286
M. Svrˇ cek
What about the objection that we can avoid all the above mentioned problems by
solving the Schrödinger equation exactly. I quote from Monkhorst’s article [5] an
analogous critique of the above mentioned PES concept: “The BO approximation
is strictly valid only near the multidimensional PES minimum. On the other hand,
the BH (Born-Huang) treatment inspired the thinking that the entire PES is usually
valid to describe the dynamics of a molecule, including dissociation into various
fragments, i.e., chemical reactions. However, this view is only acceptable if the PESassociated wave functions interact only weakly for all nuclear configurations. It is
usually quite impossible to verify this, and most consumers of the PES concept
assume it. Their only “defense” is the adiabatic nature of a molecular wave function,
where the electronic wave functions adjust “instantaneously” to the evolving nuclear
configuration. In fact, this view is central to the teachings of quantum chemistry.”
Monkhorst also describes his coupled-cluster method for the solution of the system
of electrons and nuclei on the same footing, completely evading the B-O approximation: “Therefore, when the need arises to address the limitations of the PES, it
seems valuable to remove it entirely from the toolbox of the molecular scientist. One
option is the molecular coupled-cluster (MCC) method I formulated 10 years ago
[6]. This method takes a very atomic view of a molecule: instead of fixing the nuclei
as in the BO approximation, the electrons and nuclei are both described quantumdynamically within a centrosymmetric shell structure. The coupled-cluster method,
duly generalized, is used to describe the correlation among all particles.”
Being aware of the practical computational limitations of this exact method, he
adds: “Even though the MCC method seems attractive, and (I hope) computationally
tractable once implemented, it can be only practical for small molecules. I cannot see
how the PES concept, and its attendant molecular structure ideas, will be superseded
with this method for large molecules. Its qualitative appeal, its semiquantitative
success, and its deep roots in the chemists’ minds will keep the PES as real as it is
now.”
The reasons given above are most likely why most of chemists prefer calculations
based on the B-O approximation. As far as the error caused by this calculation, on the
adiabatic level the Born-Huang (B-H) ansatz [7] is applied, and as Kutzelnigg proved
[8], the B-H ansatz fully compensates for this error. Note however that Kutzelnigg’s
proof indicates only the numerical equivalence, and solely on the adiabatic level.
There are no objective equivalences between the exact solution of the Schrödinger
equation and solutions based on the whole of the B-O approximation and B-H ansatz
at all, even on the adiabatic level.
Fortunately, a small number of scientists were inspired by Monkhorst’s idea,
and performed exact calculations of some of the simplest molecules. Cafiero and
Adamowitz wrote [9]: “The model of the molecule… is quite similar to an atom, as
has been noted by Monkhorst [5]. We have the analogue of the nucleus with the heavy
particle at the center of the internal coordinate system, and we have the analogues
of electrons in the internal particles. The main difference between this model and an
atom is that the internal particles in an atom are all electrons and in the molecular
atom or atomic molecule the internal particles may be both electrons and nuclei
(or, as we should more correctly say, pseudo-particles resembling the electron and
M. Svrˇ cek
What about the objection that we can avoid all the above mentioned problems by
solving the Schrödinger equation exactly. I quote from Monkhorst’s article [5] an
analogous critique of the above mentioned PES concept: “The BO approximation
is strictly valid only near the multidimensional PES minimum. On the other hand,
the BH (Born-Huang) treatment inspired the thinking that the entire PES is usually
valid to describe the dynamics of a molecule, including dissociation into various
fragments, i.e., chemical reactions. However, this view is only acceptable if the PESassociated wave functions interact only weakly for all nuclear configurations. It is
usually quite impossible to verify this, and most consumers of the PES concept
assume it. Their only “defense” is the adiabatic nature of a molecular wave function,
where the electronic wave functions adjust “instantaneously” to the evolving nuclear
configuration. In fact, this view is central to the teachings of quantum chemistry.”
Monkhorst also describes his coupled-cluster method for the solution of the system
of electrons and nuclei on the same footing, completely evading the B-O approximation: “Therefore, when the need arises to address the limitations of the PES, it
seems valuable to remove it entirely from the toolbox of the molecular scientist. One
option is the molecular coupled-cluster (MCC) method I formulated 10 years ago
[6]. This method takes a very atomic view of a molecule: instead of fixing the nuclei
as in the BO approximation, the electrons and nuclei are both described quantumdynamically within a centrosymmetric shell structure. The coupled-cluster method,
duly generalized, is used to describe the correlation among all particles.”
Being aware of the practical computational limitations of this exact method, he
adds: “Even though the MCC method seems attractive, and (I hope) computationally
tractable once implemented, it can be only practical for small molecules. I cannot see
how the PES concept, and its attendant molecular structure ideas, will be superseded
with this method for large molecules. Its qualitative appeal, its semiquantitative
success, and its deep roots in the chemists’ minds will keep the PES as real as it is
now.”
The reasons given above are most likely why most of chemists prefer calculations
based on the B-O approximation. As far as the error caused by this calculation, on the
adiabatic level the Born-Huang (B-H) ansatz [7] is applied, and as Kutzelnigg proved
[8], the B-H ansatz fully compensates for this error. Note however that Kutzelnigg’s
proof indicates only the numerical equivalence, and solely on the adiabatic level.
There are no objective equivalences between the exact solution of the Schrödinger
equation and solutions based on the whole of the B-O approximation and B-H ansatz
at all, even on the adiabatic level.
Fortunately, a small number of scientists were inspired by Monkhorst’s idea,
and performed exact calculations of some of the simplest molecules. Cafiero and
Adamowitz wrote [9]: “The model of the molecule… is quite similar to an atom, as
has been noted by Monkhorst [5]. We have the analogue of the nucleus with the heavy
particle at the center of the internal coordinate system, and we have the analogues
of electrons in the internal particles. The main difference between this model and an
atom is that the internal particles in an atom are all electrons and in the molecular
atom or atomic molecule the internal particles may be both electrons and nuclei
(or, as we should more correctly say, pseudo-particles resembling the electron and