Theor Chem Acc (2015) 134:134
1 3
Note, however, that if the common geometry of the tetrahedra chosen shows some symmetry, then some simplifi -
cations are possible, for example, structures of molecules
M
(2) and M
(3) may become equivalent, that can lead to a
more concise energy relation.
That is, symmetry, in this case, indeed, provides
simplifi cation.
3 Some comments about the universal molecule
aspects of potential energy hypersurface
symmetry and energy relations connected
to the reference cluster
If in a molecule all parameters are taken as continuous variables, one obtains the abstract “universal molecule model”
that connects all real molecules, even seemingly unrelated
molecules, to one another through impossible, non-physical
models within a single mathematical framework [ 18 – 21 ].
Turning discrete variables into continuous ones amounts
to “anti-quantization,” and if these variables again take up
one of the allowed (often integer) values, this amounts to
“re-quantization.” This anti-quantized and re-quantized
“universal molecule model” provides new, valid relations
on energy and other properties of real molecules, and all
actual molecules may then be regarded as special cases of
the “universal molecule.”
In a formal sense, it is possible to defi ne a universal
molecule
of N nuclei and k electrons, with nuclear confi guration
vector variable r , nuclear charge vector z , electronic state
label s , and possible additional variables w , where all these
variables are allowed to take non-integer values, even if for
some of them only integer values (and very limited values)
may actually correspond to physically possible entities.
We have seen that for the case of nuclear charges, such
a generalization provides rather quick and rigorous results.
Although taking advantage of other such variables appears
less straightforward, this general framework provides the
tools for such investigations. One fi eld that can benefi t
from such a model is using comparisons of various realizations of the universal molecule to the level of reference
clusters [ 22 , 23 ], where the potential energy surface, PES,
of the reference clusters typically shows the highest symmetry when compared to molecules of equal total nuclear
charge, as a consequence of permutational equivalences of
atoms, being the highest possible for the reference clusters.
On the other hand, as shown also by the diatomic example with N 2 being the reference cluster, the energy of the
reference cluster is typically higher than the energy of the
related other molecules with the same total nuclear charge.
M(N, k, r, z, s, w)
That is, in this case, higher symmetry is associated with
higher energy, a combination of properties that in other
contexts often considered counter-intuitive.
Specifi cally, we are concerned with two symmetry problems along PES [ 22 , 23 ]:
(a) the distribution of the three-dimensional, 3D, symmetries of molecular structures along PES
(b) the actual (3 N + 1)D symmetry of the (3 N )D PES
itself, defi ned by the 3 N laboratory-frame Cartesian
coordinates of the N nuclei; such a PES is a (3 N )D
hypersurface embedded in a (3 N + 1)D space, where
one may formally regard energy as the “vertical coordinate.”
It is often advantageous to use the (3 N − 6)D reduced
nuclear confi guration space, a metric space M obtained as
the quotient space of equivalence classes of (chemically
equivalent) internal confi gurations related to one another
by rigid translations and rigid rotations within the (3 N )D
Euclidean space of all, mass-weighted Cartesian coordinates of the nuclei. If the PES is defi ned over M , then it is a
(3 N − 6)D hypersurface embedded in a (3 N − 5)D space,
where, as before, one may formally regard energy as the
“vertical coordinate.”
In general, if the dimension of a Euclidean space is W ,
then all refl ections in such high-dimensional spaces can
be constructed by some series of refl ections with reference to some ( W − 1)D hyperplanes, and all rotations can
be generated by a series of rotations according to some
( W − 2)D rotation axes. It is advantageous to use for each
of these refl ections and rotations specifi c coordinate systems aligned with the refl ection planes and rotation axes,
in such a way that the origin falls on the refl ection plane
or rotation axis, and one coordinate axis is orthogonal to
the ( W − 1)D refl ection hyperplane, and two coordinate
axes are orthogonal to the ( W − 2)D rotation axis. Then,
refl ection is accomplished by a sign change in all coordinate values along the coordinate axis orthogonal to the
refl ection hyperplane. Rotation by some angle α is accomplished by keeping all coordinates along the ( W − 2)D
rotation axis unchanged, and by the cos( α ), sin( α ) linear
combination of the remaining two coordinate values for the
new fi rst such coordinate, and the −sin( α ), cos( α ) linear
combinations of the remaining two coordinate values for
the new second such coordinate. This is in exact analogy
with the rotation along the z axis in 3D, where x and y take
the roles of the two “remaining” coordinates, becoming
x cos( α ) + y sin( α ) for the new “fi rst” coordinate value, and
− x sin( α ) + y cos( α ) for the new “second” coordinate value.
These very tools, higher-dimensional refl ections and
rotations are those which are gradually “degraded” from
their symmetry element status as the reference cluster is
29
Reprinted from the journal
1 3
Note, however, that if the common geometry of the tetrahedra chosen shows some symmetry, then some simplifi -
cations are possible, for example, structures of molecules
M
(2) and M
(3) may become equivalent, that can lead to a
more concise energy relation.
That is, symmetry, in this case, indeed, provides
simplifi cation.
3 Some comments about the universal molecule
aspects of potential energy hypersurface
symmetry and energy relations connected
to the reference cluster
If in a molecule all parameters are taken as continuous variables, one obtains the abstract “universal molecule model”
that connects all real molecules, even seemingly unrelated
molecules, to one another through impossible, non-physical
models within a single mathematical framework [ 18 – 21 ].
Turning discrete variables into continuous ones amounts
to “anti-quantization,” and if these variables again take up
one of the allowed (often integer) values, this amounts to
“re-quantization.” This anti-quantized and re-quantized
“universal molecule model” provides new, valid relations
on energy and other properties of real molecules, and all
actual molecules may then be regarded as special cases of
the “universal molecule.”
In a formal sense, it is possible to defi ne a universal
molecule
of N nuclei and k electrons, with nuclear confi guration
vector variable r , nuclear charge vector z , electronic state
label s , and possible additional variables w , where all these
variables are allowed to take non-integer values, even if for
some of them only integer values (and very limited values)
may actually correspond to physically possible entities.
We have seen that for the case of nuclear charges, such
a generalization provides rather quick and rigorous results.
Although taking advantage of other such variables appears
less straightforward, this general framework provides the
tools for such investigations. One fi eld that can benefi t
from such a model is using comparisons of various realizations of the universal molecule to the level of reference
clusters [ 22 , 23 ], where the potential energy surface, PES,
of the reference clusters typically shows the highest symmetry when compared to molecules of equal total nuclear
charge, as a consequence of permutational equivalences of
atoms, being the highest possible for the reference clusters.
On the other hand, as shown also by the diatomic example with N 2 being the reference cluster, the energy of the
reference cluster is typically higher than the energy of the
related other molecules with the same total nuclear charge.
M(N, k, r, z, s, w)
That is, in this case, higher symmetry is associated with
higher energy, a combination of properties that in other
contexts often considered counter-intuitive.
Specifi cally, we are concerned with two symmetry problems along PES [ 22 , 23 ]:
(a) the distribution of the three-dimensional, 3D, symmetries of molecular structures along PES
(b) the actual (3 N + 1)D symmetry of the (3 N )D PES
itself, defi ned by the 3 N laboratory-frame Cartesian
coordinates of the N nuclei; such a PES is a (3 N )D
hypersurface embedded in a (3 N + 1)D space, where
one may formally regard energy as the “vertical coordinate.”
It is often advantageous to use the (3 N − 6)D reduced
nuclear confi guration space, a metric space M obtained as
the quotient space of equivalence classes of (chemically
equivalent) internal confi gurations related to one another
by rigid translations and rigid rotations within the (3 N )D
Euclidean space of all, mass-weighted Cartesian coordinates of the nuclei. If the PES is defi ned over M , then it is a
(3 N − 6)D hypersurface embedded in a (3 N − 5)D space,
where, as before, one may formally regard energy as the
“vertical coordinate.”
In general, if the dimension of a Euclidean space is W ,
then all refl ections in such high-dimensional spaces can
be constructed by some series of refl ections with reference to some ( W − 1)D hyperplanes, and all rotations can
be generated by a series of rotations according to some
( W − 2)D rotation axes. It is advantageous to use for each
of these refl ections and rotations specifi c coordinate systems aligned with the refl ection planes and rotation axes,
in such a way that the origin falls on the refl ection plane
or rotation axis, and one coordinate axis is orthogonal to
the ( W − 1)D refl ection hyperplane, and two coordinate
axes are orthogonal to the ( W − 2)D rotation axis. Then,
refl ection is accomplished by a sign change in all coordinate values along the coordinate axis orthogonal to the
refl ection hyperplane. Rotation by some angle α is accomplished by keeping all coordinates along the ( W − 2)D
rotation axis unchanged, and by the cos( α ), sin( α ) linear
combination of the remaining two coordinate values for the
new fi rst such coordinate, and the −sin( α ), cos( α ) linear
combinations of the remaining two coordinate values for
the new second such coordinate. This is in exact analogy
with the rotation along the z axis in 3D, where x and y take
the roles of the two “remaining” coordinates, becoming
x cos( α ) + y sin( α ) for the new “fi rst” coordinate value, and
− x sin( α ) + y cos( α ) for the new “second” coordinate value.
These very tools, higher-dimensional refl ections and
rotations are those which are gradually “degraded” from
their symmetry element status as the reference cluster is
29
Reprinted from the journal