Megascopic Quantum Phenomena
287
the nuclei). Formally this difference manifests itself in the effective masses of the
pseudo-particles and in the way the permutational symmetry is implemented in the
wave function.”
And further looking at the conclusion from their calculations: “Since in the nonadiabatic treatment we have included the nuclei in the wave function, we must determine
the molecular structure by calculating expectation values of the distances between the
nuclei. Since the operators representing the internuclear distances… do not commute
with the Hamiltonian…, we cannot measure these distances exactly for stationary
states of the system. Furthermore, since the ground state is spherically symmetric, we
cannot gain additional information from the wave function such as the expectation
values of the x, y, z coordinates of any of the particles, since these will average out
to zero in all cases. Lastly, the expectation values for the distances between any two
particles in each subset of identical particles (including the distance to the particle at
the origin, if that particle belongs to the subset) will be equal. This is a widely known
fact for electrons, i.e., due to antisymmetry, or indistinguishability, all of the electrons
are on average at the same distance from each other, as well as from each nucleus in
the molecule. That this same fact applies to nuclei is, perhaps, less known.”
These first calculations of their kind gave us a remarkable insight into the actual
differences between the exact and the clamped-nuclei approaches. The exact solution
communicates the concept of the “molecular atom or atomic molecule”, where we
can only see one undivided isolated composite object, while we are unable to say anything more about the individual molecules involved in such a system. The hierarchy
elementary particles → atoms → molecules doesn’t exist here as we know it from
the B-O approximation; instead it is replaced with the reduced hierarchy elementary
particles → molecular atoms or atomic molecules. The exact solution leads always
to a full symmetry picture of the molecule, and the indistinguishability principle,
valid for nuclei as well, doesn’t allow any “symmetry breaking”, such as e.g. an
isomerism. As Sutcliffe and Woolley state: “The Isolated Molecule model doesn’t
capture isomerism, nor optical activity.” It is indeed a pity that exact solutions are
so complex that till now they have only been tested on very small systems, i.e. up
to ten particles, electrons and nuclei included. It would be very interesting to test
them on something bigger at least on the simplest Jahn and Teller [10] systems. It
seems that the indistinguishability of the nuclei and their shell nature, no J-T symmetry breakings are possible at all, cf. the case of isomerism, and that we will obtain
fully symmetrical solutions for the ground states. Although standard theories of the
J-T effect [11, 12] go beyond the B-O approximation and are nonadiabatic, they are
nevertheless all of clamped-nuclei type.
Thus we can see that even exact solutions of Schrödinger’s equation do not
yield the full picture of all quantum chemical phenomena. Yet we cannot avoid
the clamped-nuclei concept in the B-O approximation, leading to a serious paradox:
how can an approximation produce results that by no means follow from the exact
solution? What is the nature of the B-O model when it is not viewed as an approximation? Where is the true origin of the clamped-nuclei concept, allowing the description
of individual molecules? This is indeed a quantum paradox, since we don’t know any
classical analogies. How do we reach the B-O approximation? Perhaps by simply
287
the nuclei). Formally this difference manifests itself in the effective masses of the
pseudo-particles and in the way the permutational symmetry is implemented in the
wave function.”
And further looking at the conclusion from their calculations: “Since in the nonadiabatic treatment we have included the nuclei in the wave function, we must determine
the molecular structure by calculating expectation values of the distances between the
nuclei. Since the operators representing the internuclear distances… do not commute
with the Hamiltonian…, we cannot measure these distances exactly for stationary
states of the system. Furthermore, since the ground state is spherically symmetric, we
cannot gain additional information from the wave function such as the expectation
values of the x, y, z coordinates of any of the particles, since these will average out
to zero in all cases. Lastly, the expectation values for the distances between any two
particles in each subset of identical particles (including the distance to the particle at
the origin, if that particle belongs to the subset) will be equal. This is a widely known
fact for electrons, i.e., due to antisymmetry, or indistinguishability, all of the electrons
are on average at the same distance from each other, as well as from each nucleus in
the molecule. That this same fact applies to nuclei is, perhaps, less known.”
These first calculations of their kind gave us a remarkable insight into the actual
differences between the exact and the clamped-nuclei approaches. The exact solution
communicates the concept of the “molecular atom or atomic molecule”, where we
can only see one undivided isolated composite object, while we are unable to say anything more about the individual molecules involved in such a system. The hierarchy
elementary particles → atoms → molecules doesn’t exist here as we know it from
the B-O approximation; instead it is replaced with the reduced hierarchy elementary
particles → molecular atoms or atomic molecules. The exact solution leads always
to a full symmetry picture of the molecule, and the indistinguishability principle,
valid for nuclei as well, doesn’t allow any “symmetry breaking”, such as e.g. an
isomerism. As Sutcliffe and Woolley state: “The Isolated Molecule model doesn’t
capture isomerism, nor optical activity.” It is indeed a pity that exact solutions are
so complex that till now they have only been tested on very small systems, i.e. up
to ten particles, electrons and nuclei included. It would be very interesting to test
them on something bigger at least on the simplest Jahn and Teller [10] systems. It
seems that the indistinguishability of the nuclei and their shell nature, no J-T symmetry breakings are possible at all, cf. the case of isomerism, and that we will obtain
fully symmetrical solutions for the ground states. Although standard theories of the
J-T effect [11, 12] go beyond the B-O approximation and are nonadiabatic, they are
nevertheless all of clamped-nuclei type.
Thus we can see that even exact solutions of Schrödinger’s equation do not
yield the full picture of all quantum chemical phenomena. Yet we cannot avoid
the clamped-nuclei concept in the B-O approximation, leading to a serious paradox:
how can an approximation produce results that by no means follow from the exact
solution? What is the nature of the B-O model when it is not viewed as an approximation? Where is the true origin of the clamped-nuclei concept, allowing the description
of individual molecules? This is indeed a quantum paradox, since we don’t know any
classical analogies. How do we reach the B-O approximation? Perhaps by simply