integration on the complex plane, using imaginary values, is advantageous in
deriving new relations for problems involving only real numbers, this additional
freedom of using continuous nuclear charges in the Universal Molecule models can
also lead to new relations between real molecules.
As one of the simplest examples, a continuous nuclear charge variation between
isoelectronic molecules N 2 and CO provides quantum chemically rigorous electronic energy inequalities, universally valid for every common bond length value
[10]. Note that, far more complicated energy relations can also be derived by this
approach [8–11].
9.4 The Extent of Identity-Preserving Deformations
of Chemical Species
A somewhat simpler aspect of the Universal Molecule model is exploited if one
considers a specific stoichiometry, that is, a given set of nuclei, as a single
“super-entity”, and all the possible molecular species which can be obtained from
this set of nuclei and a fixed number of electrons are regarded only as variants of the
same Universal Molecule. In fact, this, somewhat simplified version of the
Universal Molecule model is the closest to the potential energy surface model: if the
electronic state is also restricted, then, in fact, all realizations of this Universal
Molecule are actual species along the potential energy surface.
It is of some interest to link this model to more conventional models of chemical
species. Traditionally, molecular deformations are often considered in the context
of shape changes, for example, shape changes of the bonding pattern and the
nuclear skeleton, or the shape changes of the actual electron density cloud [4].
Shape changes are often studied in terms of symmetry, or in terms of deformations
relative to some symmetry [5–7], and the relations between local and global similarities among molecular species are relevant [7, 12]. One rather general model, as
a part of the Universal Molecule approach [8–11] that has been applied, for
example, for transformations between molecules by nuclear charge variations, as
well as in combinatorial quantum chemistry approaches [8], also describes deformations which often go beyond those which preserve chemical identity.
For a given electronic state, associated with a specified potential energy surface
E(K), the simplest model to describe identity-preserving deformations is based on
the concept of catchment regions [3]: all distorted conformations K from where an
infinitely slow, vibrationless relaxation would lead to a common critical point on
the potential energy surface, belong to the same catchment region. Note that
catchment regions can have different dimensions: for a (3N−6)-dimensional nuclear
configuration space M, the catchment region of an energy minimum is also (3N−6)dimensional, yet the catchment regions of various saddle points have dimensions
less than (3N−6).
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P.G. Mezey
deriving new relations for problems involving only real numbers, this additional
freedom of using continuous nuclear charges in the Universal Molecule models can
also lead to new relations between real molecules.
As one of the simplest examples, a continuous nuclear charge variation between
isoelectronic molecules N 2 and CO provides quantum chemically rigorous electronic energy inequalities, universally valid for every common bond length value
[10]. Note that, far more complicated energy relations can also be derived by this
approach [8–11].
9.4 The Extent of Identity-Preserving Deformations
of Chemical Species
A somewhat simpler aspect of the Universal Molecule model is exploited if one
considers a specific stoichiometry, that is, a given set of nuclei, as a single
“super-entity”, and all the possible molecular species which can be obtained from
this set of nuclei and a fixed number of electrons are regarded only as variants of the
same Universal Molecule. In fact, this, somewhat simplified version of the
Universal Molecule model is the closest to the potential energy surface model: if the
electronic state is also restricted, then, in fact, all realizations of this Universal
Molecule are actual species along the potential energy surface.
It is of some interest to link this model to more conventional models of chemical
species. Traditionally, molecular deformations are often considered in the context
of shape changes, for example, shape changes of the bonding pattern and the
nuclear skeleton, or the shape changes of the actual electron density cloud [4].
Shape changes are often studied in terms of symmetry, or in terms of deformations
relative to some symmetry [5–7], and the relations between local and global similarities among molecular species are relevant [7, 12]. One rather general model, as
a part of the Universal Molecule approach [8–11] that has been applied, for
example, for transformations between molecules by nuclear charge variations, as
well as in combinatorial quantum chemistry approaches [8], also describes deformations which often go beyond those which preserve chemical identity.
For a given electronic state, associated with a specified potential energy surface
E(K), the simplest model to describe identity-preserving deformations is based on
the concept of catchment regions [3]: all distorted conformations K from where an
infinitely slow, vibrationless relaxation would lead to a common critical point on
the potential energy surface, belong to the same catchment region. Note that
catchment regions can have different dimensions: for a (3N−6)-dimensional nuclear
configuration space M, the catchment region of an energy minimum is also (3N−6)dimensional, yet the catchment regions of various saddle points have dimensions
less than (3N−6).
252
P.G. Mezey