4.2.20. GENANO¶

GENANO is a program for determining the contraction coefficients for generally contracted basis sets [72]. They are determined by diagonalizing a density matrix, using the eigenvectors (natural orbitals) as the contraction coefficients, resulting in basis sets of the ANO (Atomic Natural Orbitals) type [73].

Some elementary theory: We can do a spectral resolution of a density matrix $$D$$

(4.2.20.1)$D=\sum_k \eta_k c_k c_k^{\text{T}}$

where $$\eta_k$$ is the $$k$$th eigenvalue (occupation value) and $$c_k$$ is the $$k$$th eigenvector (natural orbital). The occupation number for a natural orbital is a measure of how much this orbital contributes to the total one-electron density. A natural choice is to disregard the natural orbitals with small occupation numbers and use those with large occupation numbers to form contracted basis functions as

$\varphi_k=\sum_i c_{ki} \chi_i$

where $$\chi_i$$ is the $$i$$th primitive basis function.

As a generalization to this approach we can average over density matrices from several wave functions, resulting in basis sets of the density matrix averaged ANO type, see for example [74][75][76][77]. We can view the averaging of density matrices as a sequence of rank-1 updates in the same way as in equation (4.2.20.1). We have more update vectors than the rank of the matrix, but this does not really change anything. The important observation is that all $$\eta$$s are positive and no information is lost in the averaging.

The general guideline for which wave functions to include is based on what you want to be able to describe. All wave functions you want an accurate description of should be included in the averaging.

As an example, let us consider the oxygen atom. We want to be able to describe the atom by itself accurately, thus a wave function for the atom is needed, usually at the CI level. In molecular systems, oxygen usually has a negative charge, thus including $$\ce{O-}$$ is almost mandatory. A basis set derived from these two wave function is well balanced for the majority of systems containing oxygen. A logical conclusion would be that you need to include a few molecular wave functions of systems containing oxygen, but in practice this is not necessary. This is due to the fact that the degrees of freedom describing the orbital shape distortion when forming bonds are virtually identical to the lowest correlating orbitals. On the other hand, a few molecular species have oxygen with positive charge, thus it may be appropriate to include $$\ce{O+}$$ in the basis set.

A wide range of specialized basis sets can also be generated, for example a molecular basis set describing Rydberg orbitals, see the example in the “Tutorials and Examples” part, Section 5.1.5.1.2. There is a possibility to create Rydberg orbitals automatically by using the keyword RYDBERG. Here all unoccupied orbitals with negative orbital energies will be used with the associated occupation numbers

$\eta_k = e^{6.9(\epsilon_k/\epsilon_0-1)}$

where $$\epsilon_k$$ is the orbital energy of orbital $$k$$ and $$\epsilon_0$$ is the lowest orbital energy of all virtual orbitals. In order to use this option you need to use the SCF or RASSCF program to compute the orbitals for a cationic system.

You need one or more wave functions, represented by formatted orbital files, to generate the average density matrix. These natural orbital files can be produced by any of the wave function generators SCF, RASSCF, MRCI or CPF. You could also use MBPT2 or CASPT2. This approach has been used in the generation of the ANO-RCC basis sets. Your specific requirements dictate the choice of wave function generator, but MRCI would be most commonly used.

You are not restricted to atomic calculations but can mix molecular and atomic calculations freely. The restrictions are that the name of the center, for which you are constructing a basis set, must be the same in all wave functions. The center may not be “degenerate”, i.e. it may not generate other centers through symmetry operations. See the description of SEWARD on Section 4.2.48 for a more extensive discussion. For example for $$\ce{O2}$$ you cannot use $$D_{2h}$$ symmetry since this would involve one center that is mirrored into the other. Another restriction is, of course, that you must use the same primitive set in all calculations.

4.2.20.1. Dependencies¶

GENANO needs one or more wave functions in the form of natural orbitals. Thus you need to run one or more of SCF, RASSCF, MRCI or CPF. You could also use, for example, MBPT2 or CASPT2 but this is in general not recommended. GENANO also needs the one electron file ONEINT and the RUNFILE generated by SEWARD.

4.2.20.2. Files¶

Below is a list of the files that GENANO reads/writes. Files ONEnnn, RUNnnn and NATnnn must be supplied to the program. Files ANO and FIG are generated. File PROJ is an optional input file.

4.2.20.2.1. Input files¶

RUNnnn

This file contains miscellaneous information for the nnn’th wave function, generated by the program SEWARD. One file per wave function must be supplied, RUN001, RUN002, ….

ONEnnn

This file contains the one-electron integrals corresponding to the nnn’th wave function, generated by the program SEWARD. One file per wave function must be supplied, ONE001, ONE002, ….

NATnnn

This file contains the natural orbitals corresponding to the nnn’th wave function, generated by the appropriate wave function generating program. One file per wave function must be supplied, NAT001, NAT002, ….

PROJ

This file contains orbitals used for projection of the densities. Needs to be available if the keyword PROJECT is specified. It is compatible in format with the file ANO, and can thus be the the file ANO from a previous run of GENANO.

4.2.20.2.2. Output files¶

FIG

This file contains a PostScript figure file of eigenvalues.

ANO

This file contains the contraction coefficient matrix organized such that each column correspond to one contracted basis function.

4.2.20.3. Input¶

The input file must contain the line

&GENANO


right before the actual input starts. Below is a list of the available keywords. Please note that you can not abbreviate any keyword.

TITLE

This keyword starts the reading of title lines, with no limit on the number of title lines. Reading the input as title lines is stopped as soon an the input parser detects one of the other keywords. This keyword is optional.

SETS

This keyword indicates that the next line of input contains the number of sets to be used in the averaging procedure. This keyword must precede WEIGHTS if both are supplied. This keyword is optional, with one set as the default.

CENTER

This keyword is followed, on the next line, by the atom label for which the basis set is to be generated. The label must match the label you supplied to SEWARD. In previous versions of GENANO this label had to be in uppercase, but this restriction is now lifted and the case does not matter. This keyword is compulsory.

ROWWISE

This keyword makes GENANO produce the contraction coefficients row-wise instead of column-wise as is the default. This keyword is optional.

WEIGHTS

This keyword must be subsequent to keyword SETS if both are supplied. This keyword is optional, with equal weight on each of the sets as default.

PROJECT

This keyword states that you want to project out certain degrees of freedom from the density matrix. This can be useful for generating, for example, node less valence orbitals to be used with ECP’s. If this keyword is specified, you must supply the file PROJ obtained as file ANO from a previous GENANO calculation, for instance. This keyword is optional.

LIFTDEGENERACY

This keyword will modify the occupation numbers read from the orbitals files. The purpose is to lift the degeneracy of core orbitals to avoid rotations. The occupation numbers are changed according to $$\eta'=\eta(1+10^{-3}/n)$$ where $$n$$ is the sequence number of the orbital in its irreducible representation. This keyword is optional.

RYDBERG

This keyword enables automatic generation of Rydberg orbitals. With this keyword all occupied orbitals will get occupation number zero while the virtual orbitals will get a small occupation number decreasing with orbital number. Useful with a calculation on an cation where the virtual orbitals are near perfect Rydberg orbitals. Note that you must use orbitals from the SCF or RASSCF program. This keyword is optional.

NOTHRESHOLD

This keyword is used to specify the threshold for keeping NO’s (natural orbitals). Orbitals with occupation numbers less than the threshold are discarded. The threshold is read from the line following the keyword. Default value is 1.0d-8.

Below is a simple input example, where we construct an ANO basis set for the carbon atom. Two wave functions are used, the SCF wave function and the SDCI wave function for the ground state of the atom.

&SEWARD
Title
Carbon atom
Symmetry
x y z
Expert
Basis set
C..... / inline
6.0 2
10   10
5240.6353 782.20479 178.35083 50.815942 16.823562 6.1757760 2.4180490
.51190000 .15659000 .05480600
1. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 1. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 1. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 1. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 1. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 1. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 1. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 1. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 1. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 1.
6    6
18.841800 4.1592400 1.2067100 .38554000 .12194000 .04267900
1. 0. 0. 0. 0. 0.
0. 1. 0. 0. 0. 0.
0. 0. 1. 0. 0. 0.
0. 0. 0. 1. 0. 0.
0. 0. 0. 0. 1. 0.
0. 0. 0. 0. 0. 1.
3    3
1.2838000 .34400000 .09220000
1. 0. 0.
0. 1. 0.
0. 0. 1.
C  0.000000  0.000000  0.000000
End of basis

&SCF
Occupied =  2 0 0 0 0 0 0 0

&RASSCF
Symmetry =  4
Spin     =  3
nActEl   =  2 0 0
Frozen   =  0 0 0 0 0 0 0 0
Inactive =  2 0 0 0 0 0 0 0
Ras2     =  0 1 1 0 0 0 0 0
LevShft  =  0.00
LumOrb
Thrs     =  0.1d-8 0.1d-4 0.1d-4

&MOTRA
LumOrb
Frozen   =  1 0 0 0 0 0 0 0

&GUGA
Electrons =  4
Spin      =  3
Inactive  =  1 0 0 0 0 0 0 0
Active    =  0 1 1 0 0 0 0 0
CiAll     =  4

&MRCI
SDCI

>>COPY $Project.RunFile RUN001 >>COPY$Project.RunFile RUN002
>>COPY $Project.OneInt ONE001 >>COPY$Project.OneInt  ONE002
>>COPY $Project.RasOrb NAT001 >>COPY$Project.CiOrb   NAT002

&GENANO
Title
Carbon atom
Project
sets
2
Center
C
Weights
0.5 0.5
>>RM ONE001
>>RM ONE002
>>RM NAT001
>>RM NAT002