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.

Superconductivity, an amazing physical phenomenon discovered nearly 100 years ago by Kamerlingh Onnes [

Until the discovery of high-temperature superconductivity of cuprates by Bednorz and Muller in 1986 [

The range of validity of the BCS theory with respect to e-p interactions has been specified by Migdal [

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., [

The e-p interactions, which have been accepted to be responsible for electron pairing that drive transition into superconducting state for classical low-

Bell-like-shaped dependence of

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 results of high-resolution ARPES study [

Formation of the off-nodal kink (dispersion renormalization) has been attributed by the authors [

These results along with the results of neutron scattering [

The McMillan formula, which is very good approximation for

More over, new class of superconductors, for example, cuprates, fullerides, and Mg

Nonetheless, sophisticated treatment of high-

Discovery of superconductivity in a simple compound Mg

The

Nevertheless, the matter is even more complicated. It has been shown [

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

Situation for superconductors seems to be substantially different, at least in case of the Mg

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.

How to treat antiadiabatic state?

Can be system stable in antiadiabatic state?

Are the physical properties of the system in antiadiabatic state different from the corresponding properties in adiabatic state?

What is the driving force for adiabatic

How relevant is adiabatic

Is the adiabatic _{2}, or this state transition is an inherent physical mechanism which is proper also for other superconductors?

Can be adiabatic

Phonons or strong electron correlations?

What is the character of condensate-Cooper pairs or bipolarons?

Is there any relation of the adiabatic

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, [

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,

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

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

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

The

With respect to (

Until the Born approach (

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 (

In practice, physical and/or chemical properties of a many-body system in its ground electronic state

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

Hamiltonian (

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

In this context, the adiabatic electronic energies of the ground and excited electronic states

Set of

In the next step, instead of independent calculation of electronic excited states energies

for singly excited triplet state,

for singly excited singlet state,

From (

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 (

In the simplest form, with respect to (

for system at equilibrium geometry

more over,

In case of solids with quasicontinuum of states, in momentum

These inequalities have to be valid in a multiband system for each band

An important aspect should be reminded at this place. Inequalities (

In respect of it, possibility that displacement

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 (

Occurrence of such a situation means that nuclear motion (nuclear vibration, in particular, phonon mode) has induced sudden decrease of effective electron velocity

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

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

General nonrelativistic form of system Hamiltonian (

The

_{0}

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

Provided that phonon and electronic energy spectrum are well separated and (

Application of Wick’s theorem introduces renormalized Fermi vacuum, that is, the total set of orthonormal base orbitals

The electronic Hamiltonian (

The scalar quantity in this Hamiltonian, that is, zero-particle term

(II)

The one-particle term

It means that one-particle term (

In terms of orbital energies (

(III)

The third term of the electronic Hamiltonian (

_{0}

Solution of nuclear problem (

Nuclear part of Hamiltonian is

Energy and wave function of the total system in the ground electronic state are

Each eigenfunction

In second quantization, with single-bar

In a second quantization, it has the form

For fermion and boson creation and annihilation operators, the standard anticommutation, and commutation relations hold:

(

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 (

Adiabatic, nuclear displacement

Crude-adiabatic electronic wave function

Due to properties (

It can be shown [

The form of transformation relations for boson operators of system Hamiltonian has to respect the factorized form of the total system wave function (

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

For adiabatic

At solution of the problem on adiabatic (

(

As it has been mentioned in Section

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

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

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 (

Since adiabatic

The form of transformation relations for boson operators of system Hamiltonian has to respect again the factorized form of the total system wave function (

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

For nonadiabatic (

(

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:

Let us formally divide this Hamiltonian on two parts,

(

(

The system Hamiltonian (in the form (

Base functions transformations, as presented in preceding parts, introduce new dynamical variables; starting from crude-adiabatic

The system Hamiltonian (

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

Up to the second-order expansion in

In the next step, the adiabatic form (terms (

(

Like in case of the Hamiltonian transformations, also details of nonadiabatic solution are published in [

The main results of the solution can be written in the form of corrections to particular crude-adiabatic terms:

(

are coefficients of adiabatic transformation (

Approximate solution (see [

(i) True nonadiabatic or antiadiabatic state,

In this case the correction is negative,

(ii) The second limit corresponds to adiabatic level, when electronic state is only

An important aspect has to be mentioned in relation to adiabatic level. Introduction of adiabatic quasiparticles, that is,

The exact adiabatic correction to the electronic ground state is, according to (

For quasimomentum

For

(

The one-particle correction has been derived in the form

(i) Nonadiabatic polarons

The diagonal form of the one-particle correction (

Substitution for transformation coefficients (

(ii) Correction to orbital energies: gap opening in one-particle spectrum of quasidegenerate states at Fermi level.

For quasicontinuum of states at Fermi level, which is characteristic for metal-like band structures, contribution of second term in (

At these circumstances, for investigation of possible changes in the character of one-electron spectrum of system due to e-p interactions on

From (

It is trivial to show that the corrections to orbital energies are negligibly small (basically zero) for a system in adiabatic state when

(

Correction to

In Figures

Band structures of Mg

As it can be seen from Figure

In particular, for

The situation is similar for

In the case of YB

An optical phonon mode is simulated by displacements of Zn atoms in the opposite directions

Band structure of Al

As it can be seen from Figure

Instability of the electronic structure at e-p coupling is absent in respective non-superconducting analogues, such as X

In the case of Al

The Ca

In the case of deoxygenated YBCO, YB

Also the band structure of Mg remains without changes at e-p coupling (Figures

On crude-adiabatic level, total ground state electronic energy

The only correction to this energy is electron correlation energy (

_{d}

The band structures in Figure

Since for antiadiabatic state this correction is negative,

Stabilization (condensation) energy at transition from adiabatic into antiadiabatic state is

For all the presented systems at 0 K,

The highest value of the “condensation energy” (

Under these circumstances, each of the studied systems is stabilized in the antiadiabatic electronic ground state at broken symmetry with respect to the adiabatic equilibrium high-symmetry structure. In my opinion, it can be identified experimentally by ARPES as a kink on momentum distribution curve at FL in form of the band curvature at the ACP when it approaches FL (see Figure

Calculated dispersion of Cu

Nontrivial and very important property of system which is stabilized in the antiadiabatic state should be stressed. Due to translation symmetry of the lattice, the created antiadiabatic electronic ground state is geometrically degenerate in distorted geometry (

In transition to the antiadiabatic state,

Replacement of discrete summation by integration,

For some of the studied compounds, the calculated corrected DOS (

Corrected DOS and gap formation near

In particular, YB

Two gaps, in

A small gap opens on pd-band in the

The corrections to orbital energies (

The calculated values of

Presented results naturally provoke the question if the antiadiabatic state is related in some way to superconducting state. Crucial in this respect are thermodynamic properties of antiadiabatic state, that is, at temperatures

(

As it has already been mentioned, in the antiadiabatic state, ground-state total electronic energy of system is geometrically degenerate. Distorted nuclear structure, related to couple of nuclei in the phonon mode

Due to the geometric degeneracy of the ground state energy, the involved atoms can circulate over perimeters of the circles without the energy dissipation.

The dissipation-less motion of the couple of nuclei implies, however, that e-p coupling of involved phonon mode and electrons of corresponding band has to be zero.

Let as shown that in the antiadiabatic state this aspect is fulfilled.

The effective e-p interactions which cover the

Iso-density line of highest electron density at equilibrium nuclear geometry

In the adiabatic state, properties of the electrons are in sharp contrast with the properties of electrons in antiadiabatic state. The electrons in this case are in a valence band more or less, tightly bound to respective nuclei at adiabatic equilibrium positions and theirs motion in conducting band is restricted by scattering with interacting phonon modes. It corresponds to situation at

For extreme adiabatic limit

II.

It has been shown that at formation of the antiadiabatic ground-state, electronic energy is decreased and for involved band(s) the gap in one-particle spectrum has been opened (shift of orbital energies). This fact has to be reflected by change of related thermodynamic properties. In particular, for electronic specific heat,

From the (

III.

System in superconducting state can exhibit absolute diamagnetism and Meissner effect only if inside the system

It can be shown that antiadiabatic state exhibits this property.

From thermodynamics for critical magnetic field in this case, it follows that

Correction

Let us see if the correction to correlation energy due to e-p coupling

From (

In case of strong antiadiabatic regime

Like for nonadiabatic

Along with crude-adiabatic correlation energy (

At finite temperature, the product of Fermi-Dirac occupation factors has to be introduced into derived equations:

Let us turn attention to the Fröhlich effective Hamiltonian of e-e interactions [

In the limit of extreme antiadiabaticity

With respect to the fact that effective attractive e-e interaction is a pure correction to electron correlation energy, the Cooper’s pair idea within the BCS treatment is without any doubts formally correct, nevertheless, it is rather a model treatment how to solve the instability problem with severe restriction to fixed nuclear framework, that is, within the crude-adiabatic BOA which requires for total system validity of the adiabatic condition

From the theory of the antiadiabatic ground state, instead of Cooper’s pairs, formation of mobile bipolarons results in a natural way as a consequence of translation symmetry breakdown on antiadiabatic level. The bipolarons arise as polarized intersite charge density distribution that can move over lattice without dissipation due to geometric degeneracy (fluxional structure) of the antiadiabatic ground state at distorted nuclear configurations. Formation of polarized intersite charge density distribution at transition from adiabatic into antiadiabatic state is reflected by corresponding change of the wave function. For spinorbital (band)

In antiadiabatic state, for particular

Experimental results are always crucial for any theory which aims to formulate basic physics behind observed phenomenon or property. However, an experiment always cover much wider variety of different influences which have impact on results of experimental observation than any theory can account for, mainly if theory is formulated on microscopic level and some unnecessary approximations and assumptions are usually incorporated. On the other hand, interpretation of many experimental results is based on particular theoretical model. This is also the case of ARPES experiments at reconstruction of Fermi surface for electronic structure determination of high-

In case of high-

In this sense, unexpected results have been published recently [

The order parameter of (

In spite that there is a general agreement among different DFT-based band structure calculations in overall character of band structure for particular cuprate, there are small, but important differences in details concerning topology of some bands at FL. In particular, for YBCO, DFT-based all-electron band structure calculation [

With respect to topology of Cu-O chain band, as discussed above, this band should be the best candidate to be considered for gap opening in both, off-nodal (

Nonetheless, an interpretation of the effects seen in the discussed ARPES from the stand-point of antiadiabatic theory should be of interest. The basic aspects concerning YBCO have been predicted [

Coupling to

It should be mentioned that considerably smaller gap (

In conclusion, it can be summarized that based on the ab initio theory of complex electronic ground state of superconductors, it can be concluded that e-p coupling in superconductors induces the temperature-dependent electronic structure instability related to fluctuation of analytic critical point (ACP––maximum, minimum, or saddle point of dispersion) of some band across FL, which results in breakdown of the adiabatic BOA. When ACP approaches FL, chemical potential

An analysis of e-p interaction Hamiltonian has shown that an effective attractive e-e interaction, which is, the basis of Cooper’s pair formation, is in fact the correction to electron correlation energy at transition from adiabatic into antiadiabatic ground electronic state. In this respect, increased electron correlation is not the primary reason for transition into superconducting state, but it is a consequence of antiadiabatic state formation which is stabilized by nonadiabatic e-p interactions at broken translation symmetry. It has been shown that thermodynamic properties of system in the antiadiabatic state correspond to thermodynamics of superconducting state.

The author acknowledges the support of the grant VEGA1/0013/08.