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invariance conditions, gauge transformations, variational calculus
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% versione del giorno 11-3-1999                                                       %
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%\input boldgreek 
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\documentstyle[12pt]{article}     
\begin{document}     
\renewcommand{\thepage}{\hglue 9.3cm\arabic{page}\hfill}     
\renewcommand{\thesection}{\Roman{section}}       
\newcounter{numeq}      
\title{\large \bf Some mathematical properties of 
gauge transformations with respect to the Coulomb's gauge: variational analysis
of an energy functional}
\author{\normalsize
 M. P. B\'eccar Varela, M. C. Caputo,  M. B. Ferraro $^{\ddag}$\\     
%\phantom{riga bianca}\\     
\normalsize Departamento de F\'{\i}sica.\\
\normalsize Facultad de Ciencias Exactas y     
 Naturales\\   
\normalsize Universidad de Buenos Aires\\     
\normalsize Ciudad Universitaria. Pab. I\\     
%\phantom{riga bianca}\\     
\normalsize (1428) Buenos Aires, Argentina\\
\phantom{riga bianca}\\
\normalsize P. Lazzeretti$^{\dag}$,\\
%\phantom{riga bianca}\\
\normalsize Dipartimento di Chimica dell'Universit\`a degli Studi di Modena \\
\normalsize Via Campi 183, 41100 Modena, Italy\\
\phantom{riga bianca}\\
\normalsize  M. C. Mariani$^{\ddag}$ and D. Rial \\
\normalsize Departamento de Matem\'atica.\\
\normalsize Facultad de Ciencias Exactas y     
 Naturales\\   
\normalsize Universidad de Buenos Aires\\     
\normalsize Ciudad Universitaria. Pab. I\\     
%\phantom{riga bianca}\\     
\normalsize (1428) Buenos Aires, Argentina}     
\date{\phantom{}}     
        
\maketitle     
     
\begin{abstract}     
As gauge invariance of computed magnetic properties, usually partitioned into
diamagnetic and paramagnetic terms, is not achieved within the 
algebraic approximation, unless {\sl ad hoc} techniques are adopted, 
a general variational treatment is analyzed, attempting to minimize the term
more difficult to evaluate accurately, i.e., the
paramagnetic contribution to magnetic susceptibility, by means of a gauge
transformation. It is shown that an absolute minimum in a
variational sense cannot be determined a priori. However, a `local' minimum 
of the paramagnetic contribution to
magnetic susceptibility can be arrived at by
employing general gauge transformations
of polynomial form.
\end{abstract}
\noindent     
\begin{description}     
\item $^{\dag}$To whom correspondence should be addressed.
\item $^{\ddag}$Members of Carrera del Investigador del CONICET.    
\end{description}     
\thispagestyle{empty}                        % produce una pagina senza numero     
%\vfill \eject     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
The gauge problem affecting theoretical evaluation of
response properties of a molecule in the presence of an external magnetic
field arises from the 
unphysical dependence of calculated values on the non-unique definition
of the vector potential.     
     
It is well-established, from theoretical~\cite{lipscomb,epst}, as well as
computational point of view~\cite{holler,plzold}, that coupled
Hartree-Fock (CHF) methods~{\cite{lipscomb}} yield  magnetic susceptibility
and nuclear magnetic shielding invariant to a gauge transformation only
in the limit of complete atomic basis sets.
This requirement brings in serious difficulties, for, in most cases,
use of large basis sets is mandatory to achieve near Hartree-Fock accuracy,
and, at the same time, a satisfactory degree of gauge independence.

Several computational procedures have been proposed so far
to circumvent the problem of gauge translation, and related invariance
of calculated magnetic properties in a change of coordinate 
system. Current methods exploit
atomic basis sets explicitly depending on a gauge factor
(GIAO)~\cite{giao}, individual gauges for localized molecular orbitals 
(IGLO)~\cite{iglo}, and (LORG)~\cite{lorg}.
%, or different origins
%for different pairs of orbitals~{\cite{levy}}.
    
Theoretical determinations of magnetic properties of a molecule perturbed by
an external magnetic field, assumed, for the sake of simplicity, spatially 
uniform and time independent, are
usually obtained by retaining the Coulomb gauge for the  vector
potential ${\bf A}^{\cal C}$~{\cite{lan1}:   
%\be     
%{\bf A}^{\cal C}({\bf r})={1\over2}{\bf B}\times{\bf r}.     
%\ee     
\bea 
{\bf A}^{\cal C}({\bf r})={1\over 2}{\bf B}\times{\bf r},
\hspace{1cm} \nabla\cdot{\bf A}^{\cal C}=0. 
\label{vcoulomb} 
\eea
     
The flux of the external magnetic field
${\bf B}={\nabla}\times{\bf A}^{\cal C}$ is obviously
invariant in a gauge transformation  ~{\cite{lan1}
\be     
{\bf A}^{\cal C} \rightarrow {\bf A}^{\cal C}+{\nabla}\Lambda,     
\label{avect}
\ee     
where $\Lambda({\bf r})$ is an arbitary function of position,
well behaved for $ r\rightarrow{\infty}$.
When a gauge transformation is carried out, the wave function
$\psi$ and the Hamiltonian $h$ of a particle with
charge $q$ are transformed according to the equations
\be     
\psi\rightarrow\psi^\prime=\psi\exp\left({iq\over\hbar c}\Lambda\right)
\hspace{1cm}\,      h\rightarrow h^\prime     
=\exp\left({iq\over\hbar c}\Lambda\right) h\exp\left(-{iq\over\hbar c}
\Lambda\right),     
\label{waveinv}     
\ee     
leaving invariant the one particle  Schr\"odinger equation
$h\psi=\epsilon\psi$.  If the basis set is complete, also the calculated energy
of the system is gauge-invariant, and the electronic current density
is conserved, i.e., ${\nabla}\cdot {\bf j}=0$~\cite{epst}.
For wavefunctions that are exact eigenfunctions to some model hamiltonian,
the choice of gauge is immaterial, because variations of
the diamagnetic and paramagnetic contributions arising from a change of gauge
cancel out one another, leaving invariant total magnetic properties.
More generally, in the case of actual
calculations relying on the algebraic approximation,
selecting a gauge, e.g., specifying the ``best'' form of the vector potential,
introduces serious problems: the fulfillment of constraints for gauge
invariance constitutes a severe test of reliability for the computational 
scheme, and a hallmark of quality of the molecular wave function.

The gauge transformations which received most attention so far are
usually related to a change of origin of coordinate system, 
${\bf r}^\prime\rightarrow{\bf r}^{\prime\prime} ={\bf r}^\prime+{\bf d}$,
with ${\bf d}$ an arbitrary vector. They are limited to a class of
$\Lambda$ functions such that    
\be     
{\bf A}^{{\cal C}^{\prime\prime}}={\bf A}^{{\cal C}^\prime}     
+{\bf \nabla}\Lambda, \qquad     
\Lambda\equiv{\bf d}\cdot{\bf A}^{{\cal C}^\prime}, \qquad     
{\bf A}^{{\cal C}^\prime}     
={\bf A}^{\cal C}({\bf r}-{\bf r}^\prime).     
\ee     

More general gauge transformations have been also proposed. The Landau
gauge~\cite{lan1} for computing magnetic properties was studied in
a series of articles~\cite{pladvcp}-\cite{sumrul}.

Within Coulomb and Landau gauges, the vector potential
is purely transverse; an alternative form of
vector potential containing a longitudinal component has also been 
considered~\cite{long}. Some interesting features of these gauges were
pointed out elsewhere~\cite{pladvcp}-~\cite{long}.

Besides the mere theoretical interest, the use of Landau and
longitudinal gauges is characterized by some remarkable features in numerical
applications, as they provide clear-cut intrinsic criteria for
assessing the accuracy of a calculation a priori, e.g., these
alternative gauges furnish the tools for estimating Hartree-Fock limits
to magnetic properties.

%Quite different procedures have been adopted by Geertsen~\cite{geer} by
%defining polarization propagator expressions for the diamagnetic
%contribution to magnetic susceptibility and nuclear magnetic shieldings.
%They have been used in previous calculations on a series of small
%molecules~\cite{cax}.  Closely related methods have been described
%within the framework of a continuous transformation of origin of the%
%current density (CTOCD)~\cite{dz1}, whereby either diamagnetic or
%paramagnetic contributions to the current density are formally
%annihilated~\cite{dz1,ijqc}. The merit of these approaches is that they
%obtain exact invariance in a gauge translation for nuclear magnetic
%shielding, and virtual invariance for magnetic susceptibility, as well
%as important information on the degree of current conservation, in
%calculations based on the algebraic approximation.

Chan and Das~\cite{chd} have shown that, within the Coulomb gauge, the
paramagnetic contribution to susceptibility reaches a minimum when the origin
of the vector potential is fixed in the electronic centroid.
Sadlej~\cite{sad} suggested another scheme for choosing the optimal gauge
origin in a finite basis set calculation of magnetic
properties, by minimizing the basis set errors affecting computed
second-order energy, and proved that the best gauge origin may not
coincide with the electronic centroid. He took into account
completeness criteria for a given basis set in calculating molecular magnetic
properties. The advantage of this scheme over that of
Chan and Das is that the latter does not provide any criterion for the
gauge origin in the case of nuclear magnetic shielding constants.

For a given molecule, and for the same coordinate origin,
the paramagnetic contribution to magnetic susceptibility evaluated within 
Landau and longitudinal gauges is systematically larger than Coulomb's,
(which suggests that Coulomb gauge should preferably be used in actual calculations, {\sl vide infra}).
Therefore the question naturally arises whether another
gauge may be adopted, such that the paramagnetic contribution to
susceptibility becomes smaller than Coulomb's. Accordingly,
the present work is aimed at investigating whether it is formally
possible to choose a special form of the vector potential such that,
allowing for the customary
partition of magnetic susceptibility into diamagnetic and
paramagnetic contributions, the latter is an absolute
minimum. The interest of such a gauge, if it were available,
is evident, since it would imply that total magnetic susceptibility can be 
calculated with higher precision.
In fact, the diamagnetic term is an expectation value over the reference
state, which is usually calculated with great
accuracy, whereas a satisfactory description of excited states, needed to
evaluate the paramagnetic contribution, is much more difficult to obtain.

In the present article the general physical requirement of gauge invariance of
molecular magnetic properties is tackled by a variational approach, by
considering a complete set of trial gauge transformations,
$\{ \Lambda_i\}$, in order to check if a minimum of the paramagnetic 
contribution to the exact magnetic susceptibility can be formally attained by 
using one of these generating functions.
A two-terms functional, suitable to analyze the properties of a molecule
in the presence of an external magnetic field is studied. The first term of
this functional is related to the paramagnetic contribution to the
interaction energy within the Coulomb gauge
for the vector potential.

A gauge transformation is taken into account via the second term, defined
by a generating ${\Lambda}$ function well behaved for $ r {\rightarrow}
{\infty}$. If there existed a scalar function ${\Lambda}$ giving rise to a
maximum of the functional, then the corresponding ${\Lambda}$
would be a critical
point for the proposed functional, and a solution of the Euler-Lagrange equations.  By employing a formal procedure, it is proven
that the paramagnetic contribution to magnetic susceptibility
cannot go through an absolute minimum in the variational sense.

Sec. II of this paper deals with general invariance
conditions to be fulfilled for the exact eigenfunction of a model
Hamiltonian. 
In Sec. III  it is proven that an extremum of the
functional cannot be determined,
whereas a local minimum can be found.  Such a local minimum is
determined in Sec. IV for some small molecules. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Invariance conditions in a general change of gauge}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us consider a molecule with $n$ electrons with mass $m_e$, charge
$-e$, coordinates ${\bf r}_i$, canonical momenta ${\bf p}_i$,
%angular momenta ${\bf l}_i={\bf r}_i\times {\bf p}_i$,
$(i=1,2,....,n)$, and
$N$ nuclei, with corresponding quantities $M_{I}$, $Z_{I}e$, ${\bf R}_I$,
etc..

The ``particle'' electronic Hamiltonian is
\begin{equation}
{H}^{(0)}=\sum_{i=1}^n\left[ {{p_i}^2\over{2m_e}}-\sum_{I=1}^N{Z_{I}e^2\over{\left\vert
{\bf r}_i-{\bf R}_{I}\right\vert}}
+{1\over2}\sum_{j\not=i}^n{e^2\over{\left\vert {\bf r}_i-
{\bf r}_j\right\vert}}\right]+{1\over2}
\sum_{I}^N\sum_{J\not=I}^N{Z_{I}Z_{J}e^2\over
{\left\vert{\bf R}_{I}-{\bf R}_J\right\vert}},
\label{B0}
\end{equation}
with eigenstates
$\left |\Psi_j^{(0)}\right\rangle\equiv\left|j\right\rangle$
and energy eigenvalues $E_j^{(0)}$; the
reference state is denoted by 
$\left |\Psi_a^{(0)}\right\rangle\equiv\left|a\right\rangle$
(the notation of previous 
papers~\cite{pladvcp} is retained throughout this article, e.g.,
%$L_{\alpha}=\sum_{i=1}^n l_{i\alpha}$,
$\omega_{ja}=(E_j^{(0)} - E_a^{(0)})/\hbar$,  etc.).
In the presence of a magnetic field ${\bf B}$ with vector
potential~(\ref{vcoulomb})
in the Coulomb gauge, the ``interaction'' Hamiltonian is
\bea     
V={e\over m_ec}\sum_{i=1}^n{\bf A}_i^{\cal C}\cdot{\bf p}_i     
%+{\bf p}_i     
%\cdot{\bf A}_i^{\cal C}
+{e^2\over 2m_ec^2}\sum_{i=1}^n{\bf A}_i^{\cal C}     
\cdot{\bf A}_i^{\cal C}.      
\la{VHam}     
\eea     
If a permanent dipole $\mu_I$ on nucleus $I$ is also present, with vector potential
\bea
{\bf A}^{{\mu}_I}({\bf r})=\mu_{I} \times {{\bf r}-{\bf R}_I \over
\left|{\bf r}-{\bf R}_I\right|},     
\label{vmuI}     
\eea   
the interaction Hamiltonian (\ref{VHam}) contains the additional terms
\be
{e\over m_ec}\sum_{i=1}^n{\bf A}_i^{{\mu}_I}\cdot {\bf p}_i+{e^2\over m_ec}\sum_{i=1}^n {\bf A}_i^{\cal C}\cdot {\bf A}_i^{{\mu}_I}.
\label{vmuham}
\ee
The second-order energy
expression contains two terms,
\be     
W^{BB}=-{1\over 2}\left(\chi_{\alpha\beta}^{\rm d}
+\chi_{\alpha\beta}^{\rm p}\right)
B_{\alpha} B_{\beta} = W_d^{BB}+W_p^{BB},  
\la{secW} 
\ee         
\be     
W^{{\mu}_IB}=\mu_{I\alpha}\left(\sigma_{\alpha\beta}^{{\rm d}I}
+\sigma_{\alpha\beta}^{{\rm p}I}\right)
B_\beta=W_d^{{\mu}_IB}+     
W_p^{{\mu}_IB},
\ee     
where $\chi_{\alpha\beta}$ and $\sigma_{\alpha\beta}^I$ are the
cartesian components of magnetic susceptibility and 
magnetic shielding of nucleus $I$, which are to be invariant in
the gauge transformation~(\ref{avect}).
These second-order contributions can also be written 
\be     
W_d^{BB}+W_d^{{\mu}_IB}={e^2 \over 2m_ec^2}\left\langle a\left\vert\sum_{i=1}^n
{\bf A}_i\cdot {\bf A}_i \right\vert a \right\rangle     
\label{Wd}
\ee        
\be     
W_p^{BB}+W_p^{{\mu}_IB}=-{e^2\over m_e^2c^2\hbar}\sum_{j\ne a}     
 {1\over{\omega_{ja}}}     
\Re\left(     
\left\langle a\left\vert\sum_{i=1}^n\left(A_\alpha p_\alpha\right)_i
\right\vert j\right\rangle     
\left\langle j\left\vert\sum_{i=1}^n\left(A_\beta p_\beta\right)_i
\right\vert  a\right\rangle\right)     
\label{Wp}
\ee     
denoting by 
\be     
{\bf A}({\bf r})={\bf A}^{\cal C}({\bf r})+{\bf A}^{{\mu}_I}({\bf r})     
\la{acoulomb}     
\ee     
the total vector potential at ${\bf r}$. Repeated Greek indices in
tensor equations imply summation over $x$, $y$, and $z$ throughout this paper. 

In a gauge transformation the contributions which define the magnetic susceptibility tensor transform according to the expressions
\be    
W_d^{BB}\rightarrow W_d^{\prime BB}= W_d^{BB} + \Delta_d^{BB},
\la{sotransd}
\ee
\be
W_p^{BB}\rightarrow W_p^{\prime BB}= W_p^{BB} + \Delta_p^{BB}
\label{sotrans}
\ee
where
\be     
W_d^{BB}={e^2 \over 2m_ec^2}\left\langle a\left\vert\sum_{i=1}^n 
\left({A}_\alpha ^{\cal C} {A}^{\cal C}_{\alpha }\right)_{i}\right  \vert a \right\rangle, 
\la{wd}
\ee     
\bea
\Delta_d^{BB}={e^2 \over {2m_ec^2}}\left[2\left\langle a\left\vert\sum_{i=1}^n\left({\bf A}^{\cal C}
\cdot{\nabla\Lambda}\right)_{i}\right
\vert a \right\rangle + \left\langle a\left\vert \sum_{i=1}^n \left(\nabla\Lambda\right)_i^2\right\vert a \right\rangle\right]. 
\label{Dd}
\eea
\bea     
 W_p^{BB}= 
-{e^2\over m_e^2c^2\hbar}\sum_{j\ne a} {1\over{\omega_{ja}}}\Re     
\left(\left\langle a\left\vert \sum_{i=1}^n\left({A}_{\alpha}^{\cal C}     
{p}_{\alpha}\right)_{i}\right\vert j     
\right\rangle \left\langle j\left\vert\sum_{i=1}^n\left({A}_{\beta}^{\cal     
C} {p}_{\beta}\right)_{i}     
\right\vert a\right\rangle\right),     
\la{wp}     
\eea     
\bea
\Delta_p^{BB}&=& 
-{e^2\over 2m_e^2c^2\hbar}\left[2 \sum_{j\ne a} {2\over{\omega_{ja}}}\Re     
\left(\left\langle a\left\vert \sum_{i=1}^n ({A}_{\alpha}^{\cal C}     
{p}_{\alpha})_{i}\right\vert j     
\right\rangle \left\langle j\left\vert\sum_{i=1}^n\left({\nabla_\beta
\Lambda}{p}_{\beta}\right)_{i}     
\right\vert a\right\rangle\right)\right.  \nn\\
&&+\left. \sum_{j\ne a} {2\over{\omega_{ja}}}\Re\left(\left\langle a\left\vert \sum_{i=1}^n\left(\nabla_{\alpha}\Lambda 
{p}_{\alpha}\right)_{i}\right\vert j \right\rangle  
\left\langle j\left\vert \sum_{i=1}^n\left(\nabla_{\beta}\Lambda
{p}_{\beta}\right)_{i}\right\vert a \right\rangle\right)\right].    
\la{deltap}     
\eea     

Therefore, under a gauge transformation of the Coulomb vector potential, general conditions for invariance of magnetic susceptibility are obtained via the identity
\be
\Delta_p^{BB}=-\Delta_d^{BB}
\label{DeqD}
\ee
in the form
\bea
\left(A_{\a}^C p_{\a},\nabla_{\b}{\l}p_{\b}\right)_{-1}
={m_e}\left\langle a\left\vert \sum_{i=1}^n\left({A}_{\alpha}^{\cal C}     
{\nabla}_{\alpha}{\l}\right)_{i}\right\vert a \right\rangle,
\label{cond1}
\eea
\bea
\left(\nabla_{\a} {\l}p_{\a},\nabla_{\b}{\l}p_{\b}\right)_{-1}
={m_e}\left\langle a\left\vert \sum_{i=1}^n \left(\nabla\Lambda\right)^2_{i}\right\vert a \right\rangle,
\label{cond2}
\eea
where the definitions
\bea
&&\left(A_{\a}^C p_{\a},\nabla_{\b}{\l}p_{\b}\right)_{-1} \nn\\
&&\phantom{00000}={1\over \hbar} 
\sum_{j\ne a} {2\over \omega_{ja}}\Re\left(\left\langle a\left\vert \sum_{i=1}^n\left({A}_{\alpha}^{\cal C}{p}_{\alpha}\right)_{i}\right\vert j \right\rangle 
\left\langle j\left\vert \sum_{i=1}^n\left(\nabla_{\b}{\l}p_{\b}\right)_{i}
\right\vert a \right\rangle\right)
\label{cond11}
\eea
\bea
&&\left(\nabla_{\a}{\l}p_{\a},\nabla_{\b}{\l}p_{\b}\right)_{-1} \nn\\
&&\phantom{00000}={1\over\hbar} \sum_{j\ne a} {2\over \omega_{ja}}\Re\left(\left\langle a\left\vert \sum_{i=1}^n\left(\nabla_{\alpha}\Lambda 
{p}_{\alpha}\right)_{i}\right\vert j \right\rangle  
\left\langle j\left\vert \sum_{i=1}^n\left(\nabla_{\b}\Lambda
{p}_{\b}\right)_{i}\right\vert a \right\rangle\right)
\label{cond12}
\eea
have been employed (corresponding conditions for invariance of nuclear magnetic shielding tensor are examined in Ref.~\cite{ sumrul}).

The same formulae are established using the hypervirial theorem, via the off-diagonal relationship
\bea
\left\langle j\left\vert \sum_{i=1}^n(\nabla\Lambda\cdot{\bf p})_i\right\vert a \right\rangle= i m_{e} \omega_{ja}\left\langle j\left\vert \sum_{i=1}^n {\Lambda}_i\right\vert a \right\rangle
\eea
and the operator equations
\be
{{im_{e}}\over \hbar} [H^{(0)}, \Lambda]= {\bf \nabla}\Lambda \cdot {\bf p},
\label{hyper1}
\ee
\be
[\Lambda,[\Lambda, H^{(0)}]]= -{{\hbar^2}\over m_{e} } (\nabla\Lambda)^2.
\label{hyper2}
\ee
These results can be used to work out explicit conditions for invariance of molecular magnetic susceptibilities corresponding to a specific form of 
$\Lambda$~\cite{pladvcp}. They are analyzed in the next section to check
whether a $\Lambda$ function can be found which would make the 
functional stationary.
The case of nuclear magnetic shielding is comparatively less interesting, as it
is immediately seen from eqs.~(\ref{Wd}) and (\ref{Wp})
that the relative functional is characterized by a cross structure which cannot be extremized. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Variational analysis of the  
paramagnetic contribution to the interaction energy in a gauge transformation.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The paramagnetic contribution~(\ref{wp}) to magnetic 
susceptibility depends on the excited states, and is much more difficult to 
estimate than the diamagnetic contribution~(\ref{wd}), which is an expectation 
value over the reference state, usually described with great accuracy
via current methods relying on the algebraic approximation.
Now, as the sum of diamagnetic and paramagnetic contributions stays the 
same in the ideal case of exact eigenfunctions to a model hamiltonian, it
is obviously advantageous to look for a gauge minimizing the size of the paramagnetic term. In such a case the bulk of the susceptibility would
be formally
obtained via the expectation value~(\ref{wd}), which is expected
to be quite good. 
Therefore, if $W_p^{BB}$ passed through a maximum, the paramagnetic
contribution to the susceptibility would reach a minimum,
and the requirement for 
accuracy would be very simply fulfilled.  
The problem becomes that of maximizing a functional representing the 
paramagnetic part of the second-order interaction energy $W^{BB}$,
and therefore minimizing the paramagnetic contribution to total average
magnetic susceptibility according to eq.~(\ref{secW}).
Such a functional can be defined as 
\be
J(\l)=W_p^{'BB}(\l)=W_p^{BB}({\bf A}^{\cal C})+\Delta_p^{BB}(\l).
\label{jota}
\ee
The ${\lambda}$ function corresponding to an extremum of
the functional~(\ref{jota}) satisfies 
the first-order necessary condition ~\cite{mar1,mar2,mar3}
\bea
\left.{\partial\over\p\vp}J(\lambda+\vp\phi)|_{\vp=0}
={\partial\over\p\vp}\Delta_p^{BB}(\lambda+\vp\phi)\right|_{\vp=0}=0
\label{dife}
\eea
(where $\epsilon$ is a parameter continuously varying in a given interval,
and $\phi=\phi({\bf r})$
is a variation function), because $W_p^{BB}$ is independent of $\Lambda$.
Working expressions for $J(\l)$ are found replacing $\Lambda$ by $\lambda+\epsilon\phi$ in eqs.~(\ref{jota}) and~(\ref{deltap}).
>From now on, the following definitions are employed to simplify the notation
\be
c_{aj}=\left\langle a\left|\sum_{i=1}^n\left(A_{\a}^{\cal C} p_{\a}
\right)_i\right|j\right\rangle
\ee
\be
z_{ja}(\l)=\left\langle j\left|\sum_{i=1}^n \left(\nabla_{\b}(\l)p_{\b}\right)_i\right|a\right\rangle
\ee
\be
z_{aj}(\l)=\left\langle a\left|\sum_{i=1}^n\left(\nabla_{\a}(\l)p_{\a}
\right)_i\right|j\right\rangle.
\ee  
Accordingly, eq.~(\ref{deltap}) takes the form
\be
\Delta_p^{BB}(\l)=-{e^2\over 2m_e^2c^2\hbar} \sum_{j\ne a}
{2\over \omega_{ja}}\Re\left[2c_{aj}z_{ja}(\l)+z_{aj}(\l)z_{ja}(\l)\right]
\ee
%%%%%%%primera posibilidad&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
%%%%%%%%%%%%%%%%fin primera posibilidad
%%%%%%%%%%%%%%%%SEGUNDA POSIBILIDAD
%%%%%%%%%%%%%%%%fin segunda posibilidad
or, writing the integrals in full and summing over repeated $\alpha=x,y,z$,
\bea
%J(\l)
\Delta_p^{BB}(\l)
&=&-{e^2\over 2m_e^2c^2\hbar}\nn\\
&&\times\left\{
2\sum_{j\ne a} {2\over \omega_{ja}}\Re
\left(
c_{aj}\sum_{k=1}^n \int
%\ldots\int
\Psi^{(0)*}_j({\bf r})
{\p\l({\bf r})\over\p{\a}_k}
{\hbar\over i}{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1
\ldots d{\bf r}_n
\right)
\right.\nn\\
&&\left.
+\sum_{j\ne a} {2\over \omega_{ja}}\Re
\left[
\left(
\sum_{k=1}^n 
\int
%\ldots\int
\Psi^{(0)*}_a({\bf r})
{\p\l({\bf r})\over\p{\a}_k}{\hbar\over i}
{\p\Psi_j^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots d{\bf r}_n
\right)
\right.
\right.\nn\\
&&\times\left.\left.\left(
\sum_{k=1}^n \int
%\ldots\int
\Psi^{(0)*}_j({\bf r})
{\p\l({\bf r})\over\p{\a}_k}{\hbar\over i} 
{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots
d{\bf r}_n\right)\right]\right\},
\eea
%%%%%%%%%%%%%%%%%%%%%%%%fin segunda posibilidad
where ${\bf r}= ({\bf r}_1, {\bf r}_2,...{\bf r}_n)$
denotes the set of electron coordinates. 
By replacing $\Lambda$ by ${\lambda}+{\vp}\phi$ in this equations, one finds
%%%%%%%SEGUNDA FORMULACION
\bea
{\partial\over\p\vp}J(\lambda &+&\left.\vp\phi)\right|_{\vp=0}=
-{e^2\over 2m_e^2c^2\hbar}\nn\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%11111111111111111111
&&\times\left\{
2\sum_{j\ne a} 
{2\over \omega_{ja}}\Re
\left(
c_{aj} \sum_{k=1}^n \int
%\ldots\int
\Psi^{(0)*}_j({\bf r})
%\sum_{\a=x,y,z}
{\p\phi({\bf r})\over\p{\a}_k}   
{\hbar\over i}{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots d{\bf r}_n
\right)
\right.\nn\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%222222222222222222
&&\left.
+\sum_{j\ne a}{2\over \omega_{ja}}\Re
\left[
\left(
\sum_{k=1}^n \int
%\ldots\int 
\Psi^{(0)*}_a({\bf r})
%\sum_{\a=x,y,z}
{\p\phi({\bf r})\over\p{\a}_k}{\hbar\over i}
{\p\Psi_j^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots d{\bf r}_n
\right)
\right.
\right.\nn\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3333333333333333333
&&\left.
\left.
\times
\left(
\sum_{k=1}^n 
\int
%\ldots\int
\Psi^{(0)*}_j({\bf r})
%\sum_{\a=x,y,z}
{\p\lambda({\bf r})\over\p{\a}_k}
%({\bf r})
{\hbar\over i} 
{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots d{\bf r}_n
\right)
\right.
\right.\nn\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%44444444444444444
&&\left.
\left.
+
\left(
\sum_{k=1}^n\int
%\ldots\int
\Psi^{(0)*}_a({\bf r})
%\sum_{\a=x,y,z}
{\p\lambda({\bf r})\over\p{\a}_k}{\hbar\over i}
{\p\Psi_j^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots d{\bf r}_n
\right)
\right.
\right.\nn\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%555555555555555
&&\times\left.
\left.
\left(
\sum_{k=1}^n    
\int
%\ldots\int 
\Psi^{(0)*}_j({\bf r})
%\sum_{\a=x,y,z}
{\p\phi({\bf r})\over\p{\a}_k}{\hbar\over i}   
{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\ldots d{\bf r}_n
\right)
\right]
\right\}\nn\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%6666666666666666
&=&-{e^2\over 2m_e^2c^2\hbar}
\left\{
2\sum_{j\ne a}   
{2\over \omega_{ja}}\Re\left[c_{aj}z_{ja}(\phi)\right]
\right.\nn\\
&&\left.
+\sum_{j\ne a}  
{2\over \omega_{ja}}\Re\left[z_{aj}(\phi)z_{ja}(\lambda)
+z_{aj}(\lambda)z_{ja}(\phi)
\right]
\right\}=0.
\la{deltajotaep}
\eea
>From the relations
\be
\left\langle i\left|O^+\right|j\right\rangle
=(O^+)_{ij}=\left\langle j\left|O\right|i\right\rangle^*={(O)_{ji}}^*
\ee
that hold for any operator $O$, and considering that
$O={\nabla\Lambda}\cdot {\bf p}$ is a hermitian operator,
$O^+=O$, it is immediately obtained that
\be
z_{aj}(\phi)z_{ja}(\lambda)+z_{aj}(\lambda)z_{ja}(\phi)=
2\Re\left[z_{ja}(\phi)z_{aj}(\lambda)\right],
\ee
and since 
\be
z_{ja}(\phi)={z_{aj}^*(\phi)},
\ee 
eq.~(\ref{deltajotaep}) becomes 
\be
-{e^2\over 2m_e^2c^2\hbar}\sum_{j\ne a}   
{2\over \omega_{ja}}2\Re\left[c_{aj}z_{ja}(\phi)+  
z_{ja}(\phi)z_{aj}(\lambda)\right]=0.
\ee
As this relationship is valid for any $\phi$ function, then, 
\be
\Re\left\{\left[c_{aj}+  
z_{aj}(\lambda)\right]z_{ja}(\phi)\right\}=0.
\ee
Defining
$z_1=a+ib=c_{aj}+z_{aj}(\l)$, and $z_2=c+id=z_{ja}(\phi)$, it is obtained that  
$ac-bd=0$: since this relation is valid for infinite values of $c$ and $d$, 
the condition $a=b=0$ must be fulfilled, and  we conclude that
\be
c_{aj}+z_{aj}(\lambda)=0,
\ee
\be
\Delta_p^{BB}(\lambda)={e^2\over 2m_e^2c^2\hbar} \sum_{j\ne a}
{2\over \omega_{ja}}\left|c_{aj}\right|^2.
\ee
From
\be
W_p^{BB}({\bf A}^{\cal C})=
-{e^2\over 2m_e^2c^2\hbar} \sum_{j\ne a}
{2\over \omega_{ja}}\left|c_{aj}\right|^2
\ee
it is finally concluded that
\be
W_p^{'BB}({\bf A^{\prime}})=W_p^{BB}({\bf A}^{\cal C})+\Delta_p^{BB}(\lambda)=0.
\ee

This relationship means that the transformed vector potential
is exactly zero, which is not possible, since ${\bf A^{\prime}}$ should
correspond to the same magnetic field as
${\bf A^{\cal C}}$. Therefore, an extremum has only been reached by
{\sl reductio ad adsurdum}. Accordingly, it is not possible to  find a ${\lambda}$ function which minimizes the paramagnetic part of the susceptibility in a variational sense, that is yielding an absolute minimum. 

The same contradiction is found if the $\Delta_p^{BB}(\l)$ expression valid
for {\sl exact} solutions of the Schr\"odinger equation,
%when ${\bf \nabla\cdot A}=0$~\cite{sumrul},
compare for eqs.~(\ref{Dd}) and~(\ref{DeqD}), is employed. In this case
\be
\Delta_p^{BB}(\l)=-{e^2\over 2m_ec^2}
\left\langle a\left|\sum_{i=1}^n\left[
2{\bf A}^{C}\cdot\nabla\Lambda+ 
(\nabla\Lambda)^2\right]_i\right|a\right\rangle.
\la{delideal}
\ee
Accordingly, owing to eq.~(\ref{DeqD}), and to the fact that
the sum of $W_d^{BB}+W_p^{BB}$ in eq.~(\ref{secW}) is invariant to a gauge 
transformation for exact eigenfunctions,
one could alternatively try to maximize functional~(\ref{jota})
via a procedure which actually amounts to minimizing the diamagnetic 
contribution
$W_d^{BB}=-(1/2)\chi_{\alpha\beta}^{\rm d}B_\alpha B_\beta$
to the energy functional, see also eqs.~(\ref{sotransd}), (\ref{wd}), and
(\ref{Dd}). Therefore, replacing again $\Lambda$ by ${\lambda}+{\vp}\phi$,
and performing the differentiation,
\be
\left.{\partial\over\partial\vp}J(\lambda+\vp\phi)\right|_{\vp=0}=
-{e^2\over 2m_e c^2}\left\langle a\left|\sum_{i=1}^n\left[
2({\bf A}^{C}+\nabla\lambda)\cdot\nabla\phi\right]_i\right|a\right\rangle =0,
\la{eq46}
\ee
i.e., $\nabla\lambda=-{\bf A}^{\cal C}$,
for any increment function $\phi$, and the conclusion would be again
that the transformed vector potential
${\bf A}^{\prime}={\bf A}^{\cal C}+\nabla\lambda$
vanishes identically all over the molecular domain, which is clearly meaningless.
Therefore one is left only with the possibility of finding {\sl local} extremum
points for a given type of ${\Lambda}$ function. In principle they can be 
determined by expressing ${\Lambda}$ via some general polynomial form.
These local extremum points would depend on the type of molecule, on
its geometry, and on
the basis set employed in the calculation of magnetic properties. Their theoretical interest is clearly limited, however an analysis is provided
in the next Section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local maxima of the diamagnetic contribution to susceptibility}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% DERIVATA CRISTINA
In deriving the results of Sec. III we used the fact that, in general, 
for $\lambda$ such that the first differential 
$d\Delta_p^{BB}(\lambda)=0$, and the second differential
$d^2\Delta_p^{BB}(\lambda)$ is a negative
definite quadratic form, the functional $\Delta_p^{BB}(\lambda)$ is
an absolute 
maximum, corresponding to a minimum of the
paramagnetic contribution to molecular susceptibility.
Let us denote the second differential by the bilinear form
%No, esa es la definicion de diferencial segunda. En general, la
%diferencial segunda de una funcion f es una forma bilineal que se calcula
%haciendo 
\be
d^2f(\mu)(\varphi,\psi)={\partial \over \partial
\epsilon}df(\mu+\epsilon\varphi)(\psi)|_{\epsilon=0}.
\ee
%En este caso estamos tomando 
%y la cuarta ecuacion es el resto del desarrollo de Taylor de
%grado 2 alrededor de \lambda
Then, choosing $\mu=\lambda$ and $\varphi=\psi=\Lambda -
\lambda$,
from the necessary condition (\ref{dife}), assuming that
identity~(\ref{DeqD}) is satisfied (which is only true in the ideal case of
exact eigenfunctions to a model hamiltonian), and
from
\be
\Delta_p^{BB}(\Lambda)=-\left\langle 0|2{\bf A}^{\cal C}\cdot\nabla\Lambda
+|\nabla\Lambda|^2|0\right\rangle,
\ee
\be
\Delta_p^{BB}(\lambda)=-\left\langle 0|2{\bf A}^{\cal C}\cdot\nabla\lambda
+|\nabla\lambda|^2|0\right\rangle,
\ee
\be
{1\over 2}d^2\Delta_p^{BB}(\lambda)(\Lambda-\lambda,\Lambda-\lambda)=
 -\left\langle 0 \left|{1\over 2} \left[\nabla(\Lambda 
-\lambda)\right]^2\right|0\right\rangle,
\ee 
and ${\bf A}^{\cal C}=-\nabla\lambda$, because $\lambda$ is a critical point, 
we obtain for any $\Lambda$:
\bea
&&\Delta_p^{BB}(\Lambda)-\Delta_p^{BB}(\lambda)-{1\over
2}d^2\Delta_p^{BB}(\lambda)(\Lambda-\lambda, \Lambda-\lambda)
\nn\\
&&=
-\left\langle 0\left|2{\bf A}^{\cal C}\cdot\nabla(\Lambda-\lambda)
+|\nabla\Lambda|^2-|\nabla\lambda|^2
-{1\over 2}\left[\nabla(\Lambda-\lambda)\right]^2\right|0\right\rangle
\nn\\
&&=-{1\over 2}\left\langle 0\left| \right[\nabla(\Lambda
-\lambda)\right]^2|0>\le 0
\eea
Hence, 
\be
\Delta_p^{BB}(\Lambda)\le \Delta_p^{BB}(\lambda)
+{1\over 2}d^2\Delta_p^{BB}(\lambda)
(\Lambda-\lambda,\Lambda-\lambda)<\Delta_p^{BB}(\lambda)
\ee
and the proof is complete.

Lets us now study some special $\Lambda$ sets suitable to maximize the 
diamagnetic contribution $W_d^{BB}$, see discussion preceding eq.~(\ref{eq46}).
Polynomial forms of the $\Lambda$ function are proposed and analyzed hereafter: 

i) For 
$\Lambda(x,y,z)=ax^2+by^2+cz^2$, 
denoting by $\left|0\right\rangle$ the wave function of the ground state, 
from the necessary condition (\ref{dife}), assuming that
identity~(\ref{DeqD}) is satisfied,
one obtains the coefficients:
\bea
a&=&{\left\langle 0|xy|0\right\rangle B_3-\left\langle 0|xz|0\right\rangle B_2 
\over 4\left\langle 0|xx|0\right\rangle}\nn\\ 
b&=&{\left\langle 0|yz|0\right\rangle B_1-\left\langle 0|xy|0\right\rangle B_3 
\over 4\left\langle 0|yy|0\right\rangle}\\ 
c&=&{\left\langle 0|xz|0\right\rangle B_2-\left\langle 0|yz|0\right\rangle B_1 
\over 4\left\langle 0|zz|0\right\rangle}\nn
\label{coefcnt}
\eea 
%%%%%%%%sbagliata precedente
%that define the generating transformation function
%\bea
%\Lambda(x,y,z)&=&{(\left\langle 0|xy|0\right\rangle B_3-\left\langle 
%0|xz|0\right\rangle B_2)x^2\over 4\left\langle 0|xx|0\right\rangle}\nn\\
%&&+{(\left\langle 0|yz|0\right\rangle B_1-\left\langle 0|xy|0\right
%\rangle B_2)y^2\over 4\left\langle 0|yy|0\right\rangle}\nn\\
%&&+{(\left\langle 0|xy|0\right\rangle B_3-\left\langle 0|yz|0\right\rangle 
%B_2)z^2 \over 4\left\langle 0|zz|0\right\rangle}
%\eea
%%%%%%%%%%fine sbagliata
%%%%%%%%%corretta
%\bea
%\Lambda(x,y,z)&=&{(\left\langle 0|xy|0\right\rangle B_3-\left\langle 
%0|xz|0\right\rangle B_2)x^2\over 4\left\langle 0|xx|0\right\rangle}\nn\\
%&&+{(\left\langle 0|yz|0\right\rangle B_1-\left\langle 0|xy|0\right
%\rangle B_3)y^2\over 4\left\langle 0|yy|0\right\rangle}\nn\\
%&&+{(\left\langle 0|xz|0\right\rangle B_2-\left\langle 0|yz|0\right\rangle 
%B_1)z^2 \over 4\left\langle 0|zz|0\right\rangle}
%\eea
%%%%%%%%%%fine
with ${\bf B}=(B_1,B_2,B_3)$ the $x$, $y$ and $z$ cartesian components of the magnetic field.
We remark that $\left\langle 0|xx|0\right\rangle $, $\left\langle 0|yy|0\right\rangle$, and $\left\langle 0|zz|0\right\rangle$ are not zero, because the integral of a positive function is zero only  on zero
measure subsets of $R^n$, and, in this case, the subset of integration
is $R^n$, that has positive measure.

For the ${\rm H}{\rm F}$, ${\rm H}_2{\rm O}$, ${\rm N}{\rm H}_3$, ${\rm C}{\rm H}_4$ molecules, assuming the principal axis system retained in the calculations
of  Ref.~\cite{landau2},
$\Lambda\equiv 0$, 
and we conclude that it is not possible to find a gauge transformation  
maximizing the diamagnetic contribution  to magnetic susceptibility
for this set of molecules via the polynomial expression
$\Lambda(x,y,z)=ax^2+by^2+cz^2$.

ii) Let us now consider another polynomial form,
$\Lambda(x,y,z)=ax^2+by^2+cz^2+dxy+eyz+fzx$, 
and the numerical values for the average second moment, obtained for
hydrogen fluoride, ammonia and methane via basis set
IV of Ref.~\cite{landau2}, and via basis set VI of Ref.~\cite{landau1}
for water.
For the ${\rm HF}$ molecule, with the $z$ axis oriented in the 
direction of the bond, one obtains the following parameters: 
$a=0$, $b=0$, $c=0$, $d=0$, $e=0.05483 B_1$, $f=-0.05483 B_2$, and 
\be
\lambda(x,y,z)=0.05483 (B_1y-B_2x)z.
\ee

For the ${\rm H}_2{\rm O}$ molecule one finds
$a=0$, $b=0$, $c=0$, $d=0.0636142 B_3$, 
$e=-0.025335 B_1$, $f=-0.0385275 B_2$, 
and
\be
\lambda(x,y,z)=0.0636142B_3xy-(0.025335B_1y+0.0385275B_2x)z.
\ee
For the NH${}_3$ molecule
$a=0$, $b=0$, $c=0$, $d=0$, 
$e=-0.395907 B_1$, $f=0.395907 B_2$, 
and
\be
 \lambda(x,y,z)=-0.395907 (B_1y-B_2x)z.
\label{ultima}
\ee

Eventually, for the  CH${}_4$ molecule, one obtains 
$a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$ by symmetry, then 
$\lambda\equiv 0$, 
and we conclude that it is not possible to find a gauge transformation  
maximizing $J(\l)$ for this molecule using a polynomial
expression of type (ii).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The functions $\lambda({a,b,c})$ with $a,b,c,$ given in
eqs.~(53)-(\ref{ultima})
correspond to a local maximum of the functional $\Delta_p^{BB}$, because the second
derivative of $\Delta_p^{BB}$ evaluated via $\lambda(abc)$ is negative.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A general form of ${\lambda}$ function satisfying the extremum
condition~(\ref{deltajotaep}), corresponding to a minimum (maximum) of the 
paramagnetic (diamagnetic)
contribution to magnetic susceptibility cannot be found. Instead, one can
find particular ${\lambda}$
transformations yielding a local minimum of the paramagnetic susceptibility
for a given molecule.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\bf  Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Financial support to the present
research from the University of Buenos Aires (UBACYT TX--063), the
Argentinian Consejo Nacional de Investigaciones Cient\'{\i}ficas y
T{\'e}cnicas (CONICET), and Agencia Nacional de Promoci{\'o}n de Ciencia y T{\'e}cnica (ANPCYT), from the Italian Consiglio Nazionale delle Ricerche 
(CNR), and Ministero dell'Universit\`a e della Ricerca Scientifica e
Tecnologica (MURST) are gratefully acknowledged. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%{\sl Chem. Phys. Lett. \bf220} (1994) 299.     
%\la{dz1}     
     
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%P. Lazzeretti and R. Zanasi,     
%{\sl Int. J. Quantum Chem \bf 60}  (1996) 249; R. Zanasi and P. Lazzeretti,
%{\sl Mol. Phys. \bf 92} (1997), 609, and references therein.     
%\la{ijqc} 

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\la{sad}   

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%M. Struwe, {\sl Plateau's Problem and the Calculus of Variations},
%Lecture Notes, Princeton University Press (1988)
%\la{S}

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%R. Adams, {\sl Sobolev Spaces}, Academic Press, New York (1975)
%\la{A}  

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\end{thebibliography}     
     
\end{document}     

_______________________________________________________________________
PRIMERA POSIBILIDAD

and, indicating explicitly the sum over $\alpha$,
\bea
J(\l)&=&-{e^2\over 2m_e^2c^2\hbar}\nn\\
&&\times\left[
{2\over \hbar}\sum_{j\ne a} {2\over \omega_{ja}}\Re
\left(
c_{aj}\sum_{k=1}^n \int\cdot\cdot\cdot\int
\Psi^{(0)*}_j({\bf r})
\sum_{\a=x,y,z}{\p\l({\bf r})\over\p{\a}_k}
{\hbar\over i}{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1
\cdot\cdot d{\bf r}_n
\right)
\right.\nn\\
&&\left.
+{1\over \hbar}\sum_{j\ne a} {2\over \omega_{ja}}\Re
\left(
\sum_{k=1}^n 
\int\cdot\cdot\cdot\int \Psi^{(0)*}_a({\bf r})
\sum_{\a=x,y,z}{\p\l({\bf r})\over\p{\a}_k}{\hbar\over i}
{\p\Psi_j^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\cdot\cdot d{\bf r}_n
\right)
\right.\nn\\
&&\times\left.\left(
\sum_{k=1}^n \int\cdot\cdot\cdot\int \Psi^{(0)*}_j({\bf r})
\sum_{\a=x,y,z}{\p\l({\bf r})\over\p{\a}_k}{\hbar\over i} 
{\p\Psi_a^{(0)}({\bf r})\over\p{\a}_k}d{\bf r}_1\cdot\cdot
d{\bf r}_n\right)\right],
\eea
 FIN PRIMERA POSIBILIDAD
------------------------------------------------------------------
%%%%FORMULACION ORIGINAL

\bea
{\partial\over\p\vp}J(\l+\vp\phi)|_{\vp=0}=\nn\\
-{e^2\over 2m_e^2c^2\hbar}\left[{2\over \hbar}\sum_{j\ne a} 
{2\over \omega_{ja}}\Re(c_{aj} 
\sum_{k=1}^n \int\cdot\cdot\cdot\int
\psi^*_j({\bf r})\right.\nn\\
\left.\sum_{\a=x,y,z}
{\p\phi\over\p{\a}_k}({\bf r})   
{\hbar\over i}{\p\psi_a\over\p{\a}_k}({\bf r})dr_1\cdot\cdot dr_n)+
{1\over \hbar}\sum_{j\ne a}
 {2\over \omega_{ja}}\Re[(\sum_{k=1}^n 
\int\cdot\cdot\cdot\int \psi^*_a({\bf r})\right.\nn\\
\left.\sum_{\a=x,y,z}{\p\phi\over\p{\a}_k}({\bf r}){\hbar\over i}
{\p\psi_j\over\p{\a}_k}({\bf r})dr_1\cdot\cdot dr_n)
(\sum_{k=1}^n 
\int\cdot\cdot\cdot\int \psi^*_j({\bf r})\right.\nn\\
\left.\sum_{\a=x,y,z}{\p\l\over\p{\a}_k}({\bf r}){\hbar\over i} 
{\p\psi_a\over\p{\a}_k}({\bf r})dr_1\cdot\cdot dr_n)\right.\nn\\
\left.+(\sum_{k=1}^n
\int\cdot\cdot\cdot\int \psi^*_a({\bf r})
\sum_{\a=x,y,z}{\p\l\over\p{\a}_k}({\bf r}){\hbar\over i}
{\p\psi_j\over\p{\a}_k}({\bf r})dr_1\cdot\cdot dr_n)\right.\nn\\
\left.(\sum_{k=1}^n    
\int\cdot\cdot\cdot\int \psi^*_j({\bf r})
\sum_{\a=x,y,z}{\p\phi\over\p{\a}_k}({\bf r}){\hbar\over i}   
{\p\psi_a\over\p{\a}_k}({\bf r})dr_1\cdot\cdot dr_n)]\right]\nn\\
=-{e^2\over 2m_e^2c^2\hbar}[{2\over \hbar}\sum_{j\ne a}   
{2\over \omega_{ja}}\Re(c_{aj}z_{ja}(\phi))\nn\\
+{1\over \hbar}\sum_{j\ne a}  
{2\over \omega_{ja}}\Re[z_{aj}(\phi)z_{ja}(\l)+z_{aj}(\l)z_{ja}(\phi)])=0
\la{deltajotaep}
\eea
%%%%FIN 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%---------------------------------------------
It is known that if ${\cal H}$ is a Hilbert space and $E:{\cal H}\to R$
is weakly lower semicontinuous and coercive on ${\cal H}$, then there exists
$x_0 \in {\cal H}$,
verifying that
\be
E(x_0)=\inf_{H} E(x)
\ee
(see, for instance, Theorem 3.3 of Ref.~\cite{S}).
In addition, 
if ${\cal H}$ is a Hilbert space and $a:{\cal H} \times {\cal H} \to R$ is
a continuous, symmetric, bilinear form on ${\cal H}$ such that $a(x,x) \ge 0$ for all $x \in {\cal H}$, then the quadratic form $E(x)=a(x,x)$ is weakly
lower semicontinuous on ${\cal H}$~\cite{S}.
In this case, $J:{\cal H}^1_0(\Omega,R) \to R$, with $\Omega$ a bounded domain in $R^3$, is weakly lower semicontinuous and coercive on
${\cal H}^1_0(\Omega,R)$, because one can suppose that $Tr(\Lambda)=0$.
In these expressions ${\cal H}^1_0(\Omega,R)$ is the usual Sovolev Space,
and ${\cal H}^1_0(\Omega,R)=\{x \in {\cal H}^1_0(\Omega,R), Tr(x)=0\}$ with $Tr:{\cal H}^1_0(\Omega,R)\to L^2(\partial\Omega,R)$ the trace
operator~\cite{A}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local Minima of the energy functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Polynomial forms of the $\Lambda$ function are proposed and analyzed hereafter: 

i) For 
$\Lambda(x,y,z)=ax^2+by^2+cz^2$, 
denoting by $\left|0\right\rangle$ the wave function of the ground state, 
from the necessary condition (\ref{dife})
one obtains the coefficients:
\bea
a&=&{\left\langle 0|xy|0\right\rangle B_3-\left\langle 0|xz|0\right\rangle B_2 
\over 4\left\langle 0|xx|0\right\rangle}\\ 
b&=&{\left\langle 0|yz|0\right\rangle B_1-\left\langle 0|xy|0\right\rangle B_3 
\over 4\left\langle 0|yy|0\right\rangle}\\ 
c&=&{\left\langle 0|xz|0\right\rangle B_2-\left\langle 0|yz|0\right\rangle B_1 
\over 4\left\langle 0|zz|0\right\rangle}
\eea 
that generate the transformation
\bea
\Lambda(x,y,z)&=&{(\left\langle 0|xy|0\right\rangle B_3-\left\langle 0|xz|0\right\rangle B_2)x^2\over 4\left\langle 0|xx|0\right\rangle}\nn\\
&&+{(\left\langle 0|yz|0\right\rangle B_1-\left\langle 0|xy|0\right
\rangle B_2)y^2\over 4\left\langle 0|yy|0\right\rangle}\nn\\
&&+{(\left\langle 0|xy|0\right\rangle B_3-\left\langle 0|yz|0\right\rangle 
B_2)z^2 \over 4\left\langle 0|zz|0\right\rangle}
\eea
with ${\bf B}=(B_1,B_2,B_3)$ the $x$, $y$ and $z$ cartesian components of the magnetic field.
We remark that $\left\langle 0|xx|0\right\rangle $, $\left\langle 0|yy|0\right\rangle$, and $\left\langle 0|zz|0\right\rangle$ are not zero, because the integral of a positive function is zero only  on zero
measure subsets of $R^n$, and, in this case, the subset of integration
is $R^n$, that has positive measure.

For the ${\rm H}{\rm F}$, ${\rm H}_2{\rm O}$, ${\rm N}{\rm H}_3$, ${\rm C}{\rm H}_4$ molecules, 
$\Lambda\equiv 0$, 
and we conclude that it is not possible to find a gauge transformation  
maximizing $J(\l)$ for this set of molecules via the polynomial expression
$\Lambda(x,y,z)=ax^2+by^2+cz^2$.


ii) Let us now consider another polynomial form,
$\Lambda(x,y,z)=ax^2+by^2+cz^2+dxy+eyz+fzx$, 
and the numerical values for the average second moment, obtained for
hydrogen fluoride, ammonia and methane via basis set
IV of Ref.~\cite{landau2}, and via basis set VI of Ref.~\cite{landau1}
for water.
For the ${\rm HF}$ molecule, with the $z$ axis oriented in the 
direction of the bond, one obtains the following parameters: 
$a=0$, $b=0$, $c=0$, $d=0$, $e=0.05483 B_1$, $f=-0.05483 B_2$, and 
\be
\lambda(x,y,z)=0.05483 (B_1y-B_2x)z.
\ee

For the ${\rm H}_2{\rm O}$ molecule one finds
$a=0$, $b=0$, $c=0$, $d=0.0636142 B_3$, 
$e=-0.025335 B_1$, $f=-0.0385275 B_2$, 
and
\be
\lambda(x,y,z)=0.0636142B_3xy-(0.025335B_1y+0.0385275B_2x)z.
\ee
For the NH${}_3$ molecule
$a=0$, $b=0$, $c=0$, $d=0$, 
$e=-0.395907 B_1$, $f=0.395907 B_2$. 
and
\be
 \lambda(x,y,z)=-0.395907 (B_1y-B_2x)z.
\ee

Eventually, for the  CH${}_4$ molecule, one obtains 
$a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$, then 
$\lambda\equiv 0$, 
and we conclude that it is not possible to find a gauge transformation  
maximizing $J(\l)$ for this molecule using a polynomial
expression of type (ii).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% DERIVATA CRISTINA
\magnification=\magstep1
The functions $\lambda_{abc}$ with $a,b,c,$ given in ec. (48)-(54)
are absolut maximum of the functional $\Delta_p$, because the second
derivative of $\Delta_p$ evaluated in $\lambda_{abc}$ is negative.

In general, we can see that for any $\lambda$, veryfing that
$d\Delta_p(\lambda)=0$, and $d^2\Delta_p(\lambda)$ is a negative
definited quadratic form, then $\lambda$ is an absolut minimum.

Proof: Indeed, from

$\Delta_p(\lambda)=-<0|2A^C\cdot\nabla\lambda+|\nabla\lambda|^2|0>$

$\Delta_p(\lambda)=-<0|2A^C\cdot\nabla\lambda+|\nabla\lambda|^2|0>$

${1\over 2}d^2\Delta_p(\lambda)(\lambda-\lambda,\lambda-\lambda)=
 -<0|{1\over 2} |\nabla(\lambda -\lambda)|^2|0>$, 
and $A^C=-\nabla\lambda$, because $\lambda$ is a critical point, 
we obtain for any $\lambda$:


No, esa es la definicion de diferencial segunda. En general, la
diferencial segunda de una funcion f es una forma bilineal que se calcula
haciendo 
$d^2f(\mu)(\varphi,\psi)={\partial \over \partial
\epsilon}df(\mu+\epsilon\varphi)(\psi)|_{\epsilon=0}$


En este caso estamos tomando $\mu=\lambda$ y $\varphi=\psi=\lambda -
\lambda$ y la cuarta ecuacion es el resto del desarrollo de Taylor de
grado 2 alrededor de \lambda


$\Delta_p(\lambda)-\Delta_p(\lambda)-{1\over
2}d^2\Delta_p(\lambda)(\lambda-\lambda, \lambda-\lambda)=
-<0|2A^C\cdot\nabla(\lambda-\lambda)+|\nabla\lambda|^2-|\nabla\lambda|^2
-{1\over 2}|\nabla(\lambda-\lambda)|^2|0>=
-{1\over 2}<0| |\nabla(\lambda-\lambda)|^2|0>\le 0$

Hence, 

$\Delta_p(\lambda)\le \Delta_p(\lambda)+{1\over 2}d^2\Delta_p(\lambda)
(\lambda-\lambda,\lambda-\lambda)<\Delta_p(\lambda)$.
 
and the proof is complete.

\bye
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Paolo Lazzeretti
---------------9904081740413--
