W Information about atomic wave-functions

A W card has the form W , label , CCSL-word, data.

The label is either

an atom-label appearing on an A card

or

a scattering-factor-label appearing on an F card.

Which of the two it is depends on the CCSL-word which follows.

The CCSL-word may be one of AMP FUNC PROD RADF ROTN.

DATA FOLLOWING ALLOWED CCSL WORDS:

AMP
Data

A product-name (see PROD below) followed by two real numbers giving the complex amplitude of the given many-electron configuration labelled product-name .

FUNC
Data

The description of a molecular orbital wave function for atom label . The data are:

A name (up to 4 characters, with which to label the function).

Two integers, $l$ and $m$, for the spherical harmonic term in the function.

Either two ($m=0$) or four ($m>0$) real numbers, being the complex amplitudes of $Y_l^m$ and $Y_l^{-m}$ repectively.

PROD
Data

A product-name of up to 4 characters followed by two real numbers giving a complex amplitude, then $n$ coded labels defining the one-electron functions in one Slater determinant for an $n$-electron or $n$-hole orbital. At present this can only be used for d-electrons($l$=2). Each label consists of a signed integer $m$ in the range $-l\le m\le l$ indicating the $z$ component of the angular momentum of the electron on the quantum axes followed without an intervening space by either $+$ or $-$ to indicate whether the spin state is parallel or antiparallel to quantum $z$.

There may be several W PROD cards with the same product-name where they are the different one-electron products making up a single configuration of the $n$ electron wave-function.

Note

The routines which read these cards are experimental and are not yet in the main Library.

RADF
Data

In this case the label is a scattering-factor-label.

Then on each W RADF card there are two integers ITYP and NVAL, followed by the coefficients $A$ and $\xi$ in the expansion of the radial wave function.

ITYP indicates the type of wave function being used:

ITYP=1

is not normalised (e.g. E Clementi & C Roetti, Atomic Data & Nuclear Data Tables, 14 183)

ITYP=2

is normalised (e.g. R E Watson, Report Solid-State Molecular Theory Group M.I.T., No. 12 )

NVAL is the power n in the Slater expansion of the form:

$$U(r)=\sum_ir^{n_i}A_ie^{-\xi_i r}$$

Notes

The units of length for $A$ and $\xi$ should be atomic units.
As many W RADF cards are needed as there are terms in the expansion.

ROTN
Data

The elements of a matrix relating the quantum axes for atom with the given atom-label to the CCSL orthogonal axes. The data are an axis label (X Y or Z ) followed by the direction cosines of the corresponding quantum axis with respect to the crystallographic axes. All three cards are needed for each relevant atom. If no W ROTN cards are given for an atom that needs them the unit matrix is assumed by default.

EXAMPLES:

W Mn1 FUNC x2y2 2 2 .70711 0.0 0.70711 0.0
W Mn RADF 1 2 .27716 3.65559
W Mn RADF 1 2 .03372 10.72370
W Mn RADF 1 2 .29360 5.53874
W Mn RADF 1 2 .45627 2.37383
W Mn RADF 1 2 .08058 1.48214
W Mn1 ROTN X 0.0000 0.0000 1.0000
W Mn1 ROTN Y .7071 -0.7071 0.0000
W Mn1 ROTN Z .7071 0.7071 0.0000
These cards define a one electron wave-function to be applied to the atom whose label is Mn1 and whose form-factor label is Mn. The angular part of the wave-function is labelled x2y2 and defined as: ${1\over\sqrt2}(Y_2^2+Y_2^{-2})$ with the $x$, $y$ and $z$ axes of the spherical harmonic functions having the direction cosines given by the ROTN X ,Y , and Z cards with respect to the CCSL orthogonal axes.
The radial wave-function to be used is defined by the RADF cards as a sum of 5 terms of the unnormalised (Clementi Roetti) type.

W Fe PROD PS12 0.8150 0.0000 2+   1+  -1+  -2+
W Fe PROD PS12 0.4101 0.0000 2+   1+   0+  -1+
W Fe PROD PS12 0.4101 0.0000 1+   0+  -1+  -2+
W Fe PROD PS32 0.7071 0.0000 2+   1+   0+  -1+
W Fe PROD PS32 -0.7071 0.0000 1+   0+  -1+  -2+
W Fe AMP PS12 0.9577
W Fe AMP PS32 0.2729
In this second example the cards describe a 4 electron wave-function for Fe. It is made up of two terms named PS12 and PS32. W Fe ROTN and W Fe RADF cards would also be needed to define the function completely.

ROUTINES WHICH READ THE CARDS:

The input of W cards is directed by various setting up routines such as WAVSET, MSETUP (not in main Library) and PFSET or by main programs such as FORFAC. These call INPUTW to read the W cards and subsequently special routines to interpret the data on each type of card: MOLORB for FUNC, RADFUN for RADF and READRT for ROTN.

NOTE:

W ROTN and W RADF cards are also used to define orientations and form-factors used in multipole calculations.