Megascopic Quantum Phenomena
349
problem in so-called J-T molecules, see Bersuker [112]: “…one of the simplest JT
E ⊗ e problems with linear vibronic coupling yields an APES (Adiabatic Potential
Energy Surface) in the form of a “Mexican hat”… For a long time, up to the last
decade, not very much attention was paid to these conical intersections; rather they
were viewed largely as a “theoretical curiosity.” This perception changed recently
in view of the latest achievements in the treatment of such systems. One of these
achievements is a generalization directly related to the JTE and now known as the
topological (geometric) phase, or the Berry-phase problem… An important feature
of the Berry-phase implications in JT problems is that the peculiar phase factor that
changes the sign of the electronic wavefunctions and makes the ground vibronic
state degenerate occurs only when one or an odd number of conical intersections
are encircled, while it retains the same sign if an even number (including zero) are
encircled. This was shown by direct calculation of the phase in the E ⊗ e problem. In
fact, the phase is φ 0 = nπ , where n is the number of conical intersections encircled;
for n = 0, 2, 4, … the sign of the wavefunction does not change.”
The Berry-phase was mentioned above. The exact definition can be found in
Berry’s original paper [133]: “A quantal system in an eigenstate, slowly transported
round a circuit C by varying parameters R in its Hamiltonian H(R), will acquire
a geometrical phase factor exp[iγ (C)] in addition to the familiar dynamical phase
factor… If C lies near a degeneracy of H, γ (C) takes a simple form which includes
as a special case the sign change of eigenfunctions of real symmetric matrices round
a degeneracy.” However, there is a problem, since the Berry-phase, based on the
definition above, cannot be applied to J-T molecules at all. For instance the BO approximation can either be derived as a result of the m/M expansion or in an
equivalent form using the hierarchical quantization process, i.e. first quantize the
electronic motion at fixed classical nuclear positions and then a posteriori quantize
the nuclear motion. Applying the Berry-phase to J-T molecules is just a product of this
type of hierarchical quantization. A more exact simultaneous quantization method,
i.e. Monkhorst’s approach, see Sect. 2, or the quantum field approach using the full
set of Goldstone bosons, presented here Sect. 11, can never lead to a meaningful
Berry-phase in J-T systems.
In Sect. 12 we mentioned the often quoted Weinberg attitude to the occurrence
of SSB: Only infinite many-body systems and fields can be spontaneously broken,
whereas finite systems due to the possible tunnelling between degenerate states cannot. But from the perspective of our analysis this attitude does not reflect the whole
truth. Physicists often reject considerations that have even if only a little sniff of
chemistry and do not realize that the simplest examples of SSB, such as isomerism
arises on the platform of quantum chemistry in finite systems and that a revealing of
the true nature of the B-O approximation is the correct response to a correct understanding of SSB. In fact, fields with infinite degrees of freedom are not necessary for
SSB occurance at all.
We have here argued that a field theoretical formulation is valid for molecules
with finite degrees of freedom as well. Moreover, this theory is able to describe isomerism, i.e. SSB. On the other hand, many-body quantum mechanical applications,
regardless of being finite or infinite, can never lead to any SSB. Based on Monkhorst’s
349
problem in so-called J-T molecules, see Bersuker [112]: “…one of the simplest JT
E ⊗ e problems with linear vibronic coupling yields an APES (Adiabatic Potential
Energy Surface) in the form of a “Mexican hat”… For a long time, up to the last
decade, not very much attention was paid to these conical intersections; rather they
were viewed largely as a “theoretical curiosity.” This perception changed recently
in view of the latest achievements in the treatment of such systems. One of these
achievements is a generalization directly related to the JTE and now known as the
topological (geometric) phase, or the Berry-phase problem… An important feature
of the Berry-phase implications in JT problems is that the peculiar phase factor that
changes the sign of the electronic wavefunctions and makes the ground vibronic
state degenerate occurs only when one or an odd number of conical intersections
are encircled, while it retains the same sign if an even number (including zero) are
encircled. This was shown by direct calculation of the phase in the E ⊗ e problem. In
fact, the phase is φ 0 = nπ , where n is the number of conical intersections encircled;
for n = 0, 2, 4, … the sign of the wavefunction does not change.”
The Berry-phase was mentioned above. The exact definition can be found in
Berry’s original paper [133]: “A quantal system in an eigenstate, slowly transported
round a circuit C by varying parameters R in its Hamiltonian H(R), will acquire
a geometrical phase factor exp[iγ (C)] in addition to the familiar dynamical phase
factor… If C lies near a degeneracy of H, γ (C) takes a simple form which includes
as a special case the sign change of eigenfunctions of real symmetric matrices round
a degeneracy.” However, there is a problem, since the Berry-phase, based on the
definition above, cannot be applied to J-T molecules at all. For instance the BO approximation can either be derived as a result of the m/M expansion or in an
equivalent form using the hierarchical quantization process, i.e. first quantize the
electronic motion at fixed classical nuclear positions and then a posteriori quantize
the nuclear motion. Applying the Berry-phase to J-T molecules is just a product of this
type of hierarchical quantization. A more exact simultaneous quantization method,
i.e. Monkhorst’s approach, see Sect. 2, or the quantum field approach using the full
set of Goldstone bosons, presented here Sect. 11, can never lead to a meaningful
Berry-phase in J-T systems.
In Sect. 12 we mentioned the often quoted Weinberg attitude to the occurrence
of SSB: Only infinite many-body systems and fields can be spontaneously broken,
whereas finite systems due to the possible tunnelling between degenerate states cannot. But from the perspective of our analysis this attitude does not reflect the whole
truth. Physicists often reject considerations that have even if only a little sniff of
chemistry and do not realize that the simplest examples of SSB, such as isomerism
arises on the platform of quantum chemistry in finite systems and that a revealing of
the true nature of the B-O approximation is the correct response to a correct understanding of SSB. In fact, fields with infinite degrees of freedom are not necessary for
SSB occurance at all.
We have here argued that a field theoretical formulation is valid for molecules
with finite degrees of freedom as well. Moreover, this theory is able to describe isomerism, i.e. SSB. On the other hand, many-body quantum mechanical applications,
regardless of being finite or infinite, can never lead to any SSB. Based on Monkhorst’s