Electronic Configuration¶

Spin: restricted vs. unrestricted¶

Unrestricted Yes/No

Unrestricted

Type

Bool

Default value

No

Description

By default, a spin-restricted calculation is performed where the spin alpha and spin beta orbitals are spatially the same.

SpinPolarization float

SpinPolarization

Type

Float

Description

The spin polarization of the system, which is 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.

In an unrestricted calculation spin-\(\alpha\) and spin-\(\beta\) MOs may be spatially different and may have different occupation numbers. 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 subkey SpinOrbitMagnetization NONCOLLINEAR or SpinOrbitMagnetization COLLINEAR of the key Relativity.

The unrestricted mode roughly doubles the computational effort. The actual numbers of spin-\(\alpha\) and spin-\(\beta\) electrons respectively are controlled by the keys SpinPolarization and OCCUPATIONS. Note carefully, that using only the keyword unrestricted, without either SpinPolarization or Occupations (or both) will 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 SpinPolarization or Occupations is applied.

The unrestricted feature is equivalent, in ab-initio terminology, to (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-\(\beta\).

An unrestricted calculation with SpinPolarization 0.0 (for instance by not specifying the spin polarization at all) is, in case one does not 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.

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 computed in ADF (note 29 in ref. 1). 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 2. The equations used in ADF to calculate the expectation value of S² can be found in Szabo and Ostlund 3. 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.

If one includes the UNRESTRICTEDFRAGMENTS key, molecules that have been calculated using the spin-unrestricted formalism can be employed as fragments. In this case, the calculation on the full complex also needs to use the spin-unrestricted formalism. See also the examples Example: unrestricted fragments: CH3I and Example: unrestricted EDA: Cu(C2H4)2.

A fair approximation to a computation with spin-unrestricted fragments can be achieved with the key FRAGOCCUPATIONS, which uses spin restricted fragments, but allows for spin-unrestricted fragment occupations. 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.

Spin: unrestricted vs restricted open shell¶

In unrestricted calculations the spin-\(\alpha\) and spin-\(\beta\) orbitals, energies and occupations may differ. In restricted open shell calculations the spin-\(\alpha\) and spin-\(\beta\) orbitals and energies should be identical, only the occupations may differ. In ADF2023 a restricted open shell method is implemented valid for high spin open shell molecules, similar as in Ref. 11, in which case the 1-determinantal wave function is an eigenfunction of S_z and S². The implementation of the method is work-in-progress. At the moment it is only valid in case of single point calculations. This method can not be used in case of spin-orbit coupling, TD-DFT, or for many other properties.

The method implemented in ADF needs integer occupation numbers, and a positive spin polarization. It requires the subkey ROSCF of the key SCF, the keyword UNRESTRICTED, and the key SPINPOLARIZATION. Part of the input could like like:

Occupations IntegerAufbau
Unrestricted Yes
SpinPolarization 2
SCF
    ROSCF
    End
End

The ROSCF options are:

SCF
   ROSCF
      Alpha float_list
   End
End

SCF

ROSCF

Type

Block

Description

Settings for the ROSCF method.

Alpha

Type

Float List

Default value

[0.5, 0.5, 0.5]

Description

Coefficients to build the alpha-spin orbital contribution to the diagonal closed-, open-, and virtual-shell blocks of the Fock matrix. The beta-spin orbital contributions are 1.0 minus the alpha ones.

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 4 5. 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. One should not specify the spin-polarization.

Collinear approximation

Unrestricted Yes
Symmetry NOSYM
Relativity
   Level Spin-Orbit
   SpinOrbitMagnetization COLLINEAR
End

The SpinOrbitMagnetization key is only relevant in the case of Spin-Orbit coupling.

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 as argument COLLINEARX for spin-polarization only in the direction of the x-axis, or as argument COLLINEARY for spin-polarization only in the direction of the y-axis.

Noncollinear approximation

Unrestricted Yes
Symmetry NOSYM
Relativity
   Level Spin-Orbit
   SpinOrbitMagnetization NONCOLLINEAR
End

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¶

The net charge of the molecule can be controlled with the key CHARGE in the System block in the AMS part of the input:

System header
   Charge float
End

System

Charge

Type

Float

Default value

0.0

GUI name

Total charge

Description

The system’s total charge in atomic units.

Aufbau, smearing, freezing¶

OCCUPATIONS Options

Options

May contain IntegerAufbau, Keeporbitals, ElectronicTemperature, Smear, 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 still 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 Smear 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.

ElectronicTemperature=T

Invokes orbital population using the Fermi-Dirac distribution with temperature T (in Kelvin). This may be used to achieve SCF convergence in an otherwise problematically converging system. Note that the default SCF acceleration method A-DIIS is not compatible with non-aufbau orbital occupations so it is disabled automatically and the classical Pulay DIIS scheme is used. For this reason it is strongly recommended to use either MESA or one of the methods from the LIST family in combination with ElectronicTemperature or Smear.

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

Another form of specifying the electronic temperature (in Hartree). The smearing parameter SmearN is related to the ElectronicTemperature in such a way that the occupation number for any orbital outside the smearing interval will deviate from an integer value (0 and 1 for spin-unrestricted, 0 and 2 for spin-restricted) by no more than 1% (0.01 electron for spin-unrestricted, 0.02 for spin-restricted). More precisely, a smearing value of 1 Hartree is equivalent to an electronic temperature of 34285 Kelvin. When a comma-delimited list of values is specified, the next value from the list is picked after SCF has converged, 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 the list of values for Smear.

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.

OptimizeSpin=Delta, OptimizeSpinRound=Delta

The spin polarization can be optimized by minimizing the difference between Fermi level values for spin-alpha and spin-beta MOs. This method works reliably only when a reasonably large finite electron temperature is used (300K is found to be sufficiently large for small transition metal clusters). The polarization value is optimized up to a specified tolerance Delta (fractional values are allowed) by step-wise reduction of the root-bracketing interval for the objective function f=FermiLevel(alpha)-FermiLevel(beta). The SCF procedure is converged for each spin polarization. When the interval becomes smaller than Delta the optimization is considered converged. After that the SCF may optionally be repeated with the spin polarization set to the nearest even or odd number (depending on the total number of electrons in the system) or, in other words, with the total number of electrons per spin rounded to the nearest integer. The method is invoked using either the OptimizeSpin or the OptimizeSpinRound Occupation option. As mentioned above, it must be used together with the ElectronicTemperature option. The OptimizeSpin and OptimizeSpinRound argument specifies the tolerance value. The OptimizeSpinRound option specifies that after polarization has converged the SCF must be completed with the total number of electrons per spin rounded to the nearest integer. It should be noted that, as usual, the Unrestricted and SpinPolarization keys must be specified as for any other spin-unrestricted calculation. The argument of the SpinPolarization key sets the initial spin polarization value from which the optimization is started. The result of the optimization may depend on the initial value because the objective function may have more than one root.

Notes about the occupations options:

• When occupation numbers are explicitly defined via the block IRREPOCCUPATIONS (see next section), the Smear 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 SpinPolarization and by any explicit occupations in the data block of occupations.

• Smearing cannot be used in combination with the keeporbitals option.

Explicit occupation numbers¶

IrrepOccupations
  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:

IrrepOccupations
   A 28 // 26
End
SpinPolarization 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 IRREPOCCUPATIONS 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 ref. 6. 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 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 symmetric displacements) 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 and SPINPOLARIZATION vs. IRREPOCCUPATIONS¶

The contents of the data block of IRREPOCCUPATIONS, 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 keys CHARGE and SPINPOLARIZATION are also used, the program will check that the 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 IRREPOCCUPATIONS, 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 the occupations were set in input.

Spin-flip method for broken symmetries¶

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 adf.rkf 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 be performed during restart by specifying the SpinFlip keyword in the Restart input block as shown below:

Restart
   File HighSpin.results/adf.rkf
   ! SpinFlip keyword is followed by the indices of the
   ! atoms for which the flipping will be performed
   SpinFlip 1
End

Modify the starting potential¶

In some applications you may want to modify the initial potential (from the restart file or the fragment files), see also the previous section. This is achieved with the block ModifyStartPotential. It allows you to scale the potential as to represent user-chosen amounts of spin-\(\alpha\) and spin-\(\beta\) 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

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 adf.rkf (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 7: start with oppositely polarized Fe centers, but with, for instance, the remaining bridge and terminal ligands unpolarized.

Prepared for bonding¶

Typically an unrestricted electron configuration for the fragments is used, such that the Pauli repulsion between the fragments is minimal, which means that the fragments are so called ‘prepared for bonding’. For example if one has two fragments which both have one unpaired electron, one would put the unpaired electron of the first fragment in the spin-\(\alpha\) orbital and the unpaired electron of the second fragment in the spin-\(\beta\) orbital. If one wants to calculate separately the electron pair bonding see key REMOVEALLFRAGVIRTUALS.

Prepared for bonding¶

Typically an unrestricted electron configuration for the fragments is used, such that the Pauli repulsion between the fragments is minimal, which means that the fragments are so called ‘prepared for bonding’. For example if one has two fragments which both have one unpaired electron, one would put the unpaired electron of the first fragment in the spin-\(\alpha\) orbital and the unpaired electron of the second fragment in the spin-\(\beta\) orbital. If one wants to calculate separately the electron pair bonding see key REMOVEALLFRAGVIRTUALS.

FRAGOCCUPATIONS¶

FRAGOCCUPATIONS
 fragtype
   irrep spin-a // spin-b
   irrep spin-a // spin-b
   ...
 subend
 fragtype
   irrep spin-a // spin-b
   ...
 subend
end

fragtype

One of the fragment types and functions as a (block type) subkey. The data block for the subkey ends with the standard end code for block type subkeys (subend).

irrep

One of the irreducible representations (irreps) for the point group symmetry that was used in the computation of that fragment.

spin-a // spin-b

Two sequences of occupation numbers, which will be applied to the spin-\(\alpha\) and spin-\(\beta\) versions of the Fragment Orbitals. The sequences must be separated by a double slash (//).

The sum of spin-\(\alpha\) and spin-\(\beta\) occupations must, for each fragment orbital in each irrep separately, be equal to the total (restricted) occupation of that orbital as it is stored on the fragment file. In other words: you can only change the distribution over spin-\(\alpha\) and spin-\(\beta\) electrons within one orbital.

(Without this restriction the spatial distribution of the total (sum over spins) fragment charge density would be changed, leading to an incorrect bonding energy analysis after the calculation).

Be aware that in more-dimensional irreps (E, T) the number of electrons in a fully occupied orbital is input as the dimension of the irrep times the one-electron orbital occupation. Compare the key IRREPOCCUPATIONS.

For irreps that are not mentioned in this input block, and hence for all irreps of fragment(type)s that are not mentioned at all, the spin-\(\alpha\) and spin-\(\beta\) occupations will be set equal, which is of course what they in fact are on the (restricted) fragment file.

For an example of applying this option see ref. 8.