Theor Chem Acc (2015) 134:134
1 3
In the present study, the earlier relations are extended
to cases involving larger number of “anchor structure”
points of the relevant nuclear charge space, that is, higherdimensional convex sets will be used than in earlier studies.
These higher-dimensional cases are not as easily visualized; however, they can provide more options for interrelations between molecules. Some new connections to the
reference cluster and the universal molecule models, especially in the context of symmetry, will also be elaborated.
A common feature of most of the earlier approaches is
the replacement of the formal nuclear charges by continuous variables, and considering the resulting values as components of a “nuclear charge vector” z in a formal nuclear
charge space Z , of dimension equal to or greater than the
number of nuclei in the molecules studied. Note that formal dummy nuclei of zero nuclear charges could always
be added without affecting the validity of the physical laws
applied, and hence very general treatments, interrelating
rather diverse molecules and ionic systems, have become
possible.
Although the specifi c linear dependence of the electronic
Hamiltonian on the nuclear charges has been the primary
tool applied, it has been also natural to extend the approach
by replacing other discrete, integer parameters of molecules
by continuous variables. Clearly, no such fractional nuclear
charges correspond to real molecules, and similarly, allowing, for example, the number of nuclei and the number
of electrons to vary continuously, many “non-physical,”
abstract entities can result. For example, just as there exists
no molecule “halfway between” N 2 and CO, as far as the
nuclear charge vectors of (7, 7) and (6, 8) are concerned,
similarly, no actual molecules exist, for example, with 10.3
nuclei or with some non-integer number of electrons. Nevertheless, just as the mathematical approaches extended to
the complex plane can provide useful shortcuts (such as
contour integration), even if no actual physical quantities
can ever take complex values, similarly, such an extension
of the family of real molecules to fi ctitious “objects” with
non-physical properties also has provided useful shortcuts.
This way, new relations can be derived between actual,
physically valid molecules, such as the simplest of these
rigorous electronic energy inequalities for N 2 and CO using
continuous nuclear charge variations, the earliest example
that will be briefl y reviewed and also extended in the next
section.
By following this idea to the extreme case, where all such
parameters of individual molecules are considered continuous variables, a rather general model has been established.
The resulting formal “universal molecule” model [ 18 – 21 ]
provides a formal link between any two actual molecules,
by some continuous changes in all those parameters which
distinguish these molecules. This universal molecule model
offers not only conceptual tools, but also new relations on
energy or other properties of actual, real molecules. In this
contribution, the advantages of this model are used to obtain
new results for both energy interpolation, as well as energy
extrapolation, resulting in both upper and lower bounds for
energies. Besides this result, the central role of the so-called
“reference cluster” [ 22 , 23 ], a formal molecular entity with
all nuclei having the same nuclear charge, is discussed.
The reference cluster has a distinguished role in terms of
its transformations to real, actual molecules, and also in
terms of the maximal symmetry properties of the associated potential energy surfaces, where somewhat counterintuitive, “opposing” trends exist between transformations
reducing symmetry and reducing energy.
2 Nuclear charge space convexity relations
and their extensions
In the Born–Oppenheimer approximation, the molecular
total energy, E t , is regarded as a sum of the nuclear repulsion energy, E n , and the electronic energy E e ,
where the latter is associated with the electronic Hamiltonian H e ( N , k , z , r ) given as
The z i nuclear charges of the N nuclei of nuclear positions
r i are collected into a formal nuclear charge vector z (usually considered as a column vector, although often the
transpose, z′ , is used), and the index t refers to the electrons
of a k -electron molecule.
In order to derive the general, ( n − 1)-dimensional convexity relation for nuclear charge vectors, consider n + 1
isoelectronic molecular systems,
in their electronic ground states (or in some common, lowest state of some electronic manifold), where the nuclear
locations match, but the nuclear charges can be different
and even zero nuclear charges, that is, “dummy nuclei” are
allowed.
For each of these molecular systems, the nuclear charge
vector z contains the individual nuclear charges as components, following some fi xed ordering.
Let us denote the corresponding nuclear charge vectors
by
(1)
E t = E n + E e
(2)
H e (N, k, z, r) = −1/2
t=1,k
t −
i=1,N
t=1,k
z i /|r i − r t |
+
t
(3)
M, M
(1) , M
(2) , . . . M
(i) , . . . M
(n)
(4)
z, z
(1) , z
(2) , . . . z
(i) , . . . z
(n) ,
26
Reprinted from the journal
1 3
In the present study, the earlier relations are extended
to cases involving larger number of “anchor structure”
points of the relevant nuclear charge space, that is, higherdimensional convex sets will be used than in earlier studies.
These higher-dimensional cases are not as easily visualized; however, they can provide more options for interrelations between molecules. Some new connections to the
reference cluster and the universal molecule models, especially in the context of symmetry, will also be elaborated.
A common feature of most of the earlier approaches is
the replacement of the formal nuclear charges by continuous variables, and considering the resulting values as components of a “nuclear charge vector” z in a formal nuclear
charge space Z , of dimension equal to or greater than the
number of nuclei in the molecules studied. Note that formal dummy nuclei of zero nuclear charges could always
be added without affecting the validity of the physical laws
applied, and hence very general treatments, interrelating
rather diverse molecules and ionic systems, have become
possible.
Although the specifi c linear dependence of the electronic
Hamiltonian on the nuclear charges has been the primary
tool applied, it has been also natural to extend the approach
by replacing other discrete, integer parameters of molecules
by continuous variables. Clearly, no such fractional nuclear
charges correspond to real molecules, and similarly, allowing, for example, the number of nuclei and the number
of electrons to vary continuously, many “non-physical,”
abstract entities can result. For example, just as there exists
no molecule “halfway between” N 2 and CO, as far as the
nuclear charge vectors of (7, 7) and (6, 8) are concerned,
similarly, no actual molecules exist, for example, with 10.3
nuclei or with some non-integer number of electrons. Nevertheless, just as the mathematical approaches extended to
the complex plane can provide useful shortcuts (such as
contour integration), even if no actual physical quantities
can ever take complex values, similarly, such an extension
of the family of real molecules to fi ctitious “objects” with
non-physical properties also has provided useful shortcuts.
This way, new relations can be derived between actual,
physically valid molecules, such as the simplest of these
rigorous electronic energy inequalities for N 2 and CO using
continuous nuclear charge variations, the earliest example
that will be briefl y reviewed and also extended in the next
section.
By following this idea to the extreme case, where all such
parameters of individual molecules are considered continuous variables, a rather general model has been established.
The resulting formal “universal molecule” model [ 18 – 21 ]
provides a formal link between any two actual molecules,
by some continuous changes in all those parameters which
distinguish these molecules. This universal molecule model
offers not only conceptual tools, but also new relations on
energy or other properties of actual, real molecules. In this
contribution, the advantages of this model are used to obtain
new results for both energy interpolation, as well as energy
extrapolation, resulting in both upper and lower bounds for
energies. Besides this result, the central role of the so-called
“reference cluster” [ 22 , 23 ], a formal molecular entity with
all nuclei having the same nuclear charge, is discussed.
The reference cluster has a distinguished role in terms of
its transformations to real, actual molecules, and also in
terms of the maximal symmetry properties of the associated potential energy surfaces, where somewhat counterintuitive, “opposing” trends exist between transformations
reducing symmetry and reducing energy.
2 Nuclear charge space convexity relations
and their extensions
In the Born–Oppenheimer approximation, the molecular
total energy, E t , is regarded as a sum of the nuclear repulsion energy, E n , and the electronic energy E e ,
where the latter is associated with the electronic Hamiltonian H e ( N , k , z , r ) given as
The z i nuclear charges of the N nuclei of nuclear positions
r i are collected into a formal nuclear charge vector z (usually considered as a column vector, although often the
transpose, z′ , is used), and the index t refers to the electrons
of a k -electron molecule.
In order to derive the general, ( n − 1)-dimensional convexity relation for nuclear charge vectors, consider n + 1
isoelectronic molecular systems,
in their electronic ground states (or in some common, lowest state of some electronic manifold), where the nuclear
locations match, but the nuclear charges can be different
and even zero nuclear charges, that is, “dummy nuclei” are
allowed.
For each of these molecular systems, the nuclear charge
vector z contains the individual nuclear charges as components, following some fi xed ordering.
Let us denote the corresponding nuclear charge vectors
by
(1)
E t = E n + E e
(2)
H e (N, k, z, r) = −1/2
t=1,k
t −
i=1,N
t=1,k
z i /|r i − r t |
+
t
(3)
M, M
(1) , M
(2) , . . . M
(i) , . . . M
(n)
(4)
z, z
(1) , z
(2) , . . . z
(i) , . . . z
(n) ,
26
Reprinted from the journal