided that the polaron. In the strong-coupling limit, one more rela- functions are obtained by the variational method. tion is added to the virial relations (3): In particular, these general relations can be E = –2F , (4) obtained for a polaron on the basis of the Frölich ph p Hamiltonian: where ប2 = ∆r +ik⋅ r + + ប † † Hp –------(Vke ak H.c. ) ωakak, (1) E = 〈Ψ | បωa a |Ψ 〉 . 2m ∑ ∑ ph p ∑ k k p k k k where m is the electron effective mass, r is the electron In this limit, Fp coincides with the total energy of † coordinate, ak and ak are the creation and annihila- the ground state of the self-consistent state of a tion operators of phonons with energy បω, polaron (the thermal ionization energy) Ep = 2 1/2 〈Ψp | Hp|Ψp 〉 and the virial relations obtained in [1] cor- e 2πបω  –1 –1 –1 V =------, ε˜ = ε∞ – ε , (2) respond to the well-known Pekar 1 : 2 : 3 : 4 theorem k 2  0 k ε˜V for a strong-coupling polaron [2]. e is the electron charge, ε∞ and ε0 are the high-fre- In the translation-invariant theory [3], the field quency and static dielectric constants, and V is the vol- operators are subject to the translation transformation: ume of the system. + , The virial relations for the polaron problem for an ak ak fk arbitrary value of the electron–phonon interaction here the function fk describes the classical field com- constant are given by [1] ponent (polaron well). In [3], based on the Hamilto- =– = = Tp Fp, Eel 3Fp, Eint 4Fp, (3) nian (1), Tulub obtained the following expression for the ground-state energy E : where p =+ = + Fp Tp Eint /2, Eel Tp Eint , = + + 2 Ep ∆E 2∑Vk fk ∑f k (5) ប2 = – ∆r k k Tp 〈Ψp | ------|Ψp 〉 , 2m (as in [3], we put here ប = ω = 1). The quantities appearing in (5) have the following E 1 2 lim ---- 0 = inf -- dr ∇Ψ (r ) meaning: 2 ∫ α Ψ, Ψ = 1 2 (12) 2 = = = 1 2 –1 2 Tp ∆E, Eint 2∑Vk fk, Eph ∑f k, – ----- r r r r – r r , ∫d 1d 2 Ψ(1 ) 1 2 Ψ(2 ) k k 2 and should satisfy the virial relations (3) and (4). The where the functional on the right-hand side is the following expression was obtained in [3] for Eint with Pekar functional obtained in the strong-coupling T 2 2 limit. A rigorous proof of formula (12) was given in [5, the use of the test function f k = – Vkexp(– k /2 a ), where a is a variational parameter: 6]. Miyake’s result has been reproduced in a large number of works (see [7, 8]) and does not raise any doubts. = = – 2 2 Eint 2∑Vk fk ------g a, (6) In our view (the results of [3] have been checked π k once again), the only possible explanation to the aris- where ing contradiction is that one uses the wave functions 2 belonging to different function classes in the transla- 2 e m g =α = ------tion-invariant theory in the strong-coupling approxi- បε˜ 2បω mation and in the strong-coupling theory based on the wave functions that minimize the functional (12). In and α is the Frölich electron–phonon coupling con- stant. Accordingly, the expression the translation-invariant theory, the wave function of a polaron with zero total momentum is given by = 1 2 Eph ------g a (7)   2π T = – k † r ˆ Ψ0 exp i∑ akak  Φ({ak } ) |0 〉 ,   (13) was obtained for Eph . k It follows from (6) and (7) that T 2 = Ψ0 const, E /E = –2 2 . (8) int ph where the explicit form of the functional Φˆ is given in This expression contradicts the virial theorem (for- [3]. A transition to a localized description in the mula (4)). polaron problem (states in the theory with spontane- In [3], the following value was obtained for the ously violated translational symmetry) was also con- ground-state energy for a single variable parameter: sidered in [3] and yielded relation (9) for energy. The approximate wave function of the ground state defined = – 4 E0 0.105 g , (9) by functional (12) belongs to the class of localized, which corresponds to a higher energy of the polaron normalizable, functions. At the same time, a rigorous compared with that obtained by Miyake [4], substantiation of the delocalized character of the true wave function of a polaron in the ground state was = – 4 E0 0.10851128 g . (10) given in [9]. It is of interest to find a minimum of the total In conclusion, note that the value (11) for the energy within the translation-invariant theory in the ground-state energy of a polaron suggests that one class of functions satisfying the virial relations. Within should re-evaluate the stability criterion with respect this problem, we have found that the choice of the test to the parameter ηc = ε∞/ε0 for the bipolaron state T Ebp < –2 Ep above which there are no bipolaron states. function f k made in [3] is not optimal. Functional (5) For the parameter ηc obtained in [10] within the same attains its minimum for a function of the type f 0 = quantum-field approach as the polaron energy given k by (11), we obtain a value of = 0.179 instead of = T ηc ηc N f k in which two variational parameters are speci- 0.2496, which is calculated with the use of the polaron fied. The variational parameter N turned out to be energy (10). Since in [10], when solving a bipolaron 0 T problem, an optimization has been carried out with equal to N = 2 . For fk = f k = 2 f k , the virial rela- respect to the additional variational parameter, the vir- tions (3) and (4) are satisfied in a strong-coupling limit ial theorem for the bipolaron [11] is satisfied in [10] (the Pekar 1 : 2 : 3 : 4 theorem [2]), and, according to automatically. (5), the ground state energy takes the value = – 4 E0 0.1257520 g , (11) ACKNOWLEDGMENTS which is much lower than the best value given by (10). This work was supported by the Russian Founda- The value of E0 given in (10) is currently a well- tion for Basic Research, project nos. 11-07-12054 and established value and is determined from the relation 10-07-00112. REFERENCES 7. E. Lieb, Stud. Appl. Math. 57 , 93 (1977). 1. L. F. Lemmens and J. T. Devreese, Solid State Com- 8. N. K. Balabaev and V. D. Lakhno, Theor. Math. Phys. mun. 12 , 1067 (1973). 45 (1), 936 (1980). 2. S. I. 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