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\title{The Singlet-Triplet Jahn-Teller Polar Centers in Copper Oxides}
\author{{\it \ A.S. Moskvin, Yu.D. Panov} \\
Department of Theoretical Physics, Ural State University, \\
620083, Lenin Ave. 51, Ekaterinburg , Russia \\
{\em E-mail: alexander.moskvin@usu.ru}
}
\date{{\small Received January 20, 1998\\ Accepted
February 4, 1998}}

\begin{document}

\pagestyle{myheadings}
\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
{\bf 2}, {\sc 1 (1997)}\hfill}
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\maketitle

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%{\Large \bf Vol. 1, \ \  No. 1, \ \ 1997}
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\vbox{\parbox{2.51in}{\LARGE \bf {\it \ \ \ Volume} 2  \\  {\it No.} 1,
1998} \vspace{.2in}}}

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\begin{center}
{\bf Abstract}
\end{center}

One of the most exciting features of the hole centers $CuO_{4}^{5-}$ in
doped cuprates is in their unusually complicated ground state as a result of
electronic quasi-degeneracy. An additional hole, doped to the basic $%
CuO_{4}^{6-}$ cluster with the $b_{1g}$ hole can occupy both the same hybrid 
$Cu3d-O2p$ orbital state resulting in a Zhang-Rice singlet $^1A_{1g}$ and
the purely oxygen $e_u$ molecular orbital resulting in singlet or triplet $%
^{1,3}E_u)$ term with the close energies. We present a detailed analysis of
the (pseudo)-Jahn-Teller effect driven by the near-degeneracy within the $%
^1A_{1g},^{1,3}E_u)$-manifold.

\begin{description}
  \item[PACS]: 71.27.+a, 74.20.s, 74.25.Ha, 76.60.-k
  \item[Keywords]: cuprates, spin fluctuations, hyperfine coupling, nuclear resonance
\end{description}
 
\bigskip
Selected for publication by Programme Committee of youth 
scientific school "Actual problems of a magnetic resonance 
and its applications: Magnetic resonance 
in high $T_c$ materials", Kazan, November 20-22, 1997

\newpage



\section{Introduction.}

Unconventional properties of the oxides like $YBa_2Cu_3O_{6+x}$, $%
La_{2-x}Sr_xCuO_4$, $(K,Ba)BiO_3$, $La_{1-x}Sr_xMnO_3$, $La_2CuO_{4+\delta }$%
, $La_2NiO_{4+\delta }$ including systems with the high-$T_c$
superconductivity and colossal magnetoresistance reflect a result of the
response of the system to the nonisovalent substitution that stabilizes
phases providing the most effective screening of the charge inhomogeneity.
These phases in oxides may involve novel unconventional molecular cluster
configurations like the Jahn-Teller $sp$-center \cite{Moskvin1993} with
anomalous high local polarizability and multi-mode behavior.

The numerous experimental investigations show that the origin of the high-$%
T_c$-super\-conduc\-ti\-vi\-ty and of the other unconventional physical
properties of the copper oxides are connected with an active interplay of
the whole set of the degrees of freedom including the charge, spin, orbital
and structural modes under conditions of the strong charge inhomogeneity and
granularity.

Indirectly this is corroborated by the failures to explain more or less
consistently the unconventional behavior of copper oxides as homogeneous
systems within the ''single-mode'' scenarios such as the spin-fluctuation,
electron-phonon or purely electronic ones.

An active interplay of the whole set of the modes is a natural element of
the ''multi-mode'' scenario based on the so called ''polar Jahn-Teller
center model'' proposed by the one of the authors earlier \cite
{Moskvin1993,Moskvin1994,Moskvin1997}. The $CuO_{4}$-clusters based copper
oxides in this model are considered as systems unstable with respect to the
disproportionation reaction 
\begin{equation}
2CuO_{4}^{6-}\rightarrow \left[ CuO_{4}^{5-}\right] _{JT}+\left[
CuO_{4}^{7-}\right] _{JT}
\end{equation}
with a formation of the system of the polar (hole - $CuO_{4}^{5-}$ or
electron - $CuO_{4}^{7-}$) pseudo-Jahn-Teller (PJT) centers. These centers
are distinguished by the so called local $S$-boson or two electrons paired
in the completely filled molecular orbital of the $CuO_{4}$-cluster. In
other words, the novel phase can be considered as a system of the local
spinless bosons moving in the lattice of the hole PJT-centers $\left[
CuO_{4}^{5-}\right] _{JT}$ or the generalized quantum lattice bose-gas (or
liquid) with the boson concentration near $N_{B}=1/2$.

Below we consider in more details a vibronic structure of the isolated PJT
centers with taking account of some effects associated with spin-orbital
coupling.

\section{Correlations and the near degeneracy effects for the $CuO_4$%
-clusters.}

At first sight an analysis of the electronic structure and of the energy
spectrum of the parent compounds such as $La_{2-x}M_xCuO_4$, $%
YBa_2Cu_3O_{6+x}$ at $x=0$ does not display any exotic peculiarities except
quasi two-dimensional antiferromagnetism determined by the strong exchange
interaction for the $b_{1g}(d_{x^2-y^2})$-holes in the ''basic'' $CuO_4^{6-}$%
-clusters. At the same time it is worth to pay attention to one important
feature namely to the exciton-band form of the fundamental absorption in the 
$1.5\div 3.0\ eV$ region strictly pronounced in the systems like $R_2CuO_4$, 
$YBa_2Cu_3O_{6+x}$, $CuO$ \cite{Moskvin1997}.

A peculiar character of this absorption connected with the allowed
charge-transfer transition $b_{1g}\rightarrow e_{u}$ between the
copper-oxygen $b_{1g}$-hybrid and the purely oxygen $e_{u}$-orbital
evidences the strongly correlated nature of the $e_{u}$-electrons with
formal emergence of two types of the $e_{u}$-states with and without strong
correlation. This peculiarity is connected with the maximal hole density
occurred for oxygen ions just in the $e_{u}$-states of the $CuO_{4}$-cluster
and can be easily explained in the framework of the non-rigid anionic
background model \cite{Hirsch1989}. This model introduces new correlation
degree of freedom with two possible states of anionic background for the
valent $O2p$-holes corresponding to two possible projections of the
correlation pseudospin $s=1/2$ and is described by the simplified
Hamiltonian 
\begin{equation}
H_{corr}=V_{1}\hat{\sigma}_{x}+V_{3}\hat{\sigma}_{z}
\end{equation}
where $V_{1,3}$ are two operators acting within the valent states. Simple
approximation used in \cite{Hirsch1989} conjectures the linear $n_{2p}$%
-dependence for $V_{1,3}$ where $n_{2p}$ is the $O2p$-hole number. According
to the optical data \cite{Moskvin1997} the correlation pseudospin splitting
can achieve the value $~\sim 0.5\ eV$.

An increase in the $O2p$-hole concentration with the hole doping $%
CuO_4^{6-}\rightarrow CuO_4^{5-}$ results in sharp increase of the $e_u$%
-correlation splitting in the hole $CuO_4^{5-}$-centers with transformation
of the $b_{1g}\rightarrow e_u$ fundamental band to the high-energy $%
(b_{1g}^{2})\rightarrow (b_{1g} e_u)$ subband and the low-energy $%
b_{1g}^{2}\rightarrow b_{1g}^{*} e_u^{*}$ subband to be well known as MIR
(mid-infrared) absorption band \cite{Moskvin1994}.

Fig.1 shows qualitatively the results of the taking account of the
considered ''$e_{u}$-correlations'' for the energy spectrum of the basic $%
CuO_{4}^{6-}$-cluster and the hole $CuO_{4}^{5-}$-cluster. It is important
to mention an appearance in our model two types of the orthogonal (!)
molecular orbitals, for instance $b_{1g}$ (upper correlation sublevel) and $%
b_{1g}^{*}$ (lower correlation sublevel ) states, $e_{u}$ and $e_{u}^{*}$.

\begin{figure}[tbph]
\centerline{\psfig{figure=figu1.eps,width=4.5in,clip=}
}
\caption{
Correlation effects in the energy spectrum of the basic $CuO_{4}^{6-}$  cluster
 and the hole  $CuO_{4}^{5-}$ center with numerical values (in eV) typical for 
oxides like $CuO$. On the right hand side we show a formation of the fundamental 
absorption spectra for the parent and hole doped oxides with peculiar MIR 
band in the latter case. 
}
\label{}
\end{figure}

Thus we come to a conclusion about a near-degeneracy for two configurations $%
b_{1g}^{*\ 2}$ and $b_{1g}^{*}e_u^{*}$ with $b_{1g}^{*}$ and $e_u^{*}$ being
a lowest correlation sublevels. This result does not drastically change with
taking into account an electrostatic interaction $V_{ee}$ between two holes.
Moreover, just the $V_{ee}$ contribution was considered earlier \cite
{Moskvin1994} as a main reason of a near degeneracy for $^1A_{1g}$
(Zhang-Rice singlet) and $^{1,3}E_u$-terms formed by $b_{1g}^{*\ 2}$ and $%
b_{1g}^{*}e_u^{*}$-configurations. So, the both correlation effects lead to
a near degeneracy in the ground state of the hole center $CuO_4^{5-}$.

Unusual properties of the $(^1A_{1g},^{1,3}E_u)$-manifold involving terms
distinguished by the spin multiplicity, parity and orbital degeneracy
provides unconventional behavior for the hole center $CuO_4^{5-}$ with
active interplay of various modes.As an extremely important result let note
that $E_u$-doublet has a nonquenched Izing like orbital moment that can be
directed only along the $C_4$-axis.

A near degeneracy within $(^1A_{1g},^{1,3}E_u)$-manifold can create
conditions for the pseudo-Jahn-Teller effect \cite{Bersuker1983} with
anomalously strong electron-lattice correlations with the active local
displacements modes of the $Q_{e_u}$-, $Q_{b_{1g}}$- and $Q_{b_{2g}}$- types.

\section{ Vibronic coupling for the $CuO_4^{5-}$ centers.}

\subsection{ The adiabatic potential.}

Below we'll make use a notation $|SM_{S}\Gamma \gamma \rangle $ for the
basis wave functions from the $(^{1}A_{1g},^{1,3}E_{u})$-manifold (see
Fig.2). Here, $S$ $(=0,1)$, $M_{S}$ are the total spin and its projection, $%
\Gamma \gamma $ ($=A_{1g}$, $E_{u}^{x}$, $E_{u}^{y}$) are symbols of the
irreducible representation of the symmetry group $D_{4h}$ for the $CuO_{4}$
center and its row, respectively, which indicate the transformation
properties of the orbital functions. Subsequently, we restrict ourselves
with the linear vibronic coupling within the $(^{1}A_{1g},^{1,3}E_{u})$%
-manifold with the JT-active vibrational coordinates of the $a_{1g}$, $%
b_{1g} $, $b_{2g}$, $e_{u}$ symmetry.

\begin{figure}[tbph]
\centerline{\psfig{figure=figu2.eps,width=4.5in,clip=}
}
\caption{
An illustration of the initial (a) $CuO_{4}^{5-}$-center energy
spectrum modification with taking into account the strong Jahn-Teller
pseudo-effect (b) and the tunnel splitting of the ground vibronic states
(c). Corresponding to different minima of AP distortions are shown in
insert.
}
\label{}
\end{figure}

Vibronic coupling for the isolated singlet or triplet $^{1,3}E_{u}$-term has
a well known for the so called $E-b_{1}-b_{2}$-problem \cite{Bersuker1983}
form diagonal in $S$ and $M_{S}$ 
\begin{equation}
\left( 
\begin{array}{cc}
V_{b_{1g}}^{\tau }Q_{b_{1g}}^{\tau } & V_{b_{2g}}^{\tau }Q_{b_{2g}}^{\tau }
\\ 
V_{b_{2g}}^{\tau }Q_{b_{2g}}^{\tau } & -V_{b_{1g}}^{\tau }Q_{b_{1g}}^{\tau }
\end{array}
\right) ,\quad \tau =\ ^{1}E_{u},\ ^{3}E_{u}.
\end{equation}

The singlet terms $^{1}E_{u}$ and $^{1}A_{1g}$ interact due to linear
vibronic coupling 
\begin{equation}
\langle 00A_{1g}|V_{vib}|00E_{u}^{i}\rangle =\sum_{e_{u}}V_{e}Q_{e_{u}^{i}}
\end{equation}
determined by active vibrational coordinates $Q_{e_{u}}^{x},Q_{e_{u}}^{y}$.
For the $CuO_{4}$ cluster there are three normal coordinates with $e_{u}$%
-symmetry, however, below we restrict ourselves with the choice of the one
active $e_{u}$-vibration with an appropriate linear vibronic coupling
constant $V_{e}$.

The sum of electronic Hamiltonian $\hat{V}_{el}$, elastic energy $\hat{V}%
_{Q} $, and vibronic Hamiltonian $\hat{V}_{vib}$ for the singlet $S=0$ spin
manifold $^{1}E_{u},^{1}A_{1g}$ could be written as: 
\begin{equation}
\hat{U}(Q)=\hat{V}_{el}+\hat{V}_{Q}+\hat{V}_{vib}=\sum_{i}\frac{\omega
_{i}^{2}Q_{i}^{2}}{2}\cdot \hat{I}+\left( 
\begin{array}{ccc}
-\Delta +V_{z}Q_{z} & V_{e}Q_{x} & V_{e}Q_{y} \\ 
V_{e}Q_{x} & V_{\alpha }Q_{\alpha } & V_{\beta }Q_{\beta } \\ 
V_{e}Q_{y} & V_{\beta }Q_{\beta } & -V_{\alpha }Q_{\alpha }
\end{array}
\right) ,
\end{equation}
where the indices both for the coupling coefficient and normal coodinates
are defined as follows: $a_{1g}\rightarrow z$, $e_{u}^{x}\rightarrow x$, $%
e_{u}^{y}\rightarrow y$, $b_{1g}\rightarrow \alpha $, $b_{2g}\rightarrow
\beta $.

An important information for the PJT center could be obtained with
examination of the adiabatic potential (AP) sheets $\varepsilon (Q)$, which
are roots of the characteristic equation for $\hat{U}(Q)$. In our case this
is reduced to a cubic equation with an extremely complicated expression for
the roots. The coordinates of the minima $Q^{0}$, energy and structure of
electronic wave function at $Q^{0}$ and curvature of energy sheet near $%
Q^{0} $ can be obtained by the Opik-Pryce method \cite{OpikPryce1957}. An
eigenvalue problem is treated in framework of this method only for extremal
points of the AP with taking account of the relation between the extremum
coordinates and the wave function in $Q^{0}$ which follows from the from the
Hellmann-Feynman theorem: 
\begin{equation}
\frac{\partial \varepsilon }{\partial Q}=\frac{\partial }{\partial Q}%
\left\langle \psi \left| \hat{U}(Q)\right| \psi \right\rangle =\left\langle
\psi \left| \frac{\partial \hat{U}(Q)}{\partial Q}\right| \psi \right\rangle
.
\end{equation}
It should be noted that the Opik-Pryce method does not permit to find
special points of the AP without definite values of the derivative, however
namely such a situation occurs for the upper sheets of the AP. The type of
minima can be derived from the curvature analysis for $\varepsilon (Q)$ near 
$Q^{0}$. The $\varepsilon (Q)$ is found as the second-order correction to $%
\varepsilon (Q^{0})$: 
\begin{equation}
\varepsilon _{-}(Q)\approx \varepsilon _{-}(Q^{0})+\left\langle \psi \left| 
\hat{U}_{2}(Q)\right| \psi \right\rangle +\sum_{i}{}^{\prime }\frac{\left|
\left\langle \psi _{i}\left| \hat{U}_{1}(Q)\right| \psi \right\rangle
\right| }{\varepsilon _{-}(Q^{0})-\varepsilon _{i}(Q^{0})}
\end{equation}
\[
\hat{U}_{1}(Q)=\sum_{i}\frac{\partial \hat{U}(Q^{0})}{\partial Q_{i}}\
q_{i},\quad \hat{U}_{2}(Q)=\frac{1}{2}\sum_{ij}\frac{\partial ^{2}\hat{U}%
(Q^{0})}{\partial Q_{i}\partial Q_{j}}\ q_{i}q_{j},\quad
q_{i}=Q_{i}-Q_{i}^{0}, 
\]
where $\left| \psi _{i}\right\rangle $ and $\varepsilon _{i}(Q^{0})$ are
eigenfunctions and eigenvalues of $\hat{U}(Q^{0})$, and the prime at sum
mark indicates the singular term is omitted.

The relations between the extremum coordinates for the AP and the wave
function coefficients are 
\begin{equation}
Q_{z}^{0}=-q_{z}^{\left( 0\right) }z^{2},\quad Q_{x}^{0}=-2q_{e}^{\left(
0\right) }zx,\quad Q_{y}^{0}=-2q_{e}^{\left( 0\right) }zy,
\end{equation}
\[
Q_{\alpha }^{0}=-q_{\alpha }^{\left( 0\right) }\left( x^{2}-y^{2}\right)
,\quad Q_{\beta }^{0}=-2q_{\beta }^{\left( 0\right) }xy, 
\]
where $q_{i}^{\left( 0\right) }=V_{i}/\omega _{i}^{2}$ and we have denoted
the coefficients of decomposition of the electronic wave function as $z$, $x$
and $y$ for $\left| 00A_{1g}\right\rangle ,\left| 00E_{u}^{x}\right\rangle $
and $\left| 00E_{u}^{y}\right\rangle $ respectively. The eigenvalue problem
for $V_{vib}$ in the exrtemum points of the AP is given by: 
\begin{equation}
\left( 
\begin{array}{ccc}
-\Delta -2E_{JT}^{z}\ z^{2} & -4E_{JT}^{e}\ zx & -4E_{JT}^{e}\ zy \\ 
-4E_{JT}^{e}\ zx & -2E_{JT}^{\alpha }\ \left( x^{2}-y^{2}\right) & 
-4E_{JT}^{\beta }\ xy \\ 
-4E_{JT}^{e}\ zy & -4E_{JT}^{\beta }\ xy & 2E_{JT}^{\alpha }\ \left(
x^{2}-y^{2}\right)
\end{array}
\right) \left( 
\begin{array}{c}
z \\ 
x \\ 
y
\end{array}
\right) =\lambda \left( 
\begin{array}{c}
z \\ 
x \\ 
y
\end{array}
\right) ,  \label{system}
\end{equation}
where $E_{JT}^{i}=\frac{1}{2}V_{i}q_{i}^{\left( 0\right) }=V_{i}^{2}/2\omega
_{i}^{2}$ is the specific JT-energy. The system (\ref{system}) supplemented
with a normalization condition $x^{2}+y^{2}+z^{2}=1$ has 13 solutions, which
are listed in the Table 1. An expressions for the corresponding quadratic
form of the AP-sheet near extreme point are given in the Table 2. These
solutions could be divided into three groups.

1. The first group NJT (non-JT)contains one solution. Electronic part of the
wave function in the exrtemal point is pure $\left| 00A_{1g}\right\rangle $.
The NJT-extremum is the minimum on the lower sheet of AP if $b<0$ $%
(b=-\Delta +4E_{JT}^{e}-2E_{JT}^{z})$ and this is the minimum on the upper
sheet of AP if $\Delta +2E_{JT}^{z}<0$. In both cases the weak
pseudo-Jahn-Teller effect takes place, when due to weakness of vibronic
interaction in comparison with initial splitting of electronic levels $%
\Delta $ there are no low symmetry distortions of the $CuO_{4}$ cluster. Due
to vibronic interaction it occurs a renormalization of the local $e_{u}$%
-vibration frequency: 
\begin{equation}
\left( \tilde{\omega}_{e}^{\left( NJT\right) }\right) ^{2}=\omega
_{e}^{2}\left( 1-\eta _{z}\right) ,\quad \eta _{z}=\frac{4E_{JT}^{e}}{\Delta
+2E_{JT}^{z}}\ .
\end{equation}

2. The second group JT$_{i}$ ($i=\alpha ,\beta $) contains four solutions.
These are similar to the results of the well known $E-b_{1}-b_{2}$-problem.
The wave function at the extremum points is a pure $E_{u}$-superposition. In
further analysis the rhombic mode with larger JT-energy will be called the
''strong'' ($\sigma $) mode, and the rhombic mode with smaller JT-energy
will be called the ''weak'' ($\sigma ^{\prime }$) one: $E_{JT}^{\sigma
}>E_{JT}^{\sigma ^{\prime }},$ $(\sigma ,\sigma ^{\prime }=\alpha ,\beta )$;
for the $E-b_{1}-b_{2}$-problem the $E_{JT}^{\sigma }$ is the Jahn-Teller
stabilization energy. From four exrtemum points only the JT$_{\sigma }$ pair
will be minima. The minima are located on the lower sheet of the AP if $%
a_{\sigma }<0$ $(a_{\sigma }=\Delta +4E_{JT}^{e}-2E_{JT}^{\sigma })$, and on
the middle one if $\Delta -2E_{JT}^{\sigma }>0$. The JT$_{\sigma ^{\prime }}$
pair represent saddle points. The wave functions at the minima are
orthogonal each other. If $E_{JT}^{\sigma }=E_{JT}^{\sigma ^{\prime }}$ the
equipotential continuum of minima exists. The frequency of the $\sigma $%
-mode does not change, and frequency of $\sigma ^{\prime }$-mode is
renormalized due to vibronic interaction: 
\begin{equation}
\tilde{\omega}_{\sigma }^{2}=\omega _{\sigma }^{2}\ ,\quad \tilde{\omega}%
_{\sigma ^{\prime }}^{2}=\omega _{\sigma ^{\prime }}^{2}\left( 1-\lambda
_{\sigma }\right) ,\quad \lambda _{\sigma }=\frac{E_{JT}^{\sigma ^{\prime }}%
}{E_{JT}^{\sigma }}\ .
\end{equation}

It should be noted here that a type ($B_{1g}$ or $B_{2g}$) of the ground
JT-mode is of principal importance for the physics of the copper oxides.
This is determined by the competition of the vibronic parameters for the $%
Cu3d-O2p$- and $O2p-O2p$-bonds minimizing the $B_{1g}$- and $B_{2g}$-modes,
respectively.

For the $e_{u}$-vibrations the JT$_{i}$ solutions correspond to the weak
pseudo-Jahn-Teller effect: only renormalization of the local $e_{u}$%
-vibration occurs. If $\alpha $-mode is strong, then the minimum coordinates
determine the rhombic distortion of the $CuO_{4}$ cluster along $x$- or $y$%
-direction (Fig.3). Accordingly, the softening of the $e_{u}^{x}$- or $%
e_{u}^{y}$-vibration occurs; but as both minima are equivalent, the
frequencies of the local $e_{u}$-modes remain twice degenerate. In a case of
''strong'' $\beta $-mode with rectangular distortion of the $CuO_{4}$
cluster, the softening of that $e$-mode occurs, which co-directs to the
cluster distortion. In all cases the expression for the renormalized local $%
e_{u}$-mode frequency is written as: 
\begin{equation}
\left( \tilde{\omega}_{e}^{\left( JT\right) }\right) ^{2}=\omega
_{e}^{2}\left( 1-\kappa _{\sigma }\right) \ ,\quad \kappa _{\sigma }=\frac{%
4E_{JT}^{e}}{-\Delta +2E_{JT}^{\sigma }}\ .
\end{equation}

\begin{figure}[tbph]
\centerline{\psfig{figure=figu3.eps,width=4.5in,clip=}
}
\caption{
Possible distortions of the $CuO_{4}$-cluster in the a) $JT_{\alpha }$%
-minimum; b) $JT_{\beta }$-minimum; c) $PJT_{\alpha }$-minimum (the dipole
moment is oriented along the $CuO_{4}$-cluster diagonal); b) $PJT_{\beta }$%
-minimum (the rotation angle value for the dipole moment with respect  to
that in $PJT_{\alpha }$-minima is $\phi =\pi /4$).
}
\label{}
\end{figure}

3. The third group PJT$_{i}$ ($i=\alpha ,\beta $) includes eight solutions
and corresponds to the most complicated case of the strong
pseudo-Jahn-Teller effect. In this case the wave functions at the extremum
points are the $A_{1g}-E_{u}$-hybrid states. The coefficients of the
appropriate linear combination depend on the initial splitiing $\Delta $ and
the JT-energies. The four from the eight PJT$_{\sigma }$-extrema ($\sigma $
- the ''strong'' rhombic mode) are the minima if $a_{\sigma }>0$ and $\dot{b}%
>0$. All the PJT$_{\sigma }$-minima are equivalent and allocated on the
lower sheet of the AP. An arrangement of the minima in a space of the normal
coordinates of the $CuO_{4}$ cluster is schematically shown in Fig.4. The
wave functions at the minima are not orthogonal each other, that is a
characteristic feature of the strong pseudo-Jahn-Teller effect. The four PJT$%
_{\sigma ^{\prime }}$-extrema ($\sigma ^{\prime }$ - the ''weak'' rhombic
mode) are saddle points. A specific case of degeneracy for the JT-energies
of rhombic modes should be investigated separately.

\begin{figure}[tbph]
\centerline{\psfig{figure=figu4.eps,width=4.5in,clip=}
}
\caption{
The allocation of the $PJT_{\sigma }$-minima in a space of symmetric
coordinates $Q_{\sigma },Q_{1},Q_{2}$ ($\sigma $ is the ''strong'' rhombic
mode). If $\sigma =\alpha $, then $Q_{1}=Q_{x}$, $Q_{2}=Q_{y}$; if $\sigma
=\beta $, then $Q_{1}=\frac{Q_{x}+Q_{y}}{2}$, $Q_{2}=\frac{-Q_{x}+Q_{y}}{2}$.
}
\label{}
\end{figure}


The rhombic distortion of the $CuO_{4}$ cluster at the minimum points with
the nonzero plane quadrupole moment is accompanied by the co-directed $e_{u}$%
-distortion with the electric dipole moment. The possible PJT$_{\sigma }$
distortions are shown in Fig.3.

Close to minimum $M$ the equipotential surface of the quadratic form $%
\varepsilon (Q-Q^{(0,M)})$ is a five-dimensional ellipsoid with the center
located at $Q^{(0,M)}$ and principal axes to be turned with regard to basic
ones. The $\sigma $-mode at minimum is mixed with co-directed $e_u$-mode and 
$a_{1g}$-mode giving rise to three local hybrid modes. The $\sigma ^{\prime
} $-mode is mixed with the second $e_u$-mode giving rise to two local hybrid
modes. The mixing coefficients are proportional to the appropriate vibronic
coupling constants.

In case $V_{z}=0$ the frequencies of the normal local hybrid modes are
written as: 
\begin{equation}
\left( Q_{\sigma ^{\prime }},\tilde{Q}_{1}\right) :\qquad \omega _{\pm }^{2}=%
\frac{A+B}{2}\pm \sqrt{\left( \frac{A-B}{2}\right) ^{2}+C^{2}}\ ,
\end{equation}
where 
\[
A=\omega _{e}^{2}\left( 1-\upsilon _{\sigma }^{2}\right) ,\ B=\omega
_{\sigma ^{\prime }}^{2}\left( 1-\rho _{\sigma }^{2}\right) ,\ C=\omega
_{e}\omega _{\sigma ^{\prime }}\upsilon _{\sigma }\rho _{\sigma }, 
\]
\[
\upsilon _{\sigma }=\sqrt{\frac{E_{JT}^{e}\ a_{\sigma }}{E_{JT}^{e}\
a_{\sigma }+E_{JT}^{\sigma }\ b}},\ \rho _{\sigma }=\sqrt{\frac{%
E_{JT}^{\sigma ^{\prime }}\ b}{E_{JT}^{e}\ a_{\sigma }+E_{JT}^{\sigma }\ b}}%
, 
\]
and 
\begin{equation}
\left( Q_{\sigma },\tilde{Q}_{2}\right) :\qquad \omega _{\pm }^{\prime \ 2}=%
\frac{D+E}{2}\pm \sqrt{\left( \frac{D-E}{2}\right) ^{2}+F^{2}}\ ,
\end{equation}
where 
\[
D=\omega _{e}^{2}\left( 1-\nu _{\sigma }^{2}\right) ,\ B=\omega _{\sigma
}^{2}\left( 1-\mu _{\sigma }^{2}\right) ,\ F=\omega _{e}\omega _{\sigma }\nu
_{\sigma }\mu _{\sigma },\ 
\]
\[
\mu _{\sigma }=\sqrt{\frac{E_{JT}^{\sigma }\ a_{\sigma }b}{E_{JT}^{e}\
\left( a_{\sigma }+b\right) ^{2}}},\ \nu _{\sigma }=\frac{a_{\sigma }-b}{%
a_{\sigma }+b}. 
\]

For minima 1 and 3 $\tilde{Q}_{1}=Q_{x}$ if $\sigma =\alpha $ ($Q_{1}$ if $%
\sigma =\beta $), $\tilde{Q}_{2}=Q_{y}$ if $\sigma =\alpha $ ($Q_{2}$ if $%
\sigma =\beta $); for minima 2 and 4 $\tilde{Q}_{1}=Q_{y}$ if $\sigma
=\alpha $ ($Q_{2}$ if $\sigma =\beta $), $\tilde{Q}_{2}=Q_{x}$ if $\sigma
=\alpha $ ($Q_{1}$ if $\sigma =\beta $). Due to equivalence of minima all
local frequencies coincide.

Conditions for the emergence of the certain type minima on the lower sheet
of the AP will be determined mainly by the following quantities: 
\begin{equation}
a_{\sigma }=\Delta +4E_{JT}^{e}-2E_{JT}^{\sigma },\quad b=-\Delta
+4E_{JT}^{e}-2E_{JT}^{z}\ .
\end{equation}

The type of minima on the lower sheet of the AP for various combinations of
signs for the $a_{\sigma }$ and $b$ is shown in Table 3. The diagram of
states of the lower sheet of the AP in a space of parameters $\Delta $, $%
E_{JT}^{\sigma }$ and $E_{JT}^{e}$ at constant value of $E_{JT}^{z,0}$ is
shown in Fig.5. A cross-section of the parameter space for a constant value
of $E_{JT}^{e}$ is shown in Fig.6. If $\Delta >0$ the lowest $^{1}A_{1g}$
level is well isolated and the corresponding lower sheet of the AP has a
rather trivial NJT minimum. In contrast, at $\Delta <0$ the $^{1}E_{u}$ term
becomes the lower and an usual $E-b_{1}-b_{2}$-problem with two $JT_{\sigma
} $ minima on the lower sheet of the AP occurs. With coming together the $%
^{1}A_{1g}$ and $^{1}E_{u}$ terms are mixed by $e_{u}$-mode and $e_{u}$%
-frequency is renormalized. An appropriate process depending on magnitude of 
$E_{JT}^{\sigma }$ is accompanied by a formation of four PJT$_{\sigma }$%
-minima or three (NJT+JT$_{\sigma }$)-minima. These two possibilities
correspond to lines $E_{JT}^{\sigma }=E_{JT}^{\sigma ,1}$ and $%
E_{JT}^{\sigma }=E_{JT}^{\sigma ,2}$ in Fig.7.

\begin{figure}[tbph]
\centerline{\psfig{figure=figu5.eps,width=4.5in,clip=}
}
\caption{
The diagram of the lower AP sheet states in the $(\Delta
,E_{JT}^{\sigma },E_{JT}^{e})$-space at certain fixed $%
E_{JT}^{z}=E_{JT}^{z,0}$ value. The planes $a_{\sigma }=0$ and $b=0$ divide
the $(\Delta ,E_{JT}^{\sigma },E_{JT}^{e})$-space into four parts
corresponding to $NJT$-, $JT_{\sigma }$-, $PJT_{\sigma }$- or $%
NJT+JT_{\sigma }$-type of the lower AP sheet minima. In the $NJT+JT_{\sigma }
$ region the plane  is shown, where the $NJT$- and $JT_{\sigma }$-minimum
energies are equal.
}
\label{}
\end{figure}

\begin{figure}[tbph]
\centerline{\psfig{figure=figu6.eps,width=4.5in,clip=}
}
\caption{
The diagram of states of the lower AP-sheet on the $(\Delta
,E_{JT}^{\sigma })$-plane at certain fixed $E_{JT}^{z}$ and $E_{JT}^{e}$
value. The boundaries $a_{\sigma }=0$ and $b=0$ of the regions with
different type of the lower AP-sheet minima are shown. The lines $%
E_{JT}^{\sigma }=E_{JT}^{\sigma ,1}$ and $E_{JT}^{\sigma }=E_{JT}^{\sigma ,2}
$ correspond to the two different relative intensities of vibronic coupling 
via $\sigma $-mode and  $e_{u}$-mode.
}
\label{}
\end{figure}

\begin{figure}[tbhp]
\centerline{\psfig{figure=figu7.eps,width=4.5in,clip=}
}
\caption{
The diagram of states of the upper AP sheets on the $(\Delta
,E_{JT}^{\sigma })$-plane at certain fixed $E_{JT}^{z}$ and $E_{JT}^{e}$ 
value. In region to the left of the $\Delta =-2E_{JT}^{z}$ line there is $NJT
$-minimum on the upper AP-sheet and only the trivial non-analytical minimum
on the middle AP-sheet. In the region to the right of the $\Delta
=2E_{JT}^{\sigma }$ line there is $JT_{\sigma }$-minima on the middle
AP-sheet and the trivial non-analytical minimum on the upper AP-sheet. At $%
E_{JT}^{z}<2E_{JT}^{e}$ the regions of parameters exist where the $%
PJT_{\sigma }$-minima on the lower AP-sheet coexist with the $NJT$-minimum
on the upper sheet or with the $JT_{\sigma }$-minima on the middle sheet.
Between the $\Delta =-2E_{JT}^{z}$ and $\Delta =2E_{JT}^{\sigma }$ lines
only the trivial non-analytical minima on the upper sheets exist.
}
\label{}
\end{figure}

With a motion along line $E_{JT}^\sigma =E_{JT}^{\sigma ,1}$ from $-\infty $
up to $+\infty $ the curvature of the JT$_\sigma $-minima along $e_u$%
-directions decreases up to zero on a line $a_\sigma =0$. In this point the
JT$_\sigma $-minima transform into saddle points with simultaneous
appearence of the PJT$_\sigma $-minima (one of the JT$_\sigma $-minima
splits along $Q_1$-, another one splits along $Q_2$-direction). From the
line $a_\sigma =0$ up to $b=0$ the magnitude of rhombic distortion decreases
up to zero on a line $b=0$. The magnitude of $e_u$-coordinates of minima
firstly increases , then decreases up to zero, reaching a maximum on line $%
\Delta =-E_{JT}^\sigma +E_{JT}^z$. The $Q_z$-coordinate increases linearly ($%
\sim a_\sigma $) from zero up to $-q_z^{\left( 0\right) }$ on a line $b=0$.
Thus, the four PJT$_\sigma $-minima transform into one NJT-minimum. Further,
with $\Delta \rightarrow +\infty $ the $A_{1g}$ and $E_u$ terms interact
weaker and $\tilde{\omega }_e\rightarrow \omega _e$.

The situation with emergence of the PJT$_\sigma $-minima is realized when
the $e_u$-mode driven interaction of the $A_{1g}$ and $E_u$ terms is rather
effective. When $E_{JT}^\sigma >4E_{JT}^e-E_{JT}^z$ (for example $%
E_{JT}^\sigma =E_{JT}^{\sigma ,2}$), the $\sigma $-mode driven interaction
becomes more effective. In this case at $b=0$ the NJT minimum appears on the
lower sheet of the AP together with JT$_\sigma $-minima. All three minima
have the same energy on the line $\Delta = -E_{JT}^\sigma+E_{JT}^z$.
Further, at $\Delta \rightarrow +\infty $ JT$_\sigma $-minima become more
flat, without varying their coordinates, and at $a_\sigma =0$ only the
NJT-minimum remains.

In Fig.8 the diagram of the upper sheets of the AP is shown.

\section{Vibronic states in a presence of the spin-orbit coupling.}

Without taking account of the spin-orbit coupling the $^3E_u$ term is
isolated, and it has the six-times degenerated ground vibronic state. The
spin-orbit coupling mixes the $M_S=0$ states of the $^3E_u$ and $^1E_u$
terms. As well, this coupling splits lower vibronic states of the $^3E_u$
term which have $M_S=\pm 1$.

\subsection{ The $M_S=0$ states.}

\subsubsection{The well isolated $^1A_{1g}$ term.}

If the $^1A_{1g}$ and $^{1,3}E_u$ terms are well separated in energy ($%
\Delta _{AE}\gg \Delta _E,\lambda $) it is possible to consider the AP
within a basis of the $\left| 00E_u^x\right\rangle $, $\left|
00E_u^y\right\rangle $, $\left| 10E_u^x\right\rangle $, $\left|
10E_u^y\right\rangle $ states, and then take into account vibronic coupling
with the of $^1A_{1g}$ term as perturbation. The potential energy matrix has
a form:
\begin{equation}
\begin{array}{ll}
\lefteqn{ \hat{U}(Q)  =  \sum_{i}\frac{\omega _{i}^{2}Q_{i}^{2}}{2}\cdot \hat{I}+} 
%\nonumber 
\\
+ \left(
\begin{array}{cccc}
-\Delta _{E}+V_{\alpha }Q_{\alpha } & V_{\beta }Q_{\beta } & 0 & -i\lambda
\\ 
V_{\beta }Q_{\beta } & -\Delta _{E}-V_{\alpha }Q_{\alpha } & i\lambda & 0 \\ 
0 & -i\lambda & \Delta _{E}+V_{\alpha }Q_{\alpha } & V_{\beta }Q_{\beta } \\ 
i\lambda & 0 & V_{\beta }Q_{\beta } & \Delta _{E}-V_{\alpha }Q_{\alpha }
\end{array}
\right) ,
\end{array}
\end{equation}
where $\lambda $ is a submatrix element of the spin-orbit coupling. The
eigenvalues and eigenvectors of $\hat{U}(Q)$ are written as 
\begin{equation}
\begin{array}{ll}
\varepsilon _{1}=\Sigma -\sqrt{\left( \Delta _{E}+\rho \right) ^{2}+\lambda
^{2}}\, & \left| 1\right\rangle =-i\sin \eta _{1}\left| -\rho
,0\right\rangle +\cos \eta _{1}\left| \rho ,1\right\rangle \ , \\ 
\varepsilon _{2}=\Sigma -\sqrt{\left( \Delta _{E}-\rho \right) ^{2}+\lambda
^{2}}\, & \left| 2\right\rangle =i\sin \eta _{2}\left| \rho ,0\right\rangle
+\cos \eta _{2}\left| -\rho ,1\right\rangle \ , \\ 
\varepsilon _{3}=\Sigma +\sqrt{\left( \Delta _{E}-\rho \right) ^{2}+\lambda
^{2}}\, & \left| 3\right\rangle =\cos \eta _{2}\left| \rho ,0\right\rangle
+i\sin \eta _{2}\left| -\rho ,1\right\rangle \ , \\ 
\varepsilon _{4}=\Sigma +\sqrt{\left( \Delta _{E}+\rho \right) ^{2}+\lambda
^{2}}\, & \left| 4\right\rangle =\cos \eta _{1}\left| -\rho ,0\right\rangle
-i\sin \eta _{1}\left| \rho ,1\right\rangle \ ,
\end{array}
\end{equation}
where 
\[
\Sigma =\sum_{i}\frac{\omega _{i}^{2}Q_{i}^{2}}{2},\quad \rho =\sqrt{\left(
V_{\alpha }Q_{\alpha }\right) ^{2}+\left( V_{\beta }Q_{\beta }\right) ^{2}}, 
\]
\[
\left| \rho ,S\right\rangle =\cos \theta \left| S0E_{u}^{x}\right\rangle
+\sin \theta \left| S0E_{u}^{y}\right\rangle ,\quad \left| -\rho
,S\right\rangle =-\sin \theta \left| S0E_{u}^{x}\right\rangle +\cos \theta
\left| S0E_{u}^{y}\right\rangle , 
\]
\[
\tan 2\theta =\frac{V_{\beta }Q_{\beta }}{V_{\alpha }Q_{\alpha }}\ ,\quad
\tan 2\eta _{1}=\frac{\lambda }{-\Delta _{E}-\rho }\ ,\quad \tan 2\eta _{2}=%
\frac{\lambda }{-\Delta _{E}+\rho }\ . 
\]

The minima of the lower sheet of the AP are located on $Q_{\sigma }$-axis
(with $E_{JT}^{\sigma }>E_{JT}^{\sigma ^{\prime }}$, $\sigma ,\sigma
^{\prime }=\alpha ,\beta $) in points, which represent solutions of the
equation: 
\begin{equation}
\frac{\left| Q_{\sigma }\right| }{q_{\sigma }^{\left( 0\right) }}=\left( 1+%
\frac{\lambda ^{2}}{\left( V_{\sigma }\left| Q_{\sigma }\right| +\Delta
_{E}\right) ^{2}}\right) ^{-\frac{1}{2}}.
\end{equation}

The $Q_{\Gamma \gamma }=0$ is a point of discontinuity of the derivative.
The non-trivial minima, which correspond to the low-symmetric distortions of
the cluster, exist for the arbitrary large $\lambda $, but with $\lambda
\rightarrow \infty $ the minima depth and the distortions magnitude become
negligibly small.

A ground vibronic state is twice degenerated. In the strong coupling scheme
it is realized due to orthogonality of elecrtonic states belonging to
different minima of the AP. In a case $\lambda \ll E_{JT}^\sigma $ the
expressions for the minimum points and their energy are written as:

\begin{equation}
Q_{\sigma }=\pm \tilde{q}_{\sigma }^{\left( 0\right) },\ \tilde{q}_{\sigma
}^{\left( 0\right) }=q_{\sigma }^{\left( 0\right) }\left( 1-\frac{\lambda
^{2}}{2\left( \Delta _{E}+2E_{JT}^{\sigma }\right) ^{2}}\right) ,
\end{equation}
\[
\varepsilon _{0}=-\Delta _{E}-E_{JT}^{\sigma }-\frac{\lambda ^{2}}{2\left(
\Delta _{E}+2E_{JT}^{\sigma }\right) }\ . 
\]

The mixing coefficient for the triplet spin states with the $^{1}A_{1g}$
term wave funtion is proportional to 
\begin{equation}
\frac{\lambda }{\Delta _{E}+\Delta _{AE}+E_{JT}^{e}}\ .
\end{equation}

\subsubsection{The strong PJT-effect for the singlet spin states.}

In this case it is neccessary to consider a mixing of the tunnel states with
lower vibronic states of $^3E_u$ term due to the spin-orbit coupling. The
only non-zero matrix elements are those between the tunnel $E_u$-states and
the lower vibronic states of $^3E_u$ term with $M_S=0$. The effective
Hamiltonian matrix is

\begin{equation}
\left( 
\begin{array}{cccc}
-\Delta _{E}^{\prime } & 0 & 0 & -i\lambda ^{\prime } \\ 
0 & -\Delta _{E}^{\prime } & i\lambda ^{\prime } & 0 \\ 
0 & -i\lambda ^{\prime } & \Delta _{E}^{\prime } & 0 \\ 
i\lambda ^{\prime } & 0 & 0 & \Delta _{E}^{\prime }
\end{array}
\right) ,
\end{equation}
where $2\Delta ^{\prime }$ is an appropriate energy splitting, $\lambda
^{\prime }$ is a modified matrix element of the spin-orbit coupling: 
\begin{equation}
\lambda ^{\prime }=\ \lambda \ N_{E_{u}}^{-1}\sqrt{2}d_{\sigma }\left\langle
\chi _{M}^{\left( \sigma \right) }|\tilde{\chi}_{0}^{\left( \sigma \right)
}\right\rangle \ ,
\end{equation}
where $\chi _{M}^{\left( \sigma \right) }$ and $\tilde{\chi}_{0}^{\left(
\sigma \right) }$ are the ground state vibrational functions, which are
centered in the minimum $M$ of PJT$_{\sigma }$-type and in the minimum of
the AP of the $E-b_{1}-b_{2}$-problem with opposite to $M$ sign of $\sigma $%
-mode, respectively.

\subsubsection{The singlet-triplet asymmetry of the vibronic coupling.}

Let the difference in the linear vibronic coupling constants for singlet $%
^1E_u$ and $^3E_u$ triplet states corresponds to the following inequalities
for the JT-energies: $E_{JT}^\alpha (^1E_u)>E_{JT}^\beta (^3E_u)$ and $%
E_{JT}^\alpha (^1E_u)<E_{JT}^\beta (^3E_u)$.

The the minima of the AP of the $^1E_u$ term are located on the $Q_\alpha $%
-axis, and those of the $E_u$ therm are located on the $Q_\beta $-axis. If
the sheets of the AP for different terms intersect, taking account of the
spin-orbit coupling could result in a complicated form of the AP with four
minima. The states in minima located on the same axis are orthogonal to each
other, and those located on different axis are not. Hence, even if $\Delta
_E=0$ the lower vibronic states are two doublets, which are separated by the
tunnel splitting $\tilde{\Delta }$. The frequency related to $\tilde{\Delta }
$ corresponds to the combined pulsing motion of the electronic and nuclear
density between $b_{1g}$- and $b_{2g}$-distortions of $CuO_4$-cluster.

\subsection{The $M_{S}=\pm 1$ states.}

A joint operation of the vibronic and spin-orbital coupling for the $%
M_{S}=\pm 1$ states within the $^{3}E_{u}$ manifold is described by the
matrix 
\begin{equation}
V_{\alpha }Q_{\alpha }\hat{\sigma}_{z}+V_{\beta }Q_{\beta }\hat{\sigma}%
_{x}+iM_{S}\lambda _{1}\hat{\sigma}_{y}\ ,
\end{equation}
where $\lambda _{1}$ is a submatrix element for the spin-orbital coupling
within the orbital part of the $^{3}E_{u}$ manifold, $\hat{\sigma}_{i}$ are
the Pauli matrices, and the energy is counted off the $^{3}E_{u}$ manifold.

The lower sheet of the adiabatic potential has four extrema at points 
\begin{equation}
q_{i}^{\left( 0,\pm \right) }=\pm l_{i}\sqrt{2\tilde{E}_{JT}^{i}\left(
1-p_{i}^{2}\right) }\ ,\quad i=\alpha ,\beta ,
\end{equation}
where $l_{i}=\sqrt{\hbar /\omega _{i}}$, $\tilde{E}_{JT}^{i}=E_{JT}^{i}/%
\hbar \omega _{i}$, $p_{i}=\lambda _{1}/2E_{JT}^{i}$, and $q_{i}^{\left(
0,\pm \right) }=0$ at $\left| p_{i}\right| \geq 1$. The parameter $p_{i}$
equals to the ratio of the $E_{u}$-level splitting due to the spin-orbital
coupling ($2\lambda _{1}$) to that of due to the vibronic coupling ($%
4E_{JT}^{i}$ ). At $\left| p_{i}\right| \geq 1$ ($i=\alpha ,\beta $) we come
to the weak pseudo-Jahn-Teller effect with the only minimum at $%
q_{i}^{\left( 0,\pm \right) }=0$ ($i=\alpha ,\beta $) and with the effective
local vibration frequencies 
\begin{equation}
\tilde{\omega}_{i}^{2}=\omega _{i}^{2}\left( 1-\left| p_{i}\right|
^{-1}\right) .
\end{equation}

At $\left| p_{i}\right| \leq 1$ ( $E_{JT}^{\sigma }>E_{JT}^{\sigma ^{\prime
}}$) the strong pseudo-Jahn-Teller effect occurs with the AP-minima at $%
q_{\sigma }^{\left( 0,\pm \right) }$, and with the AP-saddle points at $%
q_{\sigma ^{\prime }}^{\left( 0,\pm \right) }$. The effective frequencies
for the local vibrations at the AP-minima and corresponding energies are
derived as follows: 
\begin{equation}
\begin{array}{c}
\tilde{\omega}_{\sigma }^{2}=\omega _{\sigma }^{2}\left( 1-p_{\sigma
}^{2}\right) , \\ 
\tilde{\omega}_{\sigma ^{\prime }}^{2}=\omega _{\sigma ^{\prime }}^{2}\left(
1-\lambda _{\sigma }\right) ,\quad \lambda _{\sigma }=\frac{E_{JT}^{\sigma
^{\prime }}}{E_{JT}^{\sigma }}\ , \\ 
\varepsilon _{-}\left( q_{\sigma }^{\left( 0,\pm \right) }\right)
=-E_{JT}^{\sigma }-\hbar \omega _{\sigma }p_{\sigma }\ .
\end{array}
\end{equation}
As is seen, unlike the usual $E-b_{1}-b_{2}$-problem, the $\sigma $-mode
frequency is renormalized due to the spin-orbital coupling.

The electronic wave functions $\varphi _{\pm }^{\left( M_{S}\right) }$ in
the $q_{\sigma }^{\left( 0,\pm \right) }$ minima are reduced to the
following form: 
\begin{equation}
\begin{array}{c}
\varphi _{+}^{\left( M_{S}\right) }=\frac{1}{\sqrt{2}}\left( i\ M_{S}\sqrt{1-%
\sqrt{1-p_{\sigma }^{2}}}\left| E_{u}^{\left( 1\right) }\right\rangle +\sqrt{%
1+\sqrt{1-p_{\sigma }^{2}}}\left| E_{u}^{\left( 2\right) }\right\rangle
\right) , \\ 
\varphi _{-}^{\left( M_{S}\right) }=\frac{1}{\sqrt{2}}\left( \sqrt{1+\sqrt{%
1-p_{\sigma }^{2}}}\left| E_{u}^{\left( 1\right) }\right\rangle +i\ M_{S}%
\sqrt{1-\sqrt{1-p_{\sigma }^{2}}}\left| E_{u}^{\left( 2\right)
}\right\rangle \right) .
\end{array}
\end{equation}
In contrast to the above considered $E-b_{1}-b_{2}$-problem, taking account
of the spin-orbital coupling results in the non-orthogonality of the wave
functions for the minima of the lower sheet of the adiabatic potential: 
\begin{equation}
\left\langle \varphi _{+}^{\left( M_{S}\right) }|\varphi _{-}^{\left(
M_{S}\right) }\right\rangle =-i\ M_{S}\ p_{\sigma }
\end{equation}
that leads to the tunnel splitting.

Varying the energy functional with the basis functions 
\begin{equation}
\Psi =a\ \varphi _{+}^{\left( M_{S}\right) }\chi _{+}+b\ \varphi
_{-}^{\left( M_{S}\right) }\chi _{-},
\end{equation}
where $\chi _{\pm }$ are the vibration functions centered at $q_{\sigma
}^{\left( 0,\pm \right) }$, results in a matrix equation: 
\begin{equation}
\left( 
\begin{array}{cc}
H_{++} & H_{+-} \\ 
H_{+-}^{*} & H_{++}
\end{array}
\right) \left( 
\begin{array}{c}
c_{1} \\ 
c_{2}
\end{array}
\right) =E\ \left( 
\begin{array}{cc}
1 & S_{+-} \\ 
S_{+-}^{*} & 1
\end{array}
\right) \left( 
\begin{array}{c}
c_{1} \\ 
c_{2}
\end{array}
\right) ,
\end{equation}
where 
\begin{equation}
\begin{array}{c}
S=\left\langle \varphi _{+}^{\left( M_{S}\right) }\chi _{+}|\varphi
_{-}^{\left( M_{S}\right) }\chi _{-}\right\rangle =-i\ M_{S}\ p_{\sigma }\
e^{-2\tilde{E}_{JT}^{\sigma }\left( 1-p_{\sigma }^{2}\right) ^{\frac{3}{2}}},
\\ 
H_{++}=\hbar \omega _{\sigma }\left[ \frac{2-p_{\sigma }^{2}}{4\sqrt{%
1-p_{\sigma }^{2}}}-\tilde{E}_{JT}^{\sigma }\left( 1+p_{\sigma }^{2}\right)
\right] +\hbar \omega _{\sigma ^{\prime }}\frac{2-\lambda _{\sigma }}{4\sqrt{%
1-\lambda _{\sigma }}}\ , \\ 
H_{+-}=\left( H_{++}-\hbar \omega _{\sigma }B\right) S\ ,\quad B=\tilde{E}%
_{JT}^{\sigma }\left( 1-p_{\sigma }^{2}\right) \left( 2-p_{\sigma
}^{2}\right) .
\end{array}
\label{tunnel matrix elements}
\end{equation}

The tunnel level energies and tunnel splitting are determined as follows: 
\begin{equation}
\begin{array}{c}
E_{g}=H_{++}-\frac{\hbar \omega _{\sigma }B\left| S\right| \ }{1+\left|
S\right| }\ ,\quad E_{u}=H_{++}+\frac{\hbar \omega _{\sigma }B\left|
S\right| \ }{1-\left| S\right| }\ , \\ 
\Delta _{1}=E_{g}-E_{u}=2\hbar \omega _{\sigma }\frac{B\left| S\right| \ }{%
1-\left| S\right| ^{2}}\ .
\end{array}
\end{equation}
and appropriate vibronic wave functions can be represented as: 
\begin{equation}
\begin{array}{llll}
\Psi _{g}^{\left( 1\right) }=\frac{1}{\sqrt{2\left( 1+\left| S\right|
\right) }}\left( \varphi _{+}^{\left( 1\right) }\chi _{+}+i\varphi
_{-}^{\left( 1\right) }\chi _{-}\right) ; \\
 \Psi _{u}^{\left( 1\right) }=%
\frac{1}{\sqrt{2\left( 1-\left| S\right| \right) }}\left( i\varphi
_{+}^{\left( 1\right) }\chi _{+}+\varphi _{-}^{\left( 1\right) }\chi
_{-}\right) ; \\ 
\Psi _{g}^{\left( -1\right) }=\frac{1}{\sqrt{2\left( 1+\left| S\right|
\right) }}\left( i\varphi _{+}^{\left( -1\right) }\chi _{+}+\varphi
_{-}^{\left( -1\right) }\chi _{-}\right) ; \\
 \Psi _{u}^{\left( -1\right) }=%
\frac{1}{\sqrt{2\left( 1-\left| S\right| \right) }}\left( \varphi
_{+}^{\left( -1\right) }\chi _{+}+i\varphi _{-}^{\left( -1\right) }\chi
_{-}\right) .
\end{array}
\label{tunnel states}
\end{equation}
In the limit $\left| p_{\sigma }\right| \ll 1$ for the magnitude of the
tunnel splitting one obtains: 
\begin{equation}
\Delta _{1}=2\lambda _{1}e^{-2\tilde{E}_{JT}^{\sigma }}\ ,
\end{equation}
that might be interpreted as a result of the vibronic reduction for the
purely spin orbital splitting.

\section{Spin Hamiltonian of the PJT center.}

A detailed pattern of the energy spectrum in the external magnetic field and
the magneto-resonance properties of the PJT-centers are substantially depend
on the bare $A-E$-splitting $\Delta $, vibronic parameters and relative
magnitude of the vibronic and spin-orbital coupling.

However, some common features are determined only by the symmetry
considerations, in particular, by the $C_4$ axial symmetry and the specific
properties of the bare electronic basis functions. So, the orbital doublet $%
^{1,3}E_u$ terms are characterized by the nonquenched highly anisotropic
(Izing like) orbital moment $\tilde{l}=1/2$ oriented only along the $C_4$
axis.

Without account of vibronic coupling the effective spin Hamiltonian for the $%
^{1,3}E_{u}$ terms could be represented as follows: 
\begin{equation}
H=\lambda _{1}\tilde{l}_{z}\hat{S}_{z}+\beta H_{z}g_{l}\tilde{l}_{z}\hat{I}%
+2\beta \sum_{i=x,y,z}H_{i}\hat{S}_{i},
\end{equation}
where $\beta $ is the Bohr magneton, $\hat{S}_{i}$ are spin matrices ($S=0$
for the spin singlet $^{1}E_{u}$ term), $g_{l}$ is effective orbital $g$%
-factor for the $^{1,3}E_{u}$ term, which value is determined by the
structure of the electronic $e_{u}$-function. Here, the first term describes
the spin-orbital coupling, the second and the third terms correspond to the
orbital and spin Zeeman coupling, respectively, with the purely isotropic
spin $g$-tensor: $g_{x}=g_{y}=g_{z}=2$. Taking account of the spin-orbital $%
^{1}E_{u}$-$^{3}E_{u}$-mixing leads to an emergence of the spin anisotropy
with an additive contribution to the effective spin Hamiltonian: 
\begin{equation}
V_{an}=D\hat{S}_{z}^{2}\ ,\quad D=\Delta _{st}-\sqrt{\Delta
_{st}^{2}+\lambda ^{2}}\approx -\frac{\lambda ^{2}}{2\Delta _{st}}\ ,
\end{equation}
and the effective axial anisotropy of the spin $g$-tensor: $g_{z}=2$, $%
g_{x}=g_{y}=2\cos \theta $, where 
\begin{equation}
\cos \theta =\frac{1}{\sqrt{2}}\sqrt{1+\frac{\Delta _{st}}{\sqrt{\Delta
_{st}^{2}+\lambda ^{2}}}}\approx 1-\frac{\lambda ^{2}}{8\Delta _{st}^{2}}\ .
\end{equation}

Taking account of the vibronic coupling upon the conditions of the weak
pseudo-Jahn-Teller effect ($p_{\sigma }=\lambda _{1}/2E_{JT}^{\sigma }>1$)
does not vary a form of the effective spin Hamiltonian since the vibronic
distortions are suppressed by the spin-orbital coupling. With the strong
pseudo-Jahn-Teller effect ( $p_{\sigma }=\lambda _{1}/2E_{JT}^{\sigma }<1$)
one should make use of the vibronic states (\ref{tunnel states}), that
results in a very complicated spin-vibronic effective Hamiltonian. A
relatively simple situation occurs at $p_{\sigma }^{2}\ll 1$, and small
values of the overlap for the oscillatory functions when neglecting the
spin-orbital $^{1}E_{u}$-$^{3}E_{u}$-mixing, we come to effective
Hamiltonian: 
\begin{equation}
H=-\beta H_{z}g_{1}\left\langle \chi _{+}|\chi _{-}\right\rangle \tilde{l}%
_{z}\hat{I}-\beta \sum_{i=x,y,z}H_{i}g_{ij}\hat{S}_{j}\ ,
\end{equation}
where the overlap integral $\left\langle \chi _{+}|\chi _{-}\right\rangle $
is determined according to (\ref{tunnel matrix elements}) and for the axial $%
g$-tensor: $g_{z}=2-p_{\sigma }g_{1}$, $g_{x}=g_{y}=2$. It should be noted
the principal difference in the $g$-factor anisotropy with and without
vibronic effects.

The doublet of the states with different ($\pm $) projections of the
effective orbital moment corresponds to the functions \{$\Psi _{g}^{\left(
1\right) }$, $\Psi _{g}^{\left( 0\right) }$, $\Psi _{u}^{\left( -1\right) }$%
\} and \{$\Psi _{u}^{\left( 1\right) }$, $\Psi _{u}^{\left( 0\right) }$, $%
\Psi _{g}^{\left( -1\right) }$\}, respectively, where, along with the above
defined functions (\ref{tunnel states}), we have introduced 
\begin{equation}
\begin{array}{ll}
\Psi _{g}^{\left( 0\right) }=\frac{1}{\sqrt{2}}\left( \left| E_{u}^{\left(
2\right) }\right\rangle \ \chi _{+}^{\left( 0\right) }+i\left| E_{u}^{\left(
1\right) }\right\rangle \ \chi _{-}^{\left( 0\right) }\right) , \\ 
\Psi_{u}^{\left( 0\right) }=\frac{1}{\sqrt{2}}\left( i\left| E_{u}^{\left(
2\right) }\right\rangle \ \chi _{+}^{\left( 0\right) }+\left| E_{u}^{\left(
1\right) }\right\rangle \ \chi _{-}^{\left( 0\right) }\right) .
\end{array}
\end{equation}
An upper label for the vibration function underlines the difference in the
location of the minima for the $M_{S}=0$ and $M_{S}\neq 0$ spin $^{3}E_{u}$%
-states within a lower sheet of the adiabatic potential. It should be noted
also, that, in general, the overlap integral $\left\langle \chi _{+}|\chi
_{-}\right\rangle $ depends on the $z$-component of the magnetic field due
to the orbital Zeeman coupling which contributes to the energy of the lower
sheet of the adiabatic potential. At $\beta H_{z}\gg \lambda _{1}$ an
expression for the corresponding energy acquires a form: 
\begin{equation}
\varepsilon _{-}=\frac{1}{2}\left( \omega _{\alpha }^{2}Q_{\alpha
}^{2}+\omega _{\beta }^{2}Q_{\beta }^{2}\right) -\sqrt{\left( V_{\alpha
}Q_{\alpha }\right) ^{2}+\left( V_{\beta }Q_{\beta }\right) ^{2}+\left(
\beta H_{z}g_{1}\right) ^{2}}.
\end{equation}

\section{Some experimental evidence of the singlet-triplet PJT-centers}

Main hopes to the direct demonstration of the validity of the above model
are connected with a reveal of the isolated PJT-center. In this connection
let note the recent paper \cite{Yoshinari1996} which authors undertook the
NQR-study of the hole centers $CuO_4^{5-}$ in $La_2Cu_{0.5}Li_{0.5}O_4$.
Their results can be interpreted as convincing evidence of the
singlet-triplet PJT-structure of the hole center. This conclusion is based
on the following:

1. The authors revealed the spin singlet ground state ($S=0$) and the low
lying spin triplet state ($S=1$) with the singlet-triplet separation $%
\Delta_{st}=0.13\ eV$.

2. They observed anomalous weak temperature dependence of the relaxation
rate at low temperatures that evidences the occurrence of the spinless
multiplet structure in the $CuO_4^{5-}$-cluster ground state.

3. They revealed an appreciable spin contribution to the low temperature
relaxation indicating the simultaneous occurrence of the ground state
multiplet structure , the sufficiently low singlet-triplet separation and
the intrinsic singlet-triplet spin-orbital mixing.

4. They observed the relaxation inequivalence of the various $Cu$ sites
quite natural for the PJT-centers in the conditions of the static JT-effect.

We think that the further investigations of this system can provide a
valuable information about PJT-centers.

The Jahn-Teller hole centers like $CuO_4^{5-}$ with the singlet-triplet
quasi-de\-ge\-ne\-ra\-cy within ground state have been observed by ESR spectroscopy
in \\
\mbox{$LaSrAl_{1-x}Cu_xO_4$} which is isostructural to $La_{2-x}Sr_xCuO_4$ \cite
{Ivanova}. An important indication to the $O2p(\pi )$ nature of the doped
holes and, consequently, to occurrence of the degeneracy for configurations
like $b_{1g}^2$ and $b_{1g}e_u(\pi )$ was obtained by Yoshinari \cite{Yo}
after analysis of the $^{17}O$ Knight shift data in $YBa_2Cu_3O_{6+x}$.

Among diverse peculiarities of the spin-triplet PJT centers it should be
especially emphasized a possible appearance of the so called ''tunnel
paramagnetic centers'' \cite{FTT} which spin states $S=1$, $M_{S}=\pm 1$ and 
$S=1$, $M_{S}=0$, respectively, are localized within different wells of the
adiabatic potential. In other words, different spin states correspond to
different local distortions of the $CuO_{4}$ cluster. Spin dynamics and
relaxation for the tunnel paramagnetic centers is cruiciably dependent on
the magnitude and orientation of the external magnetic field. These centers
could be relatively simply transferred to the metastable state.

It should be noted that we did not undertake the task of reviewing all the
available experimental data confirming the PJT nature of the $CuO_4$
clusters in doped cuprates and comparing them with our model approach: it is
a separate problem. At the same time, it should be noted that the PJT
centers model is entirely based on the vast amount of the experimental
material.

In conclusion, we would like to acknowledge the stimulating discussions with
Prof. M.V.Eremin.

\begin{thebibliography}{99}
\bibitem{Moskvin1993}  A.S.Moskvin, {\it JETP Lett.} {\bf 58}, 342 (1993)

\bibitem{Moskvin1994}  A.S.Moskvin, N.N.Loshkareva, Yu.P.Sukhorukov,
M.A.Sidorov, A.A.Sa\-mo\-khva\-lov, {\it JETP} {\bf 105}, 967 (1994).

\bibitem{Moskvin1997}  A.S.Moskvin, {\it Physica} {\bf C 282-287}, 1807 (1997)

\bibitem{Hirsch1989}  J.E. Hirsch, S. Tang, {\it Phys.Rev.} {\bf B 40}, 2179 (1989)

\bibitem{Bersuker1983}  I.B.Bersuker and V.Z.Polinger. {\it Vibronic
Interactions in Molecules and Crystals} (Springer-Verlag, Berlin) 1989

\bibitem{OpikPryce1957}  U. Opik, M.H.L. Pryce. {\it Proc.Roy.Soc.} {\bf A 238},
425 (1957).

\bibitem{Yoshinari1996}  Y. Yoshinari, P.C. Hammel, J.A. Martindale, E.
Moshopoulou, J.D. Thompson, {\it Phys. Rev. Lett.} {\bf 77}, 2069 (1996)

\bibitem{Ivanova}  T.A.Ivanova, E.F.Kukovitsky, E.N.Nabiullin et al., {\it JETP
Lett.} {\bf 57}, 64 (1993)

\bibitem{Yo}  Y.Yoshinari, {\it Physica} {\bf C 276}, 147 (1997)

\bibitem{FTT}  A.S. Moskvin, I.B. Krinetsky, R. Szymczak, Yu.D. Panov, S.V.
Naumov, A.A. Samokhvalov, {\it Fizika Tverd. Tela} {\bf 39}, 474 (1997)

\end{thebibliography}

\newpage

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%\begin{document}

\centerline{\bf TABLES.}
\bigskip
{\bf Table 1.}
\bigskip
$$
\begin{array}{||c||c||c|c|c||c|c|c|c|c||}
\hline
Type & N & \left| 00A_{1g}\right\rangle  & \left| 00E_u^x\right\rangle  & \left|
00E_u^y\right\rangle  & Q_z^0 & Q_\alpha ^0 & Q_\beta ^0 & Q_x^0 & Q_y^0 \\ \hline\hline

NJT &1& 1 & 0 & 0 & -q_z^{(0)} & 0 & 0 & 0 & 0 \\ \hline\hline

JT_\alpha &1& 0 & 0 & 1 & 0 & q_\alpha ^{(0)} & 0 & 0 & 0 \\ \hline
  
          &2& 0 & 1 & 0 & 0 & -q_\alpha ^{(0)} & 0 & 0 & 0 \\ \hline\hline

JT_\beta  &1& 0 & -1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 & q_\beta ^{(0)} & 0 & 0 \\ \hline

          &2& 0 & 1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 & -q_\beta ^{(0)} & 0 & 0 \\ \hline\hline

PJT_\alpha &1& c_\alpha  & 0 & d_\alpha  & -q_z^{(0)}c_\alpha ^2 & q_\alpha
^{(0)}d_\alpha ^2 & 0 & 0 & -q_e^{(0)}2c_\alpha d_\alpha  \\ \hline

           &2& c_\alpha  & d_\alpha  & 0 & -q_z^{(0)}c_\alpha ^2 & -q_\alpha
            ^{(0)}d_\alpha ^2 & 0 & -q_e^{(0)}2c_\alpha d_\alpha  & 0 \\ \hline

           &3& c_\alpha  & 0 & -d_\alpha  & -q_z^{(0)}c_\alpha ^2 & q_\alpha
            ^{(0)}d_\alpha ^2 & 0 & 0 & q_e^{(0)}2c_\alpha d_\alpha  \\ \hline

           &4& c_\alpha  & -d_\alpha  & 0 & -q_z^{(0)}c_\alpha ^2 & -q_\alpha
            ^{(0)}d_\alpha ^2 & 0 & q_e^{(0)}2c_\alpha d_\alpha  & 0 \\ \hline\hline

PJT_\beta  &1& c_\beta  & -d_\beta /\sqrt{2} & d_\beta /\sqrt{2} & 
-q_z^{(0)}c_\beta ^2 & 0 & q_\alpha ^{(0)}d_\beta ^2 & q_e^{(0)}\sqrt{2}%
c_\beta d_\beta  & -q_e^{(0)}\sqrt{2}c_\beta d_\beta  \\ \hline

           &2& c_\beta  & d_\beta /\sqrt{2} & d_\beta /\sqrt{2} & -q_z^{(0)}c_\beta ^2 & 
           0 & -q_\alpha ^{(0)}d_\beta ^2 & -q_e^{(0)}\sqrt{2}c_\beta d_\beta  & 
           -q_e^{(0)}\sqrt{2}c_\beta d_\beta  \\ \hline

           &3& c_\beta  & d_\beta /\sqrt{2} & -d_\beta /\sqrt{2} & -q_z^{(0)}c_\beta ^2 & 
           0 & q_\alpha ^{(0)}d_\beta ^2 & -q_e^{(0)}\sqrt{2}c_\beta d_\beta  & 
           q_e^{(0)}\sqrt{2}c_\beta d_\beta  \\ \hline

           &4& c_\beta  & -d_\beta /\sqrt{2} & -d_\beta /\sqrt{2} & -q_z^{(0)}c_\beta ^2
           & 0 & -q_\alpha ^{(0)}d_\beta ^2 & q_e^{(0)}\sqrt{2}c_\beta d_\beta  & 
           q_e^{(0)}\sqrt{2}c_\beta d_\beta  \\ \hline
\end{array}
$$

 
where
$$
q_i^{(0)}=\frac{V_i}{\omega _i^2},\ 
a_\sigma =\Delta +4E_{JT}^e-2E_{JT}^\sigma ,\
b=-\Delta +4E_{JT}^e-2E_{JT}^z,\ 
c_\sigma =\sqrt{\frac{a_\sigma }{a_\sigma +b}},\ 
d_\sigma =\sqrt{\frac b{a_\sigma +b}}.
$$

\newpage
{\bf Table 2.}
\bigskip

$$
\begin{array}{||c|c|c||}
\hline
Type &N& 2\varepsilon (q)-2\varepsilon_0^\sigma -\sum_i\omega_i^2q_i^2 \\ \hline\hline

NJT &1& -\omega_e^2\eta_z\left( q_x^2+q_y^2\right)  \\ \hline\hline

JT_\alpha &1& -\omega_e^2\kappa_\alpha q_y^2-\omega_\beta ^2\lambda_\alpha q_\beta ^2 \\ \hline
  
          &2& -\omega_e^2\kappa_\alpha q_x^2-\omega_\beta ^2\lambda_\alpha q_\beta ^2 \\ \hline\hline

JT_\beta  &1& -\omega_e^2\kappa_\beta q_2^2-\omega_\alpha ^2\lambda_\beta q_\alpha ^2\\  \hline

          &2& -\omega_e^2\kappa_\beta q_1^2-\omega_\alpha ^2\lambda_\beta q_\alpha ^2 \\ \hline\hline

PJT_\alpha &1& -\left( \omega
_e\upsilon _\alpha q_x+\omega _\beta \rho _\alpha q_\beta \right) ^2-\left(
\omega _z\tau _\alpha q_z+\omega _\alpha \mu _\alpha q_\alpha -\omega _e\nu
_\alpha q_y\right) ^2 \\  \hline

           &2& -\left( \omega
_e\upsilon _\alpha q_y+\omega _\beta \rho _\alpha q_\beta \right) ^2-\left(
\omega _z\tau _\alpha q_z-\omega _\alpha \mu _\alpha q_\alpha -\omega _e\nu
_\alpha q_x\right) ^2 \\  \hline

           &3& -\left( \omega
_e\upsilon _\alpha q_x-\omega _\beta \rho _\alpha q_\beta \right) ^2-\left(
\omega _z\tau _\alpha q_z+\omega _\alpha \mu _\alpha q_\alpha +\omega _e\nu
_\alpha q_y\right) ^2 \\  \hline

           &4& -\left( \omega
_e\upsilon _\alpha q_y-\omega _\beta \rho _\alpha q_\beta \right) ^2-\left(
\omega _z\tau _\alpha q_z-\omega _\alpha \mu _\alpha q_\alpha +\omega _e\nu
_\alpha q_x\right) ^2 \\ \hline\hline

PJT_\beta  &1& -\left( \omega
_e\upsilon _\beta q_1-\omega _\alpha \rho _\beta q_\alpha \right) ^2-\left(
\omega _z\tau _\beta q_z+\omega _\beta \mu _\beta q_\beta -\omega _e\nu
_\beta q_2\right) ^2 \\  \hline

           &2& -\left( \omega _e\upsilon _\beta
q_2-\omega _\alpha \rho _\beta q_\alpha \right) ^2-\left( \omega _z\tau
_\beta q_z-\omega _\beta \mu _\beta q_\beta -\omega _e\nu _\beta q_1\right)^2 \\  \hline

           &3& -\left( \omega _e\upsilon _\beta
q_1+\omega _\alpha \rho _\beta q_\alpha \right) ^2-\left( \omega _z\tau
_\beta q_z+\omega _\beta \mu _\beta q_\beta +\omega _e\nu _\beta q_2\right)^2 \\  \hline

           &4& -\left( \omega _e\upsilon _\beta
q_2+\omega _\alpha \rho _\beta q_\alpha \right) ^2-\left( \omega _z\tau
_\beta q_z-\omega _\beta \mu _\beta q_\beta +\omega _e\nu _\beta q_1\right)^2 \\ \hline
\end{array}
$$
 
where
$$
q_1=\frac{q_x+q_y}{\sqrt{2}},\ 
q_2=\frac{-q_x+q_y}{\sqrt{2}},\ 
a_\sigma =\Delta +4E_{JT}^e-2E_{JT}^\sigma ,\
b=-\Delta +4E_{JT}^e-2E_{JT}^z,\ 
$$
$$
c_\sigma =\sqrt{\frac{a_\sigma }{a_\sigma +b}},\ 
d_\sigma =\sqrt{\frac b{a_\sigma +b}},\ 
\eta _z=\frac{4E_{JT}^e}{\Delta +2E_{JT}^z}, \ 
\kappa _\sigma =\frac{4E_{JT}^e}{-\Delta +2E_{JT}^\sigma }, \ 
\lambda _\sigma =\frac{E_{JT}^{\sigma ^{\prime }}}{E_{JT}^\sigma },
$$
$$
\upsilon _\sigma =\sqrt{\frac{E_{JT}^ea_\sigma }{E_{JT}^ea_\sigma +E_{JT}^\sigma b}}, \ 
\rho _\sigma =\sqrt{\frac{E_{JT}^{\sigma^{\prime }}b}{E_{JT}^ea_\sigma +E_{JT}^\sigma b}},
$$
$$
\tau _\sigma =\sqrt{\frac{E_{JT}^za_\sigma b}{E_{JT}^e\left( a_\sigma +b\right) ^2}}, \ 
\mu _\sigma =\sqrt{\frac{E_{JT}^\sigma a_\sigma b}{E_{JT}^e\left( a_\sigma +b\right) ^2}}, \ 
\nu _\sigma =\frac{a_\sigma -b}{a_\sigma +b},
$$
with $\sigma =\alpha $ at $\sigma ^{\prime }=\beta $, and $\sigma
=\beta $ at $\sigma ^{\prime }=\alpha $.

For NJT-extremum $\varepsilon _0^\sigma =-\Delta -E_{JT}^z$; 
for JT$_\sigma $-extrema points $\varepsilon _0^\sigma =-E_{JT}^\sigma $; 
for PJT$_\sigma $-extrema points $\varepsilon _0^\sigma =-E_{JT}^\sigma -\frac 12c_\sigma
^2a_\sigma $.


\bigskip
{\bf Table 3.}
\bigskip

$$
\begin{array}{|c|c|c|}
\hline
     & a_\sigma <0       & a_\sigma >0     \\ \hline
b<0  & NJT + JT_\sigma   &\quad NJT  \quad \\ \hline
b>0  & JT_\sigma         & PJT_\sigma      \\ \hline
\end{array}
$$

 
where $
a_\sigma =\Delta +4E_{JT}^e-2E_{JT}^\sigma ,\
b=-\Delta +4E_{JT}^e-2E_{JT}^z.
$


\end{document}
