MOLCAS manual:
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Subsections
The POLY_ANISO program is a routine which allows a semiab initio
description of the (lowlying) electronic structure and magnetic properties
of polynuclear compounds. It is based on the localized nature of the magnetic
orbitals (i.e. the d or f orbitals containing unpaired electrons [75,76]).
For many compounds of interest, the localized character of magnetic orbitals leads
to very weak character of the interactions between magnetic centers. Due to this weakness of the
interaction, the metals' orbitals and corresponding localized ground and excited states
may be optimized in the absence of the magnetic interaction at all. For this purpose, various fragmentation
models may be applied. The most commonly used fragmentation model is exemplified in Scheme 1.
Figure:
Fragmentation model of a polynuclear compound. The upper scheme shows a schematic overview of a tetranuclear compund and the resulting four mononuclear fragments obtained by diamagnetic atom substitution method. By this scheme, the neighboring magnetic centers, containing unpaired electrons are computationally replaced by their diamagnetic requivalents. As example, transition metal sites TM(II) are best replaced by either diamagnetic Zn(II) or Sc(III), in function which one is the closest. For lanthanides Ln(III) the same principle is applicable, La(III) or Lu(III) are best suited to replace a given magnetic lanthanide. Individual mononuclear metal framgents are then investigated by common CASSCF/CASPT2/RASSI/SINGLE_ANISO computational method. A single file for each magnetic site, produced by the SINGLE_ANISO run, is needed by the POLY_ANISO code as input.

Magnetic interaction between metal sites is very important for accurate description of lowlying states and their properties.
It can be considered as a sum of various interaction mechanisms: magnetic exchange, dipoledipole interaction, antisymmetric exchange, etc.
In the POLY_ANISO code we have implemented several mechanisms.
The description of the magnetic exchange interaction is done within the Lines model [77].
This model is exact in three cases:
 a) interaction between two isotropic spins (Heisenberg),
 b) interaction between one Ising spin (only S_{z} component) and one isotropic (i.e. usual) spin, and
 c) interaction between two Ising spins.
In all other cases of interaction between magnetic sites with intermediate anisotropy, the Lines model represents an
approximation. However, it was succesfully applied for a wide variety of polynuclear compounds so far.
In addition to the magnetic exchange, magnetic dipoledipole interaction can be accounted exactly, by
using the information about each metal site already computed ab initio. In the case of
strongly anisotropic lanthanide compounds, the dipoledipole interaction is usualy the dominant
one. Dipolar magnetic coupling is one kind of longrange interaction between magnetic moments.
For example, a system containing two magnetic dipoles and , separated by
distance have a total energy:

(8.11) 
where are the magnetic moments of sites 1 and 2, respectively; r is the distance between
the two magnetic dipoles, is the directional vector connecting the two magnetic dipoles (of unit length).
is the square of the Bohr magneton; with an approximative value of 0.43297 in cm^{1}/Tesla.
As inferred from the above Equation, the dipolar magnetic interaction depends on the distance and on the angle between the magnetic moments on magnetic
centers. Therefore, the cartesian coordinates of all nonequivalent magnetic centers must be provided in the input (see the keyword COOR).
The program Poly_Aniso needs the following files:
File  Contents

aniso_XX.input  This is an ASCII text file generated by the MOLCAS/SINGLE_ANISO program.
It should be provided for POLY_ANISO aniso_i.input (i=1, 2, 3 etc.): one file for each magnetic center.
In cases when the entire polynuclear cluster or molecule has exact point group symmetry, only
aniso_i.input files for crystallographically nonequivalent centers should be given.

chitexp.input  – set directly in the standard input (key TEXP)

magnexp.input  – set directly in the standard input (key HEXP)

File  Contents

zeeman_energy_xxx.txt  A series of files named zeeman_energy_xxx.txt is produced in the $WorkDir only in case keyword ZEEM is
employed (see below). Each file is an ASCII text formated and contains Zeeman spectra of the investigated
compound for each value of the applied magnetic field.

chit_compare.txt  A text file contining the experimental and calculated magnetic susceptibility data.

magn_compare.txt  A text file contining the experimental and calculated powder magnetisation data.

Files chit_compare.txt and chit_compare.txt may be used in connection with a simple GNUPLOT script
in order to plot the comparison between experimental and calculated data.
8.38.1 Input
This section describes the keywords used to control the standard input file.
Only two keywords NNEQ, PAIR (and SYMM if the polynuclear cluster has symmetry) are
mandatory for a minimal execution of the program, while the other keywords allow
customization of the execution of the POLY_ANISO.
Keywords defining the polynuclear cluster
Keyword  Meaning

NNEQ 
This keyword defines several important parameters of the calculation. On the
first line after the keyword the program reads 2 values:
1) the number of types of different magnetic centers (NONEQ) of the cluster and
2) a letter T or F in the second position of the same line.
The number of NONEQ is the total number of magnetic centers of the cluster
which cannot be related by point group symmetry.
In the second position the answer to the question: Have all NONEQ centers been computed ab initio?
is given: T for True and F for False.
On the following line the program will read NONEQ values specifying the
number of equivalent centers of each type.
On the following line the program will read NONEQ integer numbers specifying
the number of lowlying spinorbit functions from each center forming the local
exchange basis.
Some examples valid for situations where all sites have been
computed ab initio (case T, True):
NNEQ 
NNEQ 
NNEQ 
2 T 
3 T 
6 T 
1 2 
2 1 1 
1 1 1 1 1 1 
2 2 
4 2 3 
2 4 3 5 2 2 
gray!15There are two kinds of magnetic centers in the cluster; both have been computed ab initio;
the cluster consists of 3 magnetic centers: one center of the first kind and two centers
of the second kind. From each center we take into the exchange coupling only the ground
doublet. As a result the Nexch=
2^{1} x 2^{2}=8 aniso_1.input (for – type 1)
and aniso_2.input (for – type 2) files must be present. 
gray!15There are three kinds of magnetic centers in the cluster; all three have been computed ab initio;
the cluster consists of four magnetic centers: two centers of the first kind, one center of the
second kind and one center of the third kind. From each of the centers of the first kind we take
into exchange coupling four spinorbit states, two states from the second kind and three states
from the third center. As a result the Nexch=
4^{2} x 2^{1} x 3^{1}=96.
Three files aniso_i.input for each center (i=1,2,3) must be present. 
gray!15There are 6 kinds of magnetic centers in the cluster; all six have been computed ab initio;
the cluster consists of 6 magnetic centers: one center of each kind. From the center of the
first kind we take into exchange coupling two spinorbit states, four states from the second center,
three states from the third center, five states from the fourth center and two states from the
fifth and sixth centers. As a result the Nexch=
2^{1} x 4^{1} x 3^{1} x 5^{1} x 2^{1} x 2^{1}=480.
Six files aniso_i.input for each center (i=1,2,...,6) must be present. 
Only in cases when some centers have NOT been computed ab initio (i.e. for which no aniso_i.input file exists),
the program will read an additional line consisting of NONEQ letters (A or B) specifying the type of each of
the NONEQ centers:
A – the center is computed ab initio and B – the center is considered isotropic.
On the following numberofBcenters line(s) the isotropic g factors of the
center(s) defined as B are read. The spin of the B center(s) is defined: S=(N1)/2,
where N is the corresponding number of states to be taken into the exchange coupling
for this particular center.
Some examples valid for mixed situations: the system consists of centers computed ab initio and
isotropic centers (case F, False):
NNEQ 
NNEQ 
NNEQ 
2 F 
3 F 
6 F 
1 2 
2 1 1 
1 1 1 1 1 1 
2 2 
4 2 3 
2 4 3 5 2 2 
A B 
A B B 
B B A A B A 
2.3 
2.1 
2.12 

2.0 
2.43 


2.00 
gray!15There are two kinds of magnetic centers in the cluster;
the center of the first type has been computed ab initio, while the
centers of the second type are considered isotropic with g=2.3; the cluster
consists of three magnetic centers: one center of the first kind and two centers
of the second kind. Only the ground doublet state from each center is considered
for the exchange coupling. As a result the Nexch=
2^{1} x 2^{2}=8.
File aniso_1.input (for – type 1) must be present. 
gray!15There are three kinds of magnetic centers in the cluster;
the first center type has been computed ab initio, while the
centers of the second and third types are considered isotropic with g=2.1 (second type)
and g=2.0 (third type); the cluster consists of four magnetic centers: two centers
of the first kind, one center of the second kind and one center of the third kind.
From each of the centers of the first kind, four spinorbit states are considered
for the exchange coupling, two states from the second kind and three states from the
center of the third kind.
As a result the Nexch=
4^{2} x 2^{1} x 3^{1}=96.
The file aniso_1.input must be present. 
gray!15There are six kinds of magnetic centers in the cluster; only
three centers have been computed ab initio, while the other three centers
are considered isotropic; the g factor of the first center is 2.12 (
);
of the second center 2.43 (); of the fifth center 2.00 (); the entire
cluster consists of six magnetic centers: one center of each kind. From the center
of the first kind, two spinorbit states are considered in the exchange coupling,
four states from the second center, three states from the third center, five
states from the fourth center and two states from the fifth and sixth centers.
As a result the Nexch=
2^{1} x 4^{1} x 3^{1} x 5^{1} x 2^{1} x 2^{1}=480.
Three files aniso_3.input and aniso_4.input and aniso_6.input
must be present. 
There is no maximal value for NNEQ, although the calculation becomes quite heavy in case the number of
exchange functions is large.

SYMM  Specifies rotation matrices to symmetry equivalent sites. This keyword is mandatory in the case more centers of a given type are present in the calculation.
This keyword is mandatory when the calculated polynuclear compound has exact crystallographic point group symmetry. In other words, when the number of
equivalent centers of any kind i is larger than 1, this keyword must be employed. Here the rotation matrices from the one
center to all the other of the same type are declared.
On the following line the program will read the number 1 followed on the next lines by as many 3x3 rotation matrices as the total number of
equivalent centers of type 1. Then the rotation matrices of centers of type 2, 3 and so on, follow in the same format.
When the rotation matrices contain irrational numbers (e.g. (
), then more digits than presented in the examples
below are advised to be given:
.
Examples:
NNEQ 
NNEQ 
NNEQ 
2 F 
3 F 
6 F 
1 2 
2 1 1 
1 1 1 1 1 1 
2 2 
4 2 3 
2 4 3 5 2 2 
A B 
A B B 
B B A A B A 
2.3 
2.1 
2.12 

2.0 
2.43 

2.0 
2.00 
SYMM 
SYMM 

1 
1 

1.0 0.0 0.0 
1.0 0.0 0.0 

0.0 1.0 0.0 
0.0 1.0 0.0 

0.0 0.0 1.0 
0.0 0.0 1.0 

2 
0.0 1.0 0.0 

1.0 0.0 0.0 
1.0 0.0 0.0 

0.0 1.0 0.0 
0.0 0.0 1.0 

0.0 0.0 1.0 
2 

1.0 0.0 0.0 
1.0 0.0 0.0 

0.0 1.0 0.0 
0.0 1.0 0.0 

0.0 0.0 1.0 
0.0 0.0 1.0 


3 


1.0 0.0 0.0 


0.0 1.0 0.0 


0.0 0.0 1.0 

gray!15The cluster computed here is a trinuclear compound, with one center
computed ab initio, while the other two centers, related to each other by inversion,
are considered isotropic with
g_{x}=g_{y}=g_{z}=2.3.
The rotation matrix for the first center is I (identity, unity) since the center is
unique. For the centers of type 2, there are two matrices 3x3 since we have two centers
in the cluster. The rotation matrix of the first center of type 2 is Identity while the
rotation matrix for the equivalent center of type 2 is the inversion matrix. 
gray!15In this input a tetranuclear compound is defined, all centers are
computed ab initio. There are two centers of type “1”, related one to each other by C_{2}
symmetry around the Cartesian Z axis. Therefore the SYMM keyword is mandatory.
There are two matrices for centers of type 1, and one matrix (identity) for the
centers of type 2 and type 3.

gray!15In this case the computed system has no symmetry. Therefore,
the SYMM keyword may be skipped 
More examples are given in the Tutorial section.

Keywords defining the magnetic exchange interactions
This section defines the keywords used to set up the interacting pairs of magnetic centers
and the corresponding exchange interactions.
A few words about the numbering of the magnetic centers of the
cluster in the POLY_ANISO. First all equivalent centers of the type 1 are
numbered, then all equivalent centers of the type 2, etc. These labels of the magnetic
centers are used further for the declaration of the magnetic coupling.
The pseudocode is:
k=0
Do i=1, numberofnonequivalentsites
Do j=1, numberofequivalentsitesoftype(i)
k=k+1
sitenumber(i,j)=k
End Do
End Do
Keyword  Meaning

PAIR or LIN1 
Specifies the Lines interaction(s) between metal pairs. One parameter per interactiing pair is required.
LIN9
READ numberofinteractingpairs
Do i=1, numberofinteractingpairs
READ site1, site2, J
End Do

ALIN or LIN3 
Specifies the anisotropic interactions between metal pairs. Three parameters per interacting pair are required.
LIN9
READ numberofinteractingpairs
Do i=1, numberofinteractingpairs
READ site1, site2, Jxx, Jyy, Jzz
End Do
, where and are main values of the Cartesian components of the (3x3) matrix defining the exchange interaction between site1 and site2.

LIN9 
Specifies the full anisotropic interaction matrices between metal pairs. Nine parameters per interacting pair is required.
LIN9
READ numberofinteractingpairs
Do i=1, numberofinteractingpairs
READ site1, site2, Jxx, Jxy, Jxz, Jyx, Jyy, Jyz, Jzx, Jzy, Jzz
End Do
, where and are Cartesian components if the (3x3) matrix defining the exchange interaction between site1 and site2.

COOR  Specifies the symmetrized coordinates of the metal sites. This keyword enables computation of dipoledipole
magnetic interaction between metal sites defined in the keywords PAIR, ALIN, LIN1, LIN3 or LIN9.
COOR
Do i=1, numberofnonequivalentsites
READ coordinates of center 1
READ coordinates of center 2
...
End Do

Other keywords
Normally POLY_ANISO runs without specifying any of the following keywords.
Argument(s) to a keyword are always supplied on the next line of the input file.
Keyword  Meaning

MLTP  The number of molecular multiplets (i.e. groups of spinorbital eigenstates) for which
g, D and higher magnetic tensors will be calculated (default MLTP=1).
The program reads two lines: the first is the number of multiplets (NMULT) and the
second the array of NMULT numbers specifying the dimension (multiplicity) of each multiplet.
Example:
MLTP
10
2 4 4 2 2 2 2 2 2 2
POLY_ANISO will compute the g and D tensors for 10 groups of states.
The groups 1 and 410 are doublets (S̃=1/2>), while the groups 2 and 3 are quadruplets,
having the effective spin S̃=3/2>. For the latter cases, the ZFS (D) tensors will be computed.

TINT 
Specifies the temperature points for the evaluation of the magnetic susceptibility. The program will read four numbers: T_{min}, T_{max}, nT, and .
 T_{min} – the minimal temperature (Default 0.0K)
 T_{max} – the maximal temperature (Default 300.0K)
 nT – number of temperature points (Default 101)
Example:
TINT
0.0 330.0 331
POLY_ANISO will compute temperature dependence of the magnetic susceptibility in 331 points evenly distributed in temperature interval: 0.0K – 330.0K.

HINT  Specifies the field points for the evaluation of the magnetization in a certain direction. The program will read four numbers: H_{min}, H_{max} and nH
 H_{min} – the minimal field (Default 0.0T)
 H_{max} – the maximal filed (Default 10.0T)
 nH – number of field points (Default 101)
Example:
HINT
0.0 20.0 201
POLY_ANISO will compute the molar magnetization in 201 points evenly distributed in field interval: 0.0T – 20.0T.

TMAG  Specifies the temperature(s) at which the fielddependent magnetization is calculated. Default is one temperature point, T=2.0 K.
Example:
TMAG
6 1.8 2.0 2.4 2.8 3.2 4.5

ENCU  This flag is used to define the cutoff energy for the lowest states for which Zeeman interaction is taken into account exactly. The contribution to the magnetization coming from states that are higher in energy than E (see below) is done by second order perturbation theory. The program will read two integer numbers: NK and MG. Default values are: NK=100,MG=100.
The fielddependent magnetization is calculated at the (highest) temperature value defined in either TMAG or HEXP.
Example:
ENCU
250 150
If H_{max} = 10T and TMAG=1.8K, then the cutoff energy is:
This means that the magnetization coming from all spinorbit states with energy lower than
E=1013.06258 (cm^{1}) will be computed exactly.
ERAT, NCUT and ENCU are mutually exclusive.

ERAT  This flag is used to define the cutoff energy for the lowest states for which Zeeman interaction
is taken into account exactly. The contribution to the molar magnetization coming from states that
are higher in energy than E (see below) is done by second order perturbation theory.
The program reads one real number in the domain (0.01.0). Default is 1.0 ( all exchange states are
included in the Zeeman interaction).
The fielddependent magnetization is calculated at all temperature points defined in either TMAG or HEXT.
Example:
ERAT
0.75
ERAT, NCUT and ENCU are mutually exclusive.

NCUT  This flag is used to define the number of lowlying exchange states for which Zeeman interaction is taken into
account exactly. The contribution to the magnetization coming from the remaining exchange states is done by second
order perturbation theory. The program will read one integer number. The fielddependent magnetization is calculated at all temperature points defined in either TMAG or HEXT.
Example:
NCUT
125
In case the defined number is larger than the total number of exchange states in the calculation (Nexch), then nCut is set to be equal to Nexch.
ERAT, NCUT and ENCU are mutually exclusive.

MVEC  Defines the number of directions for which the magnetization vector will be computed.
On the first line below the keyword, the number of directions should be mentioned (NDIR. Default 0).
The program will read NDIR lines for cartesian coordinates specifying the direction i of the
applied magnetic field ( and ). These values may be arbitrary real numbers.
The direction(s) of applied magnetic field are obtained by normalizing the length of each vector to one.
Example:
MVEC
4
0.0000 0.0000 0.1000
1.5707 0.0000 2.5000
1.5707 1.5707 1.0000
0.4257 0.4187 0.0000
The above input requests computation of the magnetization vector in four directions of applied field.
The actual directions on the unit sphere are:
4
0.00000 0.00000 1.00000
0.53199 0.00000 0.84675
0.53199 0.53199 0.33870
0.17475 0.17188 0.00000

MAVE  This keyword specifies the grid density used for the computation of powder molar
magnetization. The program uses LebedevLaikov distribution of points on the unit sphere.[]
The program reads two integer numbers: nsym and ngrid. The nsym defines which
part of the sphere is used for averaging. It takes one of the three values: 1 (halfsphere),
2 (a quater of a sphere) or 3 (an octant of the sphere). ngrid takes values from 1
(the smallest grid) till 32 (the largest grid, i.e. the densiest). The default is to
consider integration over a halfsphere (since M(H)=M(H)): nsym=1 and ngrid=15
(i.e 185 points distributed over halfsphere). In case of symmetric compounds, powder
magnetization may be averaged over a smaller part of the sphere, reducing thus the number
of points for the integration. The user is responsible to choose the appropriate integration scheme.
Default value for gridnumber=15 (185 directions equally distributed in the given area).
Note that the programś default is rather conservative.

TEXP  This keyword allows computation of the magnetic susceptibility at experimental points.
On the line below the keyword, the number of experimental points NT is defined, and on the next NT lines the program reads the experimental temperature (in K) and the experimental magnetic susceptibility (in cm^{3}Kmol^{1} ). TEXP and TINT keywords are mutually exclusive. The magnetic susceptibility routine will also print the standard deviation from the experiment.
TEXP
READ numberofTpoints
Do i=1, numberofTpoints
READ ( susceptibility(i, Temp), TEMP = 1, numberofTpoints )
End Do

HEXP  This keyword allows computation of the molar magnetization M_{mol} (H) at experimental points.
On the line below the keyword,the number of experimental points NH is defined, and on the next NH lines the program reads the experimental field intensity (Tesla) and the experimental magnetization (in ). HEXP and HINT are mutually exclusive. The magnetization routine will print the standard deviation from the experiment.
HEXP
READ numberofTpointsforM, allTpointsforMinK
READ numberoffieldpoints
Do i=1, numberoffieldpoints
READ ( Magn(i, iT), iT=1, numberofTpointsforM )
End Do

ZJPR  This keyword specifies the value (in cm^{1}) of a phenomenological parameter of a mean molecular field acting on the spin of the complex (the average intermolecular exchange constant). It is used in the calculation of all magnetic properties (not for spin Hamiltonians) (Default is 0.0)

PRLV  This keyword controls the print level.
 2 – normal. (Default)
 3 or larger (debug)

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