Electronic Configuration¶

Spin: restricted vs. unrestricted¶

UNRESTRICTED

Specifies that spin-\(\alpha\) and spin-\(\beta\) MOs may be spatially different and may have different occupation numbers. The default (absence of the key) is spin-restricted. The key has no argument. In the case of Spin-Orbit coupling it means that Kramer’s symmetry does not have to be satisfied, in which case the key UNRESTRICTED should be used in combination with the key NONCOLLINEAR or COLLINEAR.

The unrestricted mode roughly doubles the computational effort. The actual numbers of spin-\(\alpha\) and spin-\(\beta\) electrons respectively are controlled by the keys CHARGE and OCCUPATIONS. Not e carefully, that using only the keyword unrestricted, without either Charge or Occupations (or both) would not result in any spin polarization. This implies that you would effectively perform a spin-restricted calculation, but with increased computational effort. Therefore, the program will check that in an unrestricted calculation at least one of the keys Charge and Occupations is applied.

The unrestricted feature is equivalent with, in ab-initio terminology, (Spin-)Unrestricted-Hartree-Fock (UHF); the N-particle wave function is a single determinant and not necessarily an eigenfunction of the spin operator S² .

A restricted calculation implies that the (spatial) orbitals and the occupation numbers are identical for spin-\(\alpha\) and spin-\(\beta\).

The Fock operator, both in an unrestricted and in a restricted run, commutes with the spin operator Sz, but not (unless accidentally) with S² . The obtained one-determinant wave function may for instance be a mixture of a singlet and a triplet state.

In an unrestricted calculation the expectation value of S² is now computed in ADF (note 29 in [98]). The implementation of an evaluation of S² is not quite trivial. DFT is essentially a one-particle formalism, so the S-operator for the n-particle system has to be written out in single-particle operators [99]. The equations used in ADF to calculate the expectation value of S² can be found in Szabo and Ostlund [100]. Note that the so called exact value (S_exact )² , which is printed in the ADF output, is defined as (S_exact )² = (|N_a - N_b |/2)(|N_a - N_b |/2+1), where N_a and N_b are the number of spin-\(\alpha\) and spin-\(\beta\) electrons, respectively. The expectation value of S² is not calculated in a Spin-Orbit coupled calculation.

Molecules that have been calculated using the unrestricted formalism cannot be employed as fragments. ADF will abort when you attach the TAPE21 result file from an unrestricted calculation as a fragment file.

A fair approximation to a computation with unrestricted fragments can be achieved with the key FRAGOCCUPATIONS. See also the examples Example: Spin-unrestricted Fragments: H2, Example: Bond Energy analysis meta-GGA, (meta-)hybrids: Zn2, Cr2, CrH and Example: Bond Energy analysis open-shell fragments: PCCP.

Unrestricted and Spin-Orbit Coupling¶

In the case of Spin-Orbit coupling there are two ways to do spin-polarized calculations, either using the collinear approximation or the noncollinear approximation [101, 102]. Using the unrestricted feature in order to assign different numbers of electrons to a and b spin, respectively, cannot be applied as such, if one includes Spin-Orbit coupling, since the electrons are not directly associated with spin-\(\alpha\) and spin-\(\beta\). For the collinear and noncollinear approximation one should use Symmetry NOSYM (see Symmetry Key), and each level can allocate 1 electron. Note that with the key CHARGE one should only specify one value, namely the total charge. One should not specify the spin-polarization.

COLLINEAR

This key is only relevant in the case of Spin-Orbit coupling. The key has no argument. See also the key NONCOLLINEAR.

In the collinear approximation in each point in space the spin-polarization has the same direction (default is in the direction of the z-axis). Kramer’s symmetry does not have to be satisfied. Symmetry used in the calculation should be NOSYM. The default direction of the spin-polarization can be overruled using the key SOUX (this key has no argument) for spin-polarization only in the direction of the x-axis, and the key SOUY (this key has no argument) for spin-polarization only in the direction of the y-axis. Both keys SOUX and SOUY are only relevant in the case of Spin-Orbit coupling in combination with the key COLLINEAR.

NONCOLLINEAR

This key is only relevant in the case of Spin-Orbit coupling. The key has no argument. See also the key COLLINEAR.

In the noncollinear approximation in each point in space the spin-polarization can have a different direction. Kramer’s symmetry does not have to be satisfied. Symmetry used in the calculation should be NOSYM.

Net Charge and Spin polarization¶

The net charge of the molecule and the net spin polarization can be controlled with the key CHARGE:

CHARGE {NetQ {ab}}

NetQ

The net total charge of the molecule

ab

The net total spin polarization: the number of spin-\(\alpha\) electrons in excess of spin-\(\beta\) electrons. Specification is only meaningful in a spin-unrestricted calculation. However, specification is not meaningful in an unrestricted Spin-Orbit coupled calculation using the (non-)collinear approximation.

If the key is used, the first value in the argument is assigned to netQ, the net total charge, and the second to ab. If the key is not used at all, default values apply. The default for the net total charge is the sum of fragment charges: not necessarily neutral!! The fragment charges are the net total charges that were used in the fragment runs; this information is stored in the fragment files.

The default spin polarization is zero.

An unrestricted calculation with ab=0 (for instance by not specifying the spin polarization at all) is, in the case one does spin break the spin symmetry, in fact a restricted run: it should give exactly the same as the restricted calculation, but it will use more CPU time. If one does break the spin symmetry, for example with the key MODIFYSTARTPOTENTIAL or the SPINFLIP option in the key RESTART, the solution may also be a broken spin symmetry solution. For example one may want to start a calculation in broken symmetry with spin-\(\alpha\) density on one fragment and spin-\(\beta\) density on another, e.g. in a spin-unrestricted calculation of H₂ at large separation.

Aufbau, smearing, freezing¶

OCCUPATIONS Options

Options

May contain IntegerAufbau, Keeporbitals, Smearq, Freeze, or Steep:

IntegerAufbau

Electrons are assigned to MOs according to the Aufbau principle, trying to use integer occupations, also in case of degeneracy at the Fermi level. ADF normally may use fractional occupation numbers in that case. Note that for multi-dimensional irreps, using the subkey IntegerAufbau may stil not prevent fractional occupation numers to be used.

Keeporbitals=NKeep

Until SCF cycle Nkeep electrons are assigned to MOs according to the Aufbau principle, using at each cycle the then current orbital energies of the MOs. Thereafter the KeepOrbitals feature is applied. As soon as this is activated the program will on successive SCF cycles assign electrons to the MOs that maximally resemble - in spatial form - those that were occupied in a ‘reference cycle number’. The default for Nkeep is 20, except:

1. When orbital occupations for MOs are specified explicitly in the data block of the occupations key, these apply throughout.
2. In a Create run fixed occupations are derived from a database in the program.
3. When electron smearing is explicitly turned on by the user (see the Smearq option below) Nkeep is by default 1,000,000 so the program will ‘never’ compare the spatial forms of MOs to determine the occupation numbers. The ‘reference cycle number’ is by default the previous cycle, which will suppress jumps in the spatial occupations during the SCF development while at the other hand allowing the system to let the more-or-less-frozen configuration relax to self-consistency.

Freeze

Occurrence of this word in the option list specifies that the ‘reference cycle number’ will be the cycle number on which the KeepOrbitals feature is activated: during all subsequent SCF cycles the program will assign electrons to MOs that resemble the MOs of that specific SCF cycle. This may be used when the MOs of that cycle are already reasonably close to the final ones, and it will suppress unwanted step-by-step charge-transfers from occupied to empty orbitals that are very close in energy. By default this option is not active.

Smearq=Smear1[,Smear2,Smear3,...,Smear10]

Smear N is half the energy width (in Hartree) over which electrons are smeared out over orbitals that lie around the fermi level and that are close in energy. Smearing is a trick that may help when the SCF has problems converging. One should be well aware that the physical meaning of a result obtained with smeared occupations is unclear (to express it mildly). It may be useful to get over a hurdle in a geometry optimization. By default the initial smear parameter is zero (i.e.: smearing is not applied). It is turned on automatically by the program when SCF convergence is found to be problematic, but only in an optimization-type application (simple optimization, linear transit, transition state) when the geometry is not yet converged. You can rigorously prohibit any smearing by specifying it explicitly with value zero. More generally: specifying the smear parameter makes the program to apply it always, but always with the input-specified value. When a comma-delimited list of values is specified, after SCF has converged, the next value from the list is picked and the SCF is continued. This way one can specify a list of gradually decreasing values to get sort of annealing effect. NOTE: No spaces are allowed when specifying a list of values for Smearq.

Steep=Lambda[,Nmax]

The occupation number for each orbitals are updated according to steepest-descent method (Ref: F. W. Averill and G. S. Painter, Phys. Rev. B 46, 2498 (1992)). During an SCF cycle, the occupation number for each new orbital is initially determined by decomposing the old charge density with new orbitals. Then, the occupation numbers are modified so that the total energy of the system will decrease. The Lambda parameter gives the coefficient for the charge transfer in 1/au unit. The second parameter, Nmax, is an additional limit for the amount of the charge transfer. Nmax would be useful for early steps of cycle when the Lambda parameter gives too large charge transfer. Too small Nmax results in irregular behavior in SCF convergence. In the case of difficult SCF convergence, you should make mixing and Lambda smaller. From our experience, Nmax=0.1 or 0.2 is usually OK. This method should be used with turning off DIIS method (DIIS N=0), and the choice of the mixing parameter in SCF cycle is also important. This option is especially useful for systems with many quasi-degenerate orbitals around Fermi level. For instance, cluster models of surface systems usually suffer from dangling bonds and should be converged with this method. Note though that slow convergence is an intrinsic feature of this method so one should specify a large limit for the number of SCF cycles, say 500 or even 1000, depending on the cluster size.

Notes about the occupations options:

• When occupation numbers are explicitly defined via the block form of the OCCUPATIONS keyword (see next section), the Smearq option cannot be used.
• The aufbau principle does not determine or adjust the distribution of electrons over spin-\(\alpha\) versus spin-\(\beta\) in an unrestricted calculation. This aspect is controlled by the key CHARGE and by any explicit occupations in the data block of occupations.
• When occupation numbers are not specified and no Smearing is specified either, the program will turn on smearing automatically when the SCF has serious convergence problems, in an attempt to overcome those problems, but only in a geometry optimization (including transition state, linear transit, etc.). If such happens the program restores the original situation (no smearing) at the start of each new SCF. In automatic smearing the smear parameter is initiated at 0.01 Hartree and may be varied (by the program) between 0.001 and 0.1 Hartree. The automatic use of smearing by the program can be prohibited by explicitly setting the smear option with value zero (Smearq=0).
• Smearing cannot be used in combination with the keeporbitals option. This option therefore also turns of automatic smearing in troublesome SCF ‘s during an optimization.

Explicit occupation numbers¶

OCCUPATIONS
  irrep orbitalnumbers
  irrep orbitalnumbers
  ...
End

irrep

The name of one of the irreducible representations (not a subspecies) of the point group of the system. See the Symmetry for the irrep names as they are used in ADF.

orbitalnumbers

A series of one or more numbers: the occupation numbers for one-electron valence orbitals in that irrep. The orbitals are ordered according to their energy eigenvalue; higher states than those listed get an occupation number zero.

For degenerate representations such as the 2-dimensional E-representations or the 3-dimensional T-representations, you must give the total occupation, i.e. the sum over the partner representations; ADF assigns each partner an occupation equal to the appropriate fraction of what appears here.

In an unrestricted calculation, two sequences of numbers must be specified for each irrep; the sequences are separated by a double slash (//). The first set of numbers is assigned to the spin-\(\alpha\) orbitals, the second set to the spin-\(\beta\) orbitals. Example unrestricted calculation in symmetry NOSYM with two unpaired electrons:

OCCUPATIONS
   A 28 // 26
End
CHARGE 0 2
SYMMETRY NOSYM

Note that this is not meaningful in an unrestricted Spin-Orbit coupled calculation using the (non-)collinear approximation, where one should use one sequence of occupation numbers for each irrep.

Notes about the occupations data block:

• When specifying electron configurations, all valence electrons in the calculation must be explicitly assigned to MOs and the block form of the OCCUPATIONS keyword must be used. In this context the concept valence electrons and hence valence orbitals is not necessarily identical to what you may normally assume to be the valence space of an atom or molecule. The meaning of valence is here strictly defined as whatever electrons are outside the frozen core. It depends therefore on the level of frozen core approximation applied in the calculation. This traces back to the Create runs in which the basic atoms were generated that are now used to build the molecule.

• When for some irrep there is a rather long list of occupation numbers, corresponding to * consecutive fully occupied* states, you can combine these numbers and enter their sum instead: ADF knows the maximum occupation for an irrep, and when you put a larger number the program will split it up. For instance, if you give for the p-representation (in a single atom calculation):

  P 17 3
  

  ADF will interpret this as:

  P 6 6 5 3
  

  i.e. the occupation number 17 is interpreted as denoting two fully occupied p-shells and the remaining five electrons in the next higher shell. This example also illustrates how to specify an excited state: here we have defined a hole in the third p-shell.

• Fractional occupation numbers in input are allowed. For a discussion of the interpretation of fractional occupation numbers see [103]. The program even allows you (technically) to use a non-integer total number of electrons, whatever the physical meaning of such a calculation is.

• The data block of occupations is not parsed (see the section Interpretation of Input). The program does not replace expressions by their value and it does not recognize constants or functions defined with the define key.

• In a numerical frequencies run (without the Symm argument) the symmetry used internally in the program is NOSYM, irrespective of any Schönfliess symbol in the input file. As a consequence the program will recognize only the A representation (the only irrep in nosym), but not the representations belonging to the input point group symmetry. (The symmetry in the equilibrium geometry, defined by the input Schönfliess symbol, is used to enhance efficiency and stability in the construction of the matrix of Force constants).

CHARGE vs. OCCUPATIONS¶

The contents of the data block of occupations, if used, defines the total number of valence electrons and hence the net total charge. In an unrestricted run it also defines the net spin polarization. If the key CHARGE is also used, the program will check that both specifications are consistent.

We strongly recommend to employ this and always specify the net total charge and spin polarization with charge whenever explicit occupation numbers are supplied with occupations, to that the program will check that your occupation numbers result in the total charge and spin polarization that you have in mind.

Create mode¶

In Create mode occupation numbers are predefined (see Appendix Elements of the Periodic Table), and these are applied unless you specify occupations in input yourself. Conceivably this may result in a non-aufbau configuration. In Create mode the program always operates as if occupations were set in input.

Spin-flip method for broken symmetries¶

Starting from ADF2009.01 it is possible to exchange alpha and beta electrons for selected atoms when performing a restart from a previous spin-unrestricted calculation.

In many cases, one wishes to perform a calculation of a low-spin complex where spin-density is positive on some atoms and negative on the others. It is usually very difficult to achieve SCF convergence if one starts from scratch. Sometimes, the MODIFYSTARTPOTENTIAL feature (see next section) helps with this problem but sometimes it does not. A more robust way is to first perform a high-spin calculation and then modify the resulting t21 file by “flipping” the spin on some atoms. This file then can be used to restart a subsequent low-spin calculation.

Such a “flipping” can now be performed during restart by specifying a SPINFLIP keyword in the RESTART input block as shown below:

RESTART high-spin.t21 &
   ! SpinFlip keyword is followed by atom numbers for
   ! which the flipping will be performed
   SPINFLIP 1
END

An example demonstrating the feature may be found in the examples Broken spin-symmetry: Fe4S4.

Modify the starting potential¶

In some applications you may want to modify the initial fit coefficients (from the restart file or the fragment files), see also the previous section. This is achieved with the key MODIFYSTARTPOTENTIAL. It allows you to scale them in some way so as to represent user-chosen amounts of spin-\(\alpha\) and spin-\(\beta\) fit density on some or all of the fragments. This will adjust the spin-\(\alpha\) and spin-\(\beta\) initial potentials.

This option applies only to unrestricted calculations of course. It may be used to help the program find a particular state. This might, for instance, be hard to find otherwise due to the a-b symmetry in the start-up situation. It may also be useful to speed up the SCF convergence in case you know what the final distribution of spin-\(\alpha\) and spin-\(\beta\) density over the molecule will approximately be.

MODIFYSTARTPOTENTIAL {specification}
{ frag alfa // beta
 frag alfa // beta
 ...
end }

ModifyStartPotential

A general key: it has an argument or a data block.

specification

Must be two numbers, ASPIN and BSPIN, if provided at all. They specify the (relative) amounts of spin-\(\alpha\) and spin-\(\beta\) fit density to define the spin-dependent potential at the first SCF cycle. The coefficients retrieved from the fragment files (or from the restart file in case of a SCF restart) are scaled accordingly. This will not affect the total amount of fit density: the absolute values of ASPIN and BSPIN play no role, only their ratio. In case of a restart run the restart file must have been generated in a restricted calculation, while the continuation run must be an unrestricted one.

If no argument is given a data block must be supplied with records frag alfa // beta. This is very much similar to the main option with ASPIN and BSPIN: you specify ASPIN and BSPIN now separately for each fragment. This involves somewhat more input but increases the possibilities to tune the initial potential. Again this can be applied only in an unrestricted calculation. It cannot be used in a restart: the affected fit coefficients are those from the fragment files, while in an SCF restart run these are ignored and replaced by the coefficients on the TAPE21 restart file.

Each line specifies a frag with its corresponding ASPIN and BSPIN fit partitioning. If frag is the name of a fragment type, the specified ASPIN-BSPIN is applied to all individual fragments of that type. Alternatively an individual fragment can be specified, using the format fragtype/n, where n is an index between one and the total number of fragments of that type. In such a case the ASPIN-BSPIN data applies only to that particular fragment while different values may be supplied for the other fragments of the same type.

It is allowed to specify for certain fragment types individual fragments and for other fragment types only the type. Duplicate specifications are not allowed; an individual fragment must not be specified if its fragment type is also specified as a whole. If the data block form is used, only the fit coefficients of the referenced fragments are affected. For the not-referenced fragments the fit densities are used as they are defined on the corresponding fragment files.

The SCF convergence of a spin-unrestricted calculation usually improves when you start with potentials that correspond to the correct ratio of spin-\(\alpha\) and spin-\(\beta\) electrons. By default ASPIN=BSPIN=0.5, as implied by the spin-restricted start density of the fragments or restricted molecule.

The total amount of fit density used on the first iteration is defined by the sum-of-fragment densities (or the density on the restart file). This may be different from the total nr. of electrons in the actual calculation. On the second SCF cycle the fit density will internally be normalized so as to represent the correct number of electrons.

The block-form of the key makes the start up of broken symmetry calculations easy. For example one may want to start a calculation in broken symmetry with spin-\(\alpha\) density on one fragment and spin-\(\beta\) density on another, e.g. in a spin-unrestricted calculation of H₂ at large separation. It is particularly useful for larger systems, e.g. for magnetic coupling between spin-polarized magnetic centers, as in Fe-S complexes [111]: start with oppositely polarized Fe centers, but with, for instance, the remaining bridge and terminal ligands unpolarized. See also the N2+ sample run in the examples.