#### Abstract

Complex electronic ground state of molecular and solid state system is analyzed on the ab initio level beyond the adiabatic Born-Oppenheimer approximation (BOA). The attention is focused on the band structure fluctuation (BSF) at Fermi level, which is induced by electron-phonon coupling in superconductors, and which is absent in the non-superconducting analogues. The BSF in superconductors results in breakdown of the adiabatic BOA. At these circumstances, chemical potential is substantially reduced and system is stabilized (effect of nuclear dynamics) in the antiadiabatic state at broken symmetry with a gap(s) in one-particle spectrum. Distorted nuclear structure has fluxional character and geometric degeneracy of the antiadiabatic ground state enables formation of mobile bipolarons in real space. It has been shown that an effective attractive e-e interaction (Cooper-pair formation) is in fact correction to electron correlation energy at transition from adiabatic into antiadiabatic ground electronic state. In this respect, Cooper-pair formation is not the primary reason for transition into superconducting state, but it is a consequence of antiadiabatic state formation. It has been shown that thermodynamic properties of system in antiadiabatic state correspond to thermodynamics of superconducting state. Illustrative application of the theory for different types of superconductors is presented.

#### 1. Introduction

Superconductivity, an amazing physical phenomenon discovered nearly 100 years ago by Kamerlingh Onnes [1] and his assistant Gilles Holst whose name has been basically forgotten by history (see, e.g., [2]), has been one of the most important research fields of solid-state physics of last century and it remains until the present days. What is extremely irritating is the fact that microscopic mechanism of superconducting state transition, in spite of enormous attention which has been paid to this effect, remains still unclear and represents an open challenge for theory.

Until the discovery of high-temperature superconductivity of cuprates by Bednorz and Muller in 1986 [3] and synthesis of first 90 K superconductor [4] in 1987, understanding of microscopic mechanism of superconducting (SC) state transition formulated within the BCS theory in 1957 [5] was generally accepted as a firm theoretical basis behind the physics of this phenomenon. Here, the basic physics is the idea of Cooper-pairs formation, that is, formation of boson-like particles in momentum space, which are stable in thin layer above the Fermi level and drive system into more stable-superconducting state. Sufficient condition of pair formation is whatever weak, but attractive, interaction between electrons. Real possibility of effective attractive electron-electron interactions was well known. Some years, ago it was derived by Fröhlich [6, 7] as a consequence of electron-phonon (e-p) interactions.

The range of validity of the BCS theory with respect to e-p interactions has been specified by Migdal [8] and Eliashberg [9, 10]. It can be interpreted as Migdal’s theorem and Eliashberg’s restriction (ME approximation). The first is related to validity of the condition and the second one restricts the validity only for *1*, where is e-p coupling strength and and are characteristic phonon and electron energy scales, respectively. Expressed explicitly, BCS-like theories are valid only for adiabatic systems that obey the adiabatic Born-Oppenheimer approximation (BOA): .

While for conventional (low-temperature) superconductors, the BCS theory within the ME approximation (i.e., weak coupling regime) is an excellent extension of standard theory of metals, for high-temperature cuprates in order to interpret high critical temperature and ensure the pairs condensation, beside (or instead of) the e-p interactions the important role of other interaction mechanisms has been advocated (see e.g., [11–13]). Since the copper, a transition metal with incompletely filled d-shell when chemically bounded, is a central atom of hight- cuprates, it is quite natural that the attention has been focused on strong electron correlations (in a sense of standard Coulomb-repulsive e-e interactions), magnetic interactions, and/or spin fluctuation effects. At the present, the effect of Coulomb repulsion is usually incorporated via Coulomb pseudopotential and critical temperature is calculated according to McMillan formula [14].

The e-p interactions, which have been accepted to be responsible for electron pairing that drive transition into superconducting state for classical low- superconductors, have became nearly abandoned and considered to be rather harmful for superconductivity in high- cuprates (see, e.g., [12]). Some aspects of d-wave superconductivity can be described within the models of strongly correlated electrons, for example, Hubbard-like or t-*J* models (e.g., [11, 15–19]), even without explicit account for e-p interactions. The underlying leitmotiv behind the electron correlation treatments has been to understand the phase diagram of high- cuprates, that is, the doping process. Introduction of charge carriers (holes or electrons) into the parent antiferomagnetic insulator that causes transition to superconductor (or metal) has been generally accepted to be a universal feature of high- cuprates and believed to be a matter intimately related to microscopic mechanism of superconductivity.

Bell-like-shaped dependence of on hole doping in the doping range for family of high- cuprates is well known (see, e.g., [20] and references therein). With the exception of YBCO (), the optimal hole doping with maximal is The electron doping is usually less favorable, but basically either hole or electron doping can induce superconductivity. It is, for instance, the case of infinite-layer compound CaCu (itself is an insulator) where by hole or electron doping in field-induced transistor configuration superconductivity can be induced with up to 89 K (or 34 K for electron doping) at about 0.15 charge carriers per Cu.

Without any doubts, charge doping has no-negligible impact on e-e interactions and influences, to some extent, also more subtle spin interactions. Question remains if these are the key effects behind the physics which causes superconducting state transition upon doping?

In this respect, one has to realize that like , the lattice parameters and, what is very important, the lattice dynamics is strongly influenced by doping. For instance, dependence of on lattice parameter in case of Hg-based cuprates follows similar dependence as it does on hole doping [21]. In spite that isotope effect coefficient *α* for optimally doped cuprates is very small (, exception is YBCO—), doping in underdoped as well in overdoped region for O-isotope effect (all-over the cuprates family), results in a huge changes of isotope effect coefficient (see, e.g., [22, 23]. It is important experimental evidence that doping induces very strong changes in lattice dynamics, in particular in dynamics of Cu layers. From theoretical stand point, it means that mechanism of superconducting state transition is a complex matter of electron and nuclear degrees of freedom.

The results of high-resolution ARPES study [24, 25] of wide family of different high- cuprates have brought the other experimental evidence indicating that beside doping, an abrupt change (decrease) of electron velocity near Fermi level, at about 50–80 meV in nodal direction, is the other feature common to high-cuprates. The kink in nodal direction is temperature independent. More important, from the stand point of microscopic theory of superconductivity, seems to be formation of temperature-dependent kink on momentum distribution curve (dispersion renormalization) close to Fermi level () in off-nodal direction at transition to superconducting state. It has been reported for Bi2212 [26–29]. Recently, presence of the kink in off-nodal direction has been observed at VUV ARPES study with sub-meV resolution of optimally doped untwinned YBCO in superconducting state [30].

Formation of the off-nodal kink (dispersion renormalization) has been attributed by the authors [26] to the coupling of electrons to a bosonic excitations, preferably they consider a magnetic resonance mode such as observed in some inelastic neutron scattering experiments. The inconsistency in this interpretation has been pointed out by the authors of [27, 29]. The main points are described in [29] as follows: (a) magnetic resonance has not yet been observed by neutron scattering in such a heavily doped cuprates, and (b) magnetic resonance has little spectral weight and may be too weak to cause the effect seen by ARPES. They agree, however, with the opinion of the authors [26] that the renormalization effect seen by ARPES in cuprates may indeed be related directly to the microscopic mechanism of superconductivity. The authors of [27, 29] instead of magnetic resonance mode, attribute the dispersion renormalization to coupling with phonon mode, in particular, with -buckling mode of Cu plane. The temperature dependence of dispersion renormalization they attribute to the DOS enhancement due to SC-gap opening and to the thermal broadening of the phonon self-energy in normal state.

These results along with the results of neutron scattering [31, 32] indicate that also for high- cuprates the e-p coupling has to be considered as a crucial element of microscopic mechanism of SC state transition. Expressed explicitly, also in case of cuprates the role of phonons at superconducting state transition must not be overlooked. As soon as low-Fermi energy situation occurs (), one can expect important contribution of nonadiabatic vertex corrections at SC state transition. It is beyond the standard ME approximation and this problem has been studied within the nonadiabatic theory of superconductivity [33–35]. On the other hand, as the ARPES results indicate, electron kinetic energy is decreased and importance of proper treatment of electron-electron Coulomb interactions is increased. The competition between Coulomb and e-p interactions has been intensively studied within the Holstein-Hubbard models [36–40] with both interactions of short-range character. The obtained results are not satisfactory since heavy-mass polarons are formed that yield low values of . It has been improved within the Frohlich-Coulomb model [41] that introduces long-range repulsion between charge-carriers and also long-range e-p interactions. The results show that there is a narrow window of parameters of Coulomb repulsion and e-p interactions ( / ) resulting in the light-mass bipolarons formation. In this case, according to bipolaron theory of superconductivity [42–44], coherent motion of bipolarons represents the supercarrier motion and high can be reached.

The McMillan formula, which is very good approximation for of elementary metals and their alloys [45], is often used also for calculation of critical temperature of high- superconductors within the BCS-generic framework. It has been shown [46] that in strong-coupling regime , can be as large as . However, in reality there is problem with correct estimation of Coulomb pseudopotential and with unrealistically large values of that would match high experimental of novel superconductors. It has to be stressed, however, that strong coupling regime violates adiabatic condition of the ME approximation, which is behind the derivation of the McMillan formula.

More over, new class of superconductors, for example, cuprates, fullerides, and Mg are systems that are rather pseudoadiabatic with sizeable adiabatic ratio [47], in contrast to elementary metals, where adiabatic condition is perfectly fulfilled. This situation indicates the importance of nonadiabatic contributions at calculation of e-p interactions within the BOA, an effect which is beyond the standard ME approach. As mentioned above, formulation of the nonadiabatic theory of superconductivity by Pietronero et al. [33–35, 48, 49], which accounts for vertex corrections and cross phonon scattering (beyond ME approximation), has solved this nontrivial problem by generalization of Eliashberg equations. The theory, which is nonperturbative in and perturbative in , has been applied at simulation and interpretation of different aspects of high- superconductivity [50–54]. Basically, it can be concluded that accounts for nonadiabatic effects in quasiadiabatic state is able to simulate different properties of high- superconductors, including high-value of , already at relatively moderate value of e-p coupling, Moreover, it has also been shown that increased electron correlation is important factor that makes corrections to vertex function positive, which is in this context crucial for increasing .

Nonetheless, sophisticated treatment of high- superconductivity within the nonadiabatic theory faces serious problem related to possibility of polaron collapse of the band and bipolaron formation. According to bipolaron theory of Alexandrov [42–44, 47, 55–58], polaron collapse occurs already at for uncorrelated polarons and even at smaller value for a bare e-p coupling in strongly correlated systems. For , or and for at whatever small value , the nonadiabatic polaron theory has been shown to be basically exact [58]. Bipolarons can be simultaneously small and light in suitable range of Coulomb repulsion and e-p interaction [59]. These results have important physical consequences. There are serious arguments that effect of polaron collapse cannot be covered through calculation of vertex corrections due to translation symmetry breaking and mainly, polaron collapse changes possible mechanism of pair formation, that is, instead of BCS scenario with Cooper pair formation in momentum space, the BEC with mobile bipolarons (charged bosons) in real space becomes operative.

Discovery of superconductivity in a simple compound Mg at 40 K [60] has been very surprising and has started a new revitalization of superconductivity research. Beside the many interesting aspects, discovery of the Mg superconductivity is, in my opinion, crucial for general theoretical understanding of SC state transition on microscopic level. It is related to band structure (BS) fluctuation and dramatic changes of BS topology at e-p coupling.

The *σ* bands splitting at coupling to mode in Mg has been reported [61] already in 2001 but, with exception of possible impact of anharmonicity [62], no special attention has been paid to this effect. Superconductivity in Mg has been straightforwardly interpreted [63] shortly after the discovery as a standard BCS-like, even of intermediate-strong coupling character. For clumped nuclear equilibrium geometry, the BS is of adiabatic metal-like character. The of band electrons (chemical potential ) is relatively small, , but still great enough comparing to vibration energy of phonon mode, . Thought the adiabatic ratio is sizable, it is small enough in order to interpret superconductivity within the BCS-generic framework. It is supposed that nonadiabatic effects, anharmonic contributions and/or Coulomb interactions within generalized Eliashberg approach should be important in this case. On the other hand, the value of e-p coupling, indicates that polaron collapse can be expected and superconductivity should be of nonadiabatic bipolaron character rather than the BCS-like.

Nevertheless, the matter is even more complicated. It has been shown [64, 65] that analytic critical point (ACP––maximum, minimum, or saddle point of dispersion, in case of Mg it is maximum) of band at point crosses Fermi level (FL) at vibration displacement -atom, that is, with amplitude , which is smaller than root-mean square (rms) displacement () for zero-point vibration energy in mode. It means, however, that in vibration when ACP approaches FL on the distance less than , the adiabatic Born-Oppenheimer approximation (BOA) is not valid. In this case, Fermi energy of band electrons (chemical potential ) close to point is smaller than mode vibration energy and at the moment when the ACP of the band touches Fermi level, the Fermi energy is reduced to zero, . Moreover, shift of the ACP substantially increases density of states (DOS) at FL, , and induces corresponding decrease of effective electron velocity of fluctuating band in this region of -space. From the physical stand point, it represents transition of the system from adiabatic into true nonadiabatic , or even into strong antiadiabatic state with . This effect has crucial theoretical impact. At these circumstances, not only ME approximation is not valid (including impossibility to calculate nonadiabatic vertex corrections which represent off-diagonal corrections to adiabatic ground state) but adiabatic BOA itself does not hold as well.

The BOA is crucial approximation of theoretical molecular as well as of solid-state physics. It enables to solve many-body problem via separation of electronic and nuclear motion and to study electronic problem in a field which is created by fixed nuclei.

On the level of the BOA, the motion of the electrons is a function of the instantaneous nuclear coordinates , but is not dependent on the instantaneous nuclear momenta . Usually, and in solid state physics basically always, only parametric dependence is considered, that is, nuclear coordinates are only parameters in solution of electronic problem within the clamped nuclei Hamiltonian treatment. Nuclear coordinate-dependence, when explicitly treated, modifies nuclear potential energy by so called diagonal BO correction (DBOC) that reflects an influence of small nuclear displacements out of the equilibrium positions and corrects the electronic energy of clamped nuclear structure. The DBOC enters directly into the potential energy term of nuclear motion (but leaves unchanged the nuclear kinetic energy) and in this way modifies vibration frequencies. The off-diagonal terms of the nuclear part of system Hamiltonian that mix electronic and nuclear motion through the nuclear kinetic energy operator term are neglected and it enables independent diagonalization of electronic and nuclear motion (adiabatic approximation). Neglecting the off-diagonal terms is justified only if these are very small, that is, if the energy scales of electron and nuclear motion are very different and when adiabatic condition holds, that is, *1*. If necessary, small contribution of the off-diagonal terms can be calculated by perturbation methods as so called nonadiabatic correction to the adiabatic ground state.

Situation for superconductors seems to be substantially different, at least in case of the Mg. There is considerable reduction of electron kinetic energy, which for antiadiabatic state results even in dominance of nuclear dynamics () in some region of -space. In this case, it is necessary to study electronic motion as explicitly dependent on the operators of instantaneous nuclear coordinates as well as on the operators of instantaneous nuclear momenta . It is a new aspect for many-body theory.

The electronic theory of solids has been developed with the assumption of validity of the adiabatic BOA. In this respect, it is natural that different theoretical-microscopic treatments of superconductivity based on model Hamiltonians which stress importance of one or the other type of interaction mechanism, implicitly assume validity of the BOA, and it is very seldom that possibility of the BOA breakdown at transition to SC state is risen. The notion “nonadiabatic” effects in relation to electronic structure is commonly used for contributions of the off-diagonal matrix elements of interaction Hamiltonian (e.g., e-p coupling, e-e correlations, etc.) to the adiabatic ground state electronic energy calculated in second and higher orders of perturbation theory and does not account for true nonadiabatic-antiadiabatic situation, .

In this connection, a lot of important questions arise, as the following examples.

(i)How to treat antiadiabatic state?(ii)Can be system stable in antiadiabatic state?(iii)Are the physical properties of the system in antiadiabatic state different from the corresponding properties in adiabatic state? (iv)What is the driving force for adiabatic*↔*antiadiabatic state transition, that is, which type of interaction mechanism and at which circumstances trigger this type of transition?(v)How relevant is adiabatic

*↔*antiadiabatic state transition for SC state transition in Mg?(vi)Is the adiabatic

*↔*antiadiabatic state transition an accidental effect at SC state transition which is present only in MgB

_{2}, or this state transition is an inherent physical mechanism which is proper also for other superconductors?(vii)Can be adiabatic

*↔*antiadiabatic state transition relevant for high, as well as for low-temperature superconductors?(viii)Phonons or strong electron correlations? (ix)What is the character of condensate-Cooper pairs or bipolarons? (x)Is there any relation of the adiabatic

*↔*antiadiabatic state transition to Cooper pairs formation? (xi)Cooper pairs or correction to electron correlation energy?

Theoretical aspects related to the above problems have been elaborated and discussed in details within “Ab initio theory of complex electronic ground state of superconductors’’, which has been published in the papers, [66–68]. The main theoretical point is generalization of the BOA by sequence of canonical-base functions transformations. The final electronic wave function is explicitly dependent on nuclear coordinates and nuclear momenta , or alternatively, emerging new quasiparticles-nonadiabatic fermions are explicitly dependent on nuclear dynamics. As a result, the effect of nuclear dynamics can be calculated in a form of corrections to the clamped nuclei ground state electronic energy, to the one-particle spectrum and to the two-particle term, that is, to the electron correlation energy.

*Note. *To avoid confusion, it should be stressed that the notion electron correlation energy as used in this paper stands for improvement of e-e interaction term contribution beyond the Hartree-Fock (HF) level, (), as it can be calculated, for example, by (1/r)-perturbation theory in the second and the higher orders, or by configurations interaction method. In condensed matter physics, electron correlation usually stands for an account for Coulomb e-e interaction at least on Hartree or HF level. On the HF level not only repulsive e-e term is present, like on Hartree level where spin is not considered at all, but also exchange term, fermion Coulomb-hole only for electrons with parallel spins. Correlation energy improves an account for unbalanced treatment of e-e interaction for electrons with parallel and antiparalel spins on HF level.

It has been shown that due to e-p interactions, which drive system from adiabatic into antiadiabatic state, adiabatic symmetry is broken and system is stabilized in the antiadiabatic state at distorted geometry with respect to the adiabatic equilibrium high symmetry structure. Stabilization effect is due to participation of nuclear kinetic energy term, that is, it is the effect of nuclear dynamics (dependence on ) which is absent in the adiabatic state on the level of the BOA. The antiadiabatic ground state at distorted geometry is geometrically degenerate with fluxional nuclear configuration in the phonon modes that drive system into this state. It has been shown that while system remains in antiadiabatic state, nonadiabatic polaron-renormalized phonon interactions are zero in well defined -region of reciprocal lattice. Along with geometric degeneracy of the antiadiabatic state, it enables formation of mobile bipolarons (in a form of polarized intersite charge density distribution) that can move over lattice in external electric potential as supercarriers without dissipation. Moreover, it has been shown that due to e-p interactions at transition into antiadiabatic state, -dependent gap in one-electron spectrum has been opened. Gap opening is related to the shift of the original adiabatic Hartree-Fock orbital energies and to the -dependent change of density of states of particular band(s) at the Fermi level. The shift of orbital energies determines in a unique way one-particle spectrum and thermodynamic properties of system. It has been shown that resulting one-particle spectrum yields all thermodynamic properties that are characteristic for system in superconducting state, that is, temperature dependence of the gap, specific heat, entropy, free energy, and critical magnetic field. The -dependent change of the density of states at the Fermi level in transition from adiabatic (nonsuperconducting) into antiadiabatic state (superconducting) can be experimentally verified by ARPES or tunneling spectroscopy as spectral weight transfer at cooling superconductor from temperatures above down to temperatures below .

Results of the ab initio theory of antiadiabatic state have shown that Fröhlich’s effective attractive electron-electron interaction term represents correction to electron correlation energy in transition from adiabatic into antiadiabatic state due to e-p interactions. Analysis of this term has shown that increased electron correlation is a consequence of stabilization of the system in superconducting electronic ground state, but not the reason of its formation.

In the present article, the key points of the theory are recapitulated and the adiabatic *↔* antiadiabatic state transition is shown to be operative for different types of superconductors.

#### 2. Electronic Ground State Beyond the Born-Oppenheimer Approximation

##### 2.1. Preliminary Remarks

Development of the theory of molecules and solids has been enabled due to fundamental approximation, the Born-Oppenheimer approximation (BOA). With respect to electronic structure of superconductors and transition to superconducting state, some aspects of this approximation should be outlined at the beginning.

Solution of the Schrödinger equation of many-body system composed of electrons and nuclei (total system)

with the Hamiltonian

and wave function , which is a general function of the sets of electron and nuclear Cartesian coordinates, is possible at the assumption of validity of the Born-Oppenheimer approximation (BOA). The BOA, originally formulated in the work in [69] by power expansion of potential surface for nuclear motion at equilibrium geometry with respect to displacement and electron/nuclear mass ratio , has been reformulated later by Born [70, 71] in a more practical form. According to it, the wave function of the total system (1) can be expressed in the factorized form

as a linear combination of known adiabatic electronic wave functions that are the eigenfunctions of clumped nuclei electronic Schrödinger equation

with the electronic Hamiltonian:

at fixed nuclear configuration . Expansion coefficients in (3), regarded as unknown, are nuclear wave functions for nuclear configuration with the electronic subsystem in particular adiabatic electronic state .

The -dependence in (5) is only parametric and in general it should be calculated over the full configuration space in order to calculate the adiabatic potential hypersurface of nuclear motion.

With respect to (3) and (4), the Schrödinger equation of the total system (1) for electronic state can be written in the form

with the term , which couples electronic and nuclear motions, on the right-hand side (rhs):

The term in (6):

is total adiabatic electronic energy, that is, adiabatic electronic energy plus nuclear Coulomb repulsion at nuclear configuration *R*.

Until the Born approach (3) is valid, (6) is exact and it still describes coupled motion of electrons and nuclei over the term (7), , which represents possibility of transitions between different adiabatic electronic states, , due to nuclear motion (-dependence). If such transitions are forbidden from the symmetry reasons, or if there is physically reasonably justified assumption that contributions of such transitions are negligibly small, then one can omit the rhs term, and (6) can be written in the diagonal form:

where is the only a nonzero diagonal contribution of the Λ term

The term (10) is the mean-value of the nuclear kinetic energy for adiabatic electronic state at nuclear configuration and represents so called adiabatic diagonal Born-Oppenheimer (DBOC) correction to the total adiabatic electronic energy . However, (9) is then the equation of motion of nuclei and it has the form of Schrödinger equation with Hamiltonian:

The effective-adiabatic potential for nuclear motion:

is represented by the total electronic energy (8), that is, Coulomb potential energy of the bare nuclei repulsion and adiabatic electronic energy , which is corrected by mean-value of the nuclear kinetic energy (DBOC) for the particular adiabatic electronic state (10). Contribution of the adiabatic DBOC is usually neglected as very small quantity, mean value of kinetic energy of slowly moving heavy nuclei in contrast to fast motion of light adiabatic electrons.

At these circumstances, the motion of electrons and nuclei is effectively decoupled, that is, it is possible to realize an independent diagonalization of the electronic Schrödinger equation (4) and nuclear Schrödinger equation (9), electrons and nuclei of the system behave like two statistically independent sets. Assumed small contribution of the rhs term of (6) can be calculated by some approximate way, usually by perturbation theory. Then, problem with the Hamiltonian:

is studied, where

is unperturbed part, and

is a small perturbation.

In practice, physical and/or chemical properties of a many-body system in its ground electronic state are of the prime interest. In this case, the Born approach (3) is usually restricted to the single term and the total wave function of system is a simple product of the adiabatic electronic wave function and corresponding nuclear wave function:

The Born approach in the form (16) is the adiabatic approximation. In a common sense, what is usually called the Born-Oppenheimer approximation (BOA, or clamped nuclei-crude adiabatic BOA) is the adiabatic approximation where the contribution of the DBOC (i.e., term in (12)) is also neglected and wave function (16) has the form

Contributions of the off-diagonal terms in (15) which are calculated as a small perturbation to the Hamiltonian (14) represent a nonadiabatic correction to the unperturbed adiabatic ground state. The conditions at which the nonadiabatic correction can be expected to be small, and the BOA (16, and in general 3) is valid, can be estimated by analysis of the second-order contributions to the energy of the total system, , which are small providing that

It can be derived (see, e.g., [72]), based on the expansion of the effective nuclear potential at equilibrium nuclear geometry (where the total electronic ground-state energy reaches its absolute minimum) at least up to the quadratic term in a displacement :

The result is that (18) holds and the BOA (16), (3) is valid, if inequality

is fulfilled for electronic and vibration (phonon) energy spectrum of a system. The meaning of (20) is clear, the electronic spectrum, that is, the differences between the total electronic energies of the excited electronic states and the ground-state energy has to be much greater than vibration (phonon) energy spectrum of system.

The long-time experience of theoretical molecular and solid-state physics has shown that for a ground electronic state of vast majority of molecular systems and solids at equilibrium geometry , the BOA is absolutely good approximation. In case of solids, it has enabled to derive (Bloch’s assumption of small perturbation of the periodic lattice potential for small nuclear displacement out of equilibrium) the field theory Hamiltonian of total system (13) in a well-known form, , with

The perturbation Hamiltonian represents now, instead of nuclear kinetic energy Λ-perturbation term (15), an electron-phonon (e-p) interaction term:

In this case, no special attention has been paid to the diagonal correction (10) to the total electronic ground-state energy, and at derivation of (21) and (22), this correction has been omitted as negligibly small quantity. Moreover, since perturbation is now introduced as an e-p interaction , it is immediately seen from the form of (16) and (22) that the first-order correction, that is, diagonal perturbation term, equals zero: . All interesting physics is then related to higher-order contributions with participation of excited electronic states, that is, the first possible nonzero contributions are in the second order of perturbation theory, that is, terms of the form .

Hamiltonian (21), (22) has been the starting point for theoretical study of the effects connected to e-p interactions in solids. By means of the famous unitary transformation of this Hamiltonian, Fröhlich has derived [6, 7] an effective electron-electron interaction term which is a crucial element at formulation of the BCS [5] or Migdal-Eliashberg (ME) theory of superconductivity [8–10]. The extension of the ME theory, that is, inclusion of higher-order contributions of e-p interactions by means of Feynman-Dyson perturbation expansion of (vertex corrections) is the basis for so called “nonadiabatic” theory of superconductivity, as it has been developed by Pietronero et al. [33–35, 48, 49]. It should be pointed-out that not only the above-mentioned standard and “nonadiabatic” theories of superconductivity but also other theoretical models of superconductivity which do not consider explicitly e-p interactions assume tacitly validity of the BOA as soon as the model Hamiltonian is written in the form

no matter what perturbation represents.

The reason for short sketch of the BOA in the introductory part is to attract an attention toward some aspects, which at study of solids, in particular of superconductors, are tacitly assumed to hold implicitly and seemingly there are no indications raising doubts that this class of solids should be an exception.

The main aspect is the *R*-dependence of the BOA. Whatever trivial it seems to be, it has to be stressed that the study of many-body system by means of the clumped nuclei Hamiltonian (21)–(23) requires validity of the Born ansatz (16), (3), that is, fulfillment of (20) to hold not only for equilibrium nuclear geometry but also for displaced geometry . Displacement has to be as large as to cover full configuration space experienced by nuclei in the vibration (phonon) modes of system. At 0 K temperature, has to be greater than (or at least equal to) root-mean square displacement of vibration (phonon) mode () at zero-point energy:

It means that for validity of the BOA, the inequality for the energy spectrum which, along with (20), has to be also fulfilled is

In this context, the adiabatic electronic energies of the ground and excited electronic states , at fixed nuclear geometry , need also some comment. Without the loss of generality, the Hartree-Fock (HF) calculation scheme in direct-real space orbital representation can be assumed. The adiabatic wave function of the electronic ground state is represented by the Slater determinant that satisfies requirement of antisymmetry:

where for simplicity a closed-shell, electron system, is considered. In (26), lowest laying spatial orbitals are occupied, each being occupied twice, once by an electron with *α* spin, , and once with *β* spin, . Adiabatic total electronic energy of the ground state is then the sum of adiabatic electronic energy and Coulomb repulsion of bare nuclei at fixed geometry :

Summation indices in (27) run over the occupied states .

Set of in (27), is one-electron spectrum of the adiabatic electronic ground state (26), that is, orbital energies are eigenvalues:

of the Fock operator :

and is the set of corresponding eigenfunctions (orthonormal set of optimized orbitals). In (28), is diagonal element of one-electron core Hamiltonian (integrals of electronic kinetic energy term plus electron-nuclear Coulomb attraction term), and *, * are 2-electron (1,2,3, or 4 nuclear center) Coulomb repulsion and exchange integrals.

In the next step, instead of independent calculation of electronic excited states energies , which would require new optimization of excited state wave functions (it should be extremely complicated since excited state wave functions have to be orthogonal to the ground state wave function), in practice a fairly good approach is used which is based on the orbitals already optimized for the ground-state wave function . Optimization is always performed within a basis set with some finite number of basis set functions , while . Diagonalization of (29) then yields eigenfunctions and corresponding eigenenergies , but only lowest-lying spatial orbitals are occupied and remaining are virtual—unoccupied orbitals . By promotion of electron(s) from occupied orbital(s) to virtual—unoccupied orbital(s) , excited state(s) configuration(s) as a linear combination of corresponding Slater determinants can be constructed. It can be shown that, for example, single-electron excitations yield two excited state electronic configurations—lowest lying excited state, that is, singly excited triplet state , and singly excited singlet state . Differences in the electronic energies of these excited state configurations with respect to the electronic energy of the ground state are as follows:

(i)for singly excited triplet state,(i)for singly excited singlet state,

From (30), (31), it is clear that for approximations which do not consider explicitly for two-electron terms, the differences in energies of singly excited triplet and singlet states with respect to the ground-state energy are the same:

Multiple electronic excitations can be calculated in a similar way, by generation of Slater determinants of -particle, ()-hole states in the language of particle-hole formalism.

The important point is that clumped nuclei electronic ground-state energy calculation provides approximate information about electronic excited states over the one-electron spectrum which corresponds to the electronic ground state. In particular, over the optimized set of occupied () and unoccupied-virtual () spinorbitals.

In the simplest form, with respect to (32), the inequalities (20), (25) which have to be valid for save application of the BOA, are in the terms of the optimized ground state orbital energies expressed as follows:

(i)for system at equilibrium geometry ,has to hold for the couple of frontier orbitals, that is, for highest occupied and lowest unoccupied orbitals;

(i)more over,
has to hold for the same system at displaced nuclear configuration with respect to particular vibration(phonon) mode *ν*.

In case of solids with quasicontinuum of states, in momentum *k*-space representation, these inequalities can be rewritten in the form

for equilibrium nuclear geometry, and

for the same system at displaced nuclear configuration in particular phonon mode .

These inequalities have to be valid in a multiband system for each band and energies of analytic critical points (ACP—i.e., absolute and local minima and maxima or inflex points) located on the particular band-dispersion curve of the first BZ with respect to the energy of the Fermi level .

An important aspect should be reminded at this place. Inequalities (33)–(36) have been derived on the basis of the expansion (19) of the effective nuclear potential (i.e., total electronic energy ) at . With respect to the inner structure of the total electronic energy (27), it is clear that in principle, the expansion (19) can be performed (i.e., smallness in energy change as to ensure at least harmonic approximation) even if displacement yields significant changes in the one-electron spectrum (orbital energies) of the ground state

provided that these changes are well balanced by changes and , eventually also by changes if two-electron terms are explicitly included. The main point is that for the electronic ground state, the one-electron spectrum at , , can be significantly different from the one-electron spectrum at displaced nuclear configuration for some phonon mode *ν*.

In respect of it, possibility that displacement yields for some band situation when

even if for the equilibrium geometry (35) holds, is not excluded at all.

The reason of possibility for such substantial changes of the electronic (band) structure is hidden in the chemical composition and structure of particular system. With respect to (27)–(31), it is connected to the changes of one-electron core part () and/or two-electron Coulomb repulsion and exchange integrals (). Which of these are dominant (hopping terms, onsite/intersite repulsion, electron correlations, dynamic screening, etc.) can hardly be determined on the basis of model Hamiltonians without invoking for complete chemical composition and structure of the particular system.

Occurrence of such a situation means that nuclear motion (nuclear vibration, in particular, phonon mode) has induced sudden decrease of effective electron velocity in particular band close to analytic critical point at the Fermi level . Or, what is equivalent, the effective mass of electrons have been increased in this region. In any case, however, it means that at displaced nuclear geometry , electrons, and nuclei of the system do not move as independent particles, that is, in this region, electrons are not able to follow nuclear motion adiabaticaly. The motion of electrons and nuclei is now strongly correlated. The new situation occurs, the adiabatic BOA is not valid for nuclear geometry , and wave function cannot be factorized in the form (16) and (3), that is,

and standard adiabatic form of the Hamiltonian (21)–(23) cannot be applied to study an electron-phonon problem. What is crucial, however, is the fact that the nonadiabatic effect cannot be calculated as a perturbation, that is, as a nonadiabatic correction to the adiabatic ground state at . In this case, system is in true nonadiabatic or antiadiabatic state.

##### 2.2. Generalization of the BOA Beyond Adiabatic Regime

In order to solve the problems which are sketched above, one needs to study electronic motion as explicitly dependent on nuclear dynamics, that is, dependent on instantaneous nuclear coordinates and nuclear momenta . It means that electronic state has to be explicitly dependent on nuclear operators and . Approximate solution of this problem can be found in [66]. It is based on the sequence of basis functions transformations, starting with fixed basis set (*, *-independent) over -dependent base up to -dependent basis set. A base function transformation is canonical transformation and it ensures that each transformation step preserves corresponding statistics. Moreover, since we start from fixed basis set (clamped nuclei situation) which enables factorization of total system wave function, this property is required to be preserved at each transformation step up to quadratic terms in and . Consequently, it makes the solution of , -dependence tractable. It should be stressed that base functions transformation is equivalent to the quasiparticle transformation which we have formulated in 1992 [67].

Main aspects and the results of this treatment are presented below.

General nonrelativistic form of system Hamiltonian (2) can be written in second quantization formalism as

Nuclear potential energy and one-electron core term (electron kinetic energy plus electron-nuclear coulomb attraction term) are functions of the nuclear coordinate operators (normal modes nuclear displacements out of fixed nuclear geometry ) and nuclear kinetic energy is a quadratic function of the corresponding nuclear momenta operators . For nuclear coordinate and momentum operators hold: The are standard fermion and boson annihilation and creation operators, respectively.

The -dependence of terms and in (40) can be expressed through the Taylor’s expansion at fixed nuclear configuration :

The term represents potential energy of bare nuclei at fixed nuclear configuration , is one-electron core term (electron kinetic energy plus electron-nuclear coulomb attraction term) at fixed nuclear configuration *R _{0}* and terms are related to matrix elements of electron-vibration (phonon) coupling , that is,

Two-electron terms (electron-electron coulomb repulsion and exchange integrals) do not depend explicitly on the nuclear operators.

###### 2.2.1. Crude-Adiabatic Approximation: Clamped Nuclei Solution

Since the crude-adiabatic approximation is the reference level for study the effect of nuclear dynamics on electronic structure, some details should be introduced.

With respect to sequence of transformations, which will be presented in the following parts, let as distinguish particular representation; that is, dependence of electronic states on nuclear operator only by double-bar over particular operator symbol (e.g., ,), dependence on both and will be without bar () and for fixed basis (independent on and , resp.) the operators will be with a single-bar () over the particular symbol. In this sense, the operators in crude-adiabatic approximation are written as single-bar operators, that is, ().

Provided that phonon and electronic energy spectrum are well separated and (20) holds for relevant configuration space near to , crude-adiabatic (clamped nuclei) treatment is justified. Electronic Hamiltonian is now only parametrically dependent on nuclear configuration, that is, nuclear geometry is fixed at nuclear configuration :

Application of Wick’s theorem to the product of creation and annihilation operators yields for particular terms the normal product form (N-product) with corresponding contractions:

Simultaneous diagonalization of electronic and nuclear part of system Hamiltonian, that is, factorized form of system wave function (57) within adiabatic BOA implies also validity of the following commutation relations:

(1) *Q-Dependent Adiabatic Transformation. *

In case of crude-adiabatic approximation, the electrons “see” the nuclei at theirs instantaneous positions at rest and nuclei do not “feel” internal dynamics of electrons. Within the spirit of the BOA, it would be correct if the electrons follow nuclear motion instantaneously, that is, electronic state has to dependent explicitly on instantaneous nuclear positions. In this case, the wave function of the system, instead of the form (57) with -independent electronic part should be replaced by -dependent form, that is,

Adiabatic, nuclear displacement -dependent electronic wave function in (62) assumes existence of complete orthonormal basis set , that is, validity of the following relations:

Now, electron creation and annihilation operators which correspond to the *Q*-dependent moving base are denoted as double-bar operators (). Also, the boson operators related to the -dependent moving base are written as double-bar operators:

Then,

Since adiabatic electrons remain fermions, the operators have to obey standard fermion anticommutation relations:

Shorthand notation, , has been used in (66).

Crude-adiabatic electronic wave function which does not depend on the nuclear displacements is expanded over fixed basis set with spinorbitals that are eigenfunctions of clamped nuclear electronic Hartree-Fock equations (30). This is complete and orthonormal basis set

Crude-adiabatic fermion creation and annihilation operators that correspond to the fixed basis set are written as single-bar operators (), that is,

These operators pertain to ordinary (crude-adiabatic) electrons and the standard anticommutation relations (61a) hold.

Due to properties (63), (67), the two bases are interconnected by the base transformation of the following form:

Now, for fermion operators in -dependent moving base one can write

Elements of the -dependent transformation matrix in (69), (70) are

Since

then due to closure property and orthonormality (63), (67) of the bases, it can be derived that base transformation matrix is an unitary matrix:

It can be shown [66] that the base transformation is identical with canonical transformation of operators (see (70) and (74))

It ensures preservation of statistics, that is, validity of the anticommutation relations (66) for new, -dependent, adiabatic electrons. The exponential form of canonical transformation (74) legitimates Taylor’s expansion of the matrix elements (71) and (72) of base transformation matrix, that is,

The form of transformation relations for boson operators of system Hamiltonian has to respect the factorized form of the total system wave function (62). It implies possibility of simultaneous, independent diagonalization of electron and boson subsystems. It means that transformed fermion and transformed boson operators obey not only standard anticommutation and commutation relations within the individual subsystems:

but also transformed operators of both subsystems have to commute mutually like the original operators, that is, also the following commutation relations are required to hold:

With respect to the fermion transformation relations (70), the form of transformation relations for boson operators is expressed as

Also for matrix elements of transformation matrix , the Taylor’ expansion is defined as

In order to ensure possibility of practical solution, in what follows, important restriction is imposed. The commutation relation (77) is required to hold up to quadratic terms in Taylor’s expansions. It enables to express transformation coefficients through coefficients (see [66]—Appendix ). It will be shown that covers the strength of electron-vibration (phonon) coupling up to the first order of Taylor’s expansion and determine also adiabatic correction to the electronic energy of the ground state .

The adiabatic transformation preserves total number of electrons, and nuclear coordinate operator is invariant under the transformation, that is,

Up to the first order of Taylor’s expansion, the crude-adiabatic momentum operator is transformed as

The term in (81) is nuclear momentum operator on adiabatic level.

For adiabatic *Q*-dependent spinorbitals , which are the basis functions of the adiabatic -dependent electronic wave function of the ground state , expressed over crude-adiabatic orbitals can be derived:

As it is seen from (82), adiabatic wave function is modulated by the instantaneous nuclear coordinates of particular vibration (phonon) modes with the weight proportional to transformation coefficients (coefficients of transformation matrix in the first order of Taylor’s expansion—).

At solution of the problem on adiabatic (-dependent) level, we have restricted ourselves to study total system in its electronic ground state with (62) representing wave function . The Schrödinger equation of the total system (6) for electronic ground state is then of the diagonal form (9). Seemingly we have lost the effect of electron-nuclear coupling through nuclear kinetic energy operator which is covered by terms on the rhs of equation (6). The off-diagonal terms are absent and from the diagonal terms, the only non-zero element is . It is the DBOC, , the mean value of nuclear kinetic energy in the electronic ground state which is expected to be negligibly small for systems in adiabatic state. Nonetheless, it will be shown that this term () covers the same effect of e-p coupling as it is routinely calculated in solid-state physics by perturbation theory with e-p coupling Hamiltonian (22) when electronic excited states are approximated by promotion of electrons to virtual orbitals of the electronic ground state.

(2) *QP-Dependent Nonadiabatic Transformation. *

As it has been mentioned in Section 1, study of band structure of superconductors indicates that e-p coupling induces fluctuation of some band through Fermi level. At the moment when ACP of such a band approaches Fermi level, there is considerable reduction of electron kinetic energy, which for antiadiabatic state results even for dominance of nuclear dynamics () in some region of -space. Electrons at these circumstances are not able to follow nuclear motion adiabatically. It means that electronic wave function, in order to respect this fact, should be dependent not only on instantaneous nuclear coordinates but it should also be an explicit function of the instantaneous nuclear momenta , that is, .

Let us assume that wave function of total system can be found in the following factorized form:

The form of the wave function (83) is basically *P*-dependent modification of the original -dependent BOA (3).

Like in adiabatic case, solution of the problem will be restricted to electronic ground state, that is, for total system, we have

It means that effect of nuclear momenta will be covered only in the form of -dependent diagonal correction , that is, in a similar way as it has been covered the effect of instantaneous nuclear coordinates on the adiabatic level, that is, -dependent adiabatic DBOC, .

Solution of this problem is similar to the transition from crude-adiabatic to adiabatic level as presented above. Now, however, the transition from adiabatic to antiadiabatic level is established.

Nonadiabatic, nuclear displacements, and momenta ()-dependent electronic wave function in (84) assume existence of complete orthonormal basis set , that is, validity of the following relations:

Electron creation and annihilation operators that correspond to the ()-dependent moving base are written as bar-less operators (). Boson operators related to the ()-dependent moving base are denoted also as bar-less operators, and . Then,

Since nonadiabatic electrons remain fermions, the operators obey standard fermion anticommutation relations (it follows from canonical transformation, see (91)):

In (87), shorthand notation is used, that is, and .

Since adiabatic -dependent moving base derived by adiabatic transformation is complete and orthonormal (63), then due to (85), the base transformation to nonadiabatic ()-dependent moving base can be established over the base transformation relation:

For fermion operators of second quantization, it follows that

Elements of the -dependent transformation matrix are

The -dependent transformation matrix is also unitary matrix, that is, the relations hold:

The form of transformation relations for boson operators of system Hamiltonian has to respect again the factorized form of the total system wave function (84). Also in this case, it expresses possibility of simultaneous, independent diagonalization of electron and boson subsystems. It means that transformed-nonadiabatic fermion and transformed nonadiabatic boson operators obey not only standard anticommutation and commutation relations within the individual subsystems,

but also, like the original and adiabatic operators, transformed nonadiabatic operators of both subsystems have to commute mutually, that is, also commutation relations have to hold:

With respect to the fermion transformation relations (89), the form of transformation relations for boson operators that fully respects conditions (92) is

Again, in order to enable practical solution, the commutation relation (93) is required to hold up to quadratic terms in Taylor’s expansions. This restriction enables to express transformation coefficients trough coefficients , see [66]—Appendix .

It can be shown that also this transformation preserves the total number of particles, that is,

Invariant of transformation is now momentum operator:

However, coordinate operator is transformed up to first order of Taylor’s expansion, as

that is,

For nonadiabatic ()-dependent spinorbitals which are the basis functions of the nonadiabatic ()-dependent electronic wave function of the ground state , expressed over crude-adiabatic orbitals can be derived:

Nonadiabatic wave function (99) in contrast to adiabatic wave function (82) is modulated not only by the instantaneous nuclear coordinates of particular vibration (phonon) modes but modulation is also over corresponding instantaneous nuclear momenta . The weight of momentum modulation is proportional to the -dependent transformation coefficients . It represents first derivative of matrix element with respect to nuclear momentum , that is, coefficient of transformation matrix in the first order of Taylor’s expansion. It will be shown that these coefficients reflect not only the strength of e-p coupling but mainly the magnitude of nonadiabaticity. For true nonadiabatic situation, that is, for antiadiabatic state , the weight of such -modulated state can be significant.

###### 2.2.3. Solution of Nonadiabatic Problem: Corrections to Energy Terms

(1) *Transformations of System Hamiltonian *

Base functions transformations have incorporated dependence of electronic states on operators of nuclear motion and vice versa. It implies, before the system Hamiltonian transformations, necessity to rearrange starting crude-adiabatic Hamiltonian:

into more convenient form.

Let us formally divide this Hamiltonian on two parts, and .

() The nuclear part , as we already know, is quantized on crude-adiabatic level as

In general, this part can be written as the sum of nuclear kinetic and nuclear potential energy:

whereas

The standard, usually harmonic-quadratic part of the nuclear potential energy, is corrected now by some, yet unknown, potential energy term that is, supposed to be also quadratic function of nuclear coordinate operators. Evidently, the term is absent on crude-adiabatic level. This correction originates from the interaction of vibrating nuclei with electrons on adiabatic -dependent level. In general, kinetic energy of vibration motion can also be corrected by some, yet unknown quadratic function of nuclear momenta operators-, that is,

On the adiabatic *Q*-dependent level kinetic energy correction is negligibly small and it is neglected. It becomes important only when the BOA is broken, , that is, in the case when electrons due to increased effective mass are not able to follow nuclear motion adiabatically and electronic states are -dependent. It should be stressed that both corrections are absent on crude-adiabatic level, they have been introduced just with respect to subsequent adiabatic and nonadiabatic transformation of system Hamiltonian.

() The second part of divided Hamiltonian is

It is evident that the division on the two parts is only formal

The reason of it is mainly pragmatic with respect to transformations and final solution. In this way the total system Hamiltonian is divided on quasi-bosonic (nuclear part ) and quasifermionic (electronic ) Hamiltonians.

The system Hamiltonian (in the form (105)) is now prepared for canonical transformations. The frequently used form of canonical transformation,

is applied in order to get diagonal, or “more diagonal”, final form at least up to first order of commutation expansion, . This kind of transformation changes the form of the Hamiltonian, but leaves unchanged the system variables.

Base functions transformations, as presented in preceding parts, introduce new dynamical variables; starting from crude-adiabatic , new adiabatic and nonadiabatic quasiparticles are introduced. At these circumstances, canonical transformation of Hamiltonian means introduction of new dynamical variables

Unlike to (106), now the Hamiltonian is not transformed itself, it remains of the original form, but its variables () are replaced by new variables :

The Hamiltonian written in new variables is

Since at this transformation, there is no any requirement for fulfillment of condition like at (106), the transformation does not make Hamiltonian “more” diagonal. Usually it is more complex, but very often it discloses physical aspects of the problem that are not obvious from nontransformed form with the original variables of system Hamiltonian.

The system Hamiltonian (105) is now subjected to this type of transformation.

In the first step, transformation from crude-adiabatic to adiabatic quasiparticles is realized.

The adiabatic quasi-particle transformations, up to the second order of Taylor’s expansion, generate terms and whereas , , in subscript stand now for the *i*th order of Taylor’ series expansion in adiabatic representation, the th order of the coordinate operator and the th order of momentum operator within the given order of Taylor’s expansion; that is, .

Up to the second-order expansion in *, * and in second-order expansion of the operators of quasiparticle transformation (), the and parts are transformed in the following way:

The particular form of the terms (110a)–(110e) and details of adiabatic transformation are in [66, Appendix A], [67].

In the next step, the adiabatic form (terms (110a)–(110e)) is transformed to nonadiabatic form, nonadiabatic quasiparticles are introduced (up to the 2nd order of Taylor’ series expansion):