96
3 Ab initio Electronic Structure for Few-Body Systems
contrary, the corresponding analytical evaluation requires only twice the time of a
single SCF calculation. The calculation of analytic second derivatives still requires
only three or four times the time taken by the gradient calculation.
This method is used mainly as a test of the accuracy of the calculations and for
their improvement. The comparison can be made also with a given property like the
vibrational frequencies of the molecular system or a set of properties (multiproperty
analysis) owing to the fact that each property may be more sensitive to different
features of the potential energy. Among the quantities to consider are structural and
thermodynamical properties (conformational maps, free energy maps, ionization
potentials, electronic affinities), charge distributions (Mulliken population analysis,
molecular potential fields, electronic density at the border), dynamical properties
(cross sections, energy distribution of products, vector distributions), transportation
properties (viscosity, virial).
To this end, it is important to point out here that the analytical calculation of
the derivatives of the potential energy can be significantly helpful in determining
dipole moment and polarizability derivatives. Derivatives of the potential energy
also provide invaluable information for the fitting of the calculated potential energy
values to a functional form describing the path connecting reactants and products as
we shall discuss later.
3.3.2 Density Functional Theory Methods
For large systems and a large number of molecular geometries calculations (as often
needed for the investigation of molecular collisions) electronic density funtional
theory (DFT) provides a versatile and practical means to calculate electronic energies
because there is a linearly scaling with the number of electrons (while the HF methods
scale usually as the fourth power). The key quantity of DFT is the electronic density
ρ(r) (the number of electrons per unit volume in a given space point). The method
replaces, in fact, the problem of determining the wavefunction (that is the key quantity
of traditional ab initio techniques) with that of determining the electronic density.
The method leverages on the simple idea that the electronic system of a molecule
behaves as a gas of particles subject to coulomb interactions.
The electronic density ρ(r) depends on three spatial coordinates and on the spin
independently of the dimension of the actual physical system. This has been formalized by the Hohenberg–Kohn theorems stating that for the ground state the potential
energy V (r) depends on the electronic density ρ(r) that can be computed using variational methods. At the same time ρ(r) can be easily related to the number K of
electrons as follows:
ρ(r)dr = K
(3.33)
and determines the wavefunction of the ground state as well as the related electronic
energy that is formulated as
3 Ab initio Electronic Structure for Few-Body Systems
contrary, the corresponding analytical evaluation requires only twice the time of a
single SCF calculation. The calculation of analytic second derivatives still requires
only three or four times the time taken by the gradient calculation.
This method is used mainly as a test of the accuracy of the calculations and for
their improvement. The comparison can be made also with a given property like the
vibrational frequencies of the molecular system or a set of properties (multiproperty
analysis) owing to the fact that each property may be more sensitive to different
features of the potential energy. Among the quantities to consider are structural and
thermodynamical properties (conformational maps, free energy maps, ionization
potentials, electronic affinities), charge distributions (Mulliken population analysis,
molecular potential fields, electronic density at the border), dynamical properties
(cross sections, energy distribution of products, vector distributions), transportation
properties (viscosity, virial).
To this end, it is important to point out here that the analytical calculation of
the derivatives of the potential energy can be significantly helpful in determining
dipole moment and polarizability derivatives. Derivatives of the potential energy
also provide invaluable information for the fitting of the calculated potential energy
values to a functional form describing the path connecting reactants and products as
we shall discuss later.
3.3.2 Density Functional Theory Methods
For large systems and a large number of molecular geometries calculations (as often
needed for the investigation of molecular collisions) electronic density funtional
theory (DFT) provides a versatile and practical means to calculate electronic energies
because there is a linearly scaling with the number of electrons (while the HF methods
scale usually as the fourth power). The key quantity of DFT is the electronic density
ρ(r) (the number of electrons per unit volume in a given space point). The method
replaces, in fact, the problem of determining the wavefunction (that is the key quantity
of traditional ab initio techniques) with that of determining the electronic density.
The method leverages on the simple idea that the electronic system of a molecule
behaves as a gas of particles subject to coulomb interactions.
The electronic density ρ(r) depends on three spatial coordinates and on the spin
independently of the dimension of the actual physical system. This has been formalized by the Hohenberg–Kohn theorems stating that for the ground state the potential
energy V (r) depends on the electronic density ρ(r) that can be computed using variational methods. At the same time ρ(r) can be easily related to the number K of
electrons as follows:
ρ(r)dr = K
(3.33)
and determines the wavefunction of the ground state as well as the related electronic
energy that is formulated as