Binary result files¶

Section Multipole matrix elements¶

Information about a response calculation

dipole elements

The matrix elements of the 3 dipole operator components between occupied and virtual orbitals: outer loop over the operators (in order: y, z, x), loop over virtual MOs, inner loop over occupied MOs

quadrupole elements

Similar as for dipole. Order of operators: \(\sqrt{3}\) *xy \(\sqrt{3}\) *yz z2-(x2+y2)/2 \(\sqrt{3}\) *xz \(\sqrt{3}\) *(x2-y2)/2

octupole elements

Similar as for dipole and quadrupole. Order of operators: \(\sqrt{10}\) *y*(3*x2-y2)/4 \(\sqrt{15}\) *xyz \(\sqrt{6}\) *y*(4*z2-x2-y2)/4 z*(z2-3(x2-y2)/2) \(\sqrt{6}\) *x*(4*z2-x2-y2)/4 \(\sqrt{15}\) *z*(x2-y2)/2 \(\sqrt{10}\) *x*(x2-3y2)/4

hexadecapole elements

Similar as for dipole and quadrupole. Order of operators: \(\sqrt{35}\) *xy*(x2-y2)/2 \(\sqrt{70}\) *z*(3x2y-y3)/4 \(\sqrt{5}\) *xy*(6z2-x2-y2)/2 \(\sqrt{10}\) *(4yz3-3yz*(x2+y2))/4 (8z4-24*z2*(x2+y2)+3(x4+2x2y2+y4))/8 \(\sqrt{10}\) *(4xz3-3xz*(x2+y2))/4 \(\sqrt{5}\) *(x2-y2)*(6z2-x2-y2)/4 \(\sqrt{70}\) *z*(x3-3xy2)/4 \(\sqrt{35}\) (x4-6x2y2+y4)/8

Representation of functions and frozen cores¶

ADF uses the cartesian representation for the spherical harmonics part in functions:

The angular momentum quantum number l is then given by l=a+b+c, and the main quantum number n = l +d+1.

There are (l + 1)(l + 2)/2 different combinations of (a, b, c) for a given l - value, rather than (2 l +1). The excess is caused by the presence of spurious non-l Functions in the set; a Cartesian d-set for instance consists of six functions, five of which are true d-functions while one linear combination is in fact an s-type function (x2+y2+z2). Only the five true d-combinations are actually used as degrees of freedom in the basis set, but lists of primitive basis functions (bas) for instance run over all Cartesian functions including the improper ones.

A function set in ADF is characterized by the quantum numbers l and n, and by the exponential decay factor a. A set thus represents (l +1)(l +2)/2 Cartesian functions and (2 l+1) degrees of freedom.

The atomic frozen core orbitals are described as expansions in Slater-type functions; these are not the functions of the normal basis set but another set of functions, defined on the data files you use in Create mode.

Orthogonality of the valence space to the frozen core states is enforced as follows: for each frozen core shell (characterized by the quantum numbers l and n: all orbitals with m = - l … + l are identical apart from rotation in space) the set of valence basis functions is augmented with a so-called core orthogonalization function set. You may conceptually interpret the core orthogonalization functions as single zeta expansions of the true frozen core states. Each of the normal valence basis functions is now transformed into a linear combination of that valence function with all core orthogonalization functions, where the coefficients are uniquely defined by the requirement that the resulting function is orthogonal to all true core functions.

So the list of all Cartesian basis functions is much larger than the degree of freedom of the basis: it contains the spurious non-l combinations and it contains also the core orthogonalization functions.

Evaluation of the charge density and molecular orbitals¶

adf.rkf contains all the information you need to evaluate the charge density or a Molecular Orbital (MO) in any point in space. Most of the information is located in section Basis:

A list of function characteristics (kx,ky,kz,kr,alf), including the core orthogonalization functions. This list does not run over all bas functions used in the molecule: if a particular function is used on the atoms of a given atomtype, the function occurs only once in the list, but in the molecule it occurs as many times as there are atoms of that type.

With array nbptr you locate the subsections in the function list that correspond to the different types of atoms: for atom type i the functions nbptr(i)…nbptr(i+1)-1. The distinct atom types are listed in an early section of the standard output file.

Array nqptr gives the number of atoms for type i: nqptr(i+1)-nqptr(i). With this information you construct the complete list of all functions. Repeat the subsection of type i as many times as there are atoms of that type: the complete list can be considered to be constructed as a double loop, the outer being over the atom types, the inner over the atoms that belong to that type. The total ‘overall’ list of functions you obtain in this way contains naos functions. Note that in this way we have implicitly also defined a list of all atoms, where all atoms that belong to a particular atom type are contiguous. This list is the so-called ‘internal’ atom ordering, which may not be identical to the order in which atoms were specified in input, under atoms.

For a given symmetry representation (Sections S) the array npart gives the indices of the basis functions in the overall list that are used to describe orbitals in this representation. In case of an unrestricted run the array npart applies for either spin: the same basis functions are used; the expansion coefficients for the molecular orbitals are different of course.

In the symmetry-representation sections Eigen_bas gives the expansion coefficients that describe the MOs. The expansion refer to the functions indicated by npart, and the function characteristics are given by the arrays kx,ky,kz,kr,alf, and bnorm, i.e. the expansion is in* normalized*functions.

The value of an MO is now obtained as a summation of values of primitive basis functions. For the evaluation of any such basis function you have to take into account that its characteristics are defined in the local coordinate system of its atom.

To obtain the charge density you sum all MOs, squared and multiplied by the respective occupation numbers (array froc in the appropriate irrep section).

Note that the auxiliary program densf, which is provided with the ADF package, generates orbital and density values on a user-specified grid.