Example: Spin-unrestricted Fragments: H2¶
This is a small but important example to illustrate what goes into an accurate calculation of the ‘true’ bond energy of a molecule. The (ADF-specific) problem is that in a straightforward molecular calculation, the bond energy is computed as the energy difference between at the one hand the molecule, and at the other hand the isolated spherically symmetric spin-restricted atoms. The italic-typed features imply that the reference (comparison) state is usually not the physical ground state of the reference system (isolated atoms) and hence the computed energy difference has no direct relation to experimental data. To account for the true atomic ground states, one has to add correction terms. Study this sample carefully to make sure that you fully understand the steps to take and consult the User’s Guide for details. See also the this document for a discussion of multiplet states.
See also the example, SD_Cr(NH3 )6 .
The H2 case consists of a sequence of simple calculations to demonstrate the Unrestricted Fragments option. The energy difference between an unrestricted fragment as it is used in adf and a self-consistent unrestricted fragment is also computed. This turns out to be quite small, confirming that the adf approach, although not formally exact, is adequate for practical purposes.
$ADFBIN/adf <<eor
create H file=$ADFRESOURCES/DZP/H
end input
eor
mv TAPE21 t21H
$ADFBIN/adf <<eor
title H unrestr., not self-consistent (as used in unr.frag. calcs)
scf
iterations 0 ! prohibit relaxation
end
unrestricted
charge 0 1 ! if not specified up and down electrons
! will both get 0.5 electron: in fact restricted
fragments
H t21H
end
atoms
H 0 0 0
end
endinput
eor
rm TAPE21 logfile
By setting the scf iterations to zero (a value of one (1) would give the same result) we prevent cycling to self-consistency. The energy of the ‘final’ one-electron orbitals is consequently computed in the start-up potential, i.e. the field of the restricted (basic) atom, where spin-\(\alpha\) and spin-\(\beta\) are equally occupied, in this case by 0.5 electron each. The not-self-consistent, unrestricted H atom is precisely the ‘unrestricted’ fragment as it can be used in an adf calculation with unrestricted fragments. The fragment file must be the TAPE21 result file from a restricted run, but at start-up you can specify that the Fragment Orbitals are, for purposes of reference and comparison, occupied in an unrestricted way in the final molecule.
A calculation that uses restricted fragments right away computes the bonding energy relative to the restricted fragments. The difference between using restricted and unrestricted fragments is the ‘bonding’ energy computed in the run above.
$ADFBIN/adf <<eor
title H unr. self-consistent from unr.0
unrestricted
charge 0 1
fragoccupations
H
s 1 // 0
subend
end
Atoms
H 0 0 0
end
fragments
H t21H
end
end input
eor
rm TAPE21 logfile
Here we start with the unrestricted fragment and relax to self-consistency. The ‘bonding energy’, i.e. the relaxation energy, is very small, demonstrating that using non-self-consistent unrestricted fragments involves only a small error (which, moreover, can be computed as shown here).
The key UnRestricted sets the spin-unrestricted mode. The key Charge is used to specify a net total charge of zero and a net total spin polarization by an excess of 1.0 spin-\(\alpha\) electrons over spin-\(\beta\).
$ADFBIN/adf <<eor
title H2 restricted, from restricted fragments
ATOMS
H 0 0 0.375
H 0 0 -0.375
end
fragments
H t21H
end
end input
eor
rm TAPE21 logfile
This is the simplest approach, using restricted fragments. The bonding energy must be corrected because the reference (restricted H atoms, with 0.5 electrons in spin-\(\alpha\) and 0.5 in spin-\(\beta\)) is far from the true H-atom ground state: see the previous runs on the single H atom.
$ADFBIN/adf <<eor
title H2 from unrestricted fragments
ATOMS
H.1 0 0 0.375
H.2 0 0 -0.375
end
fragments ! two different fragment types are necessary
! because the two atoms get different FragOccupations
! (see below), while the key FragOc.. addresses
! only fragmentTYPES
H.1 t21H
H.2 t21H
end
charge 0
occupations
sigma 2 ! specify the state (not always
! necessary)
end
fragoccupations
H.1
s 1 // 0
subend
H.2
s 0 // 1
subend
end
modifystartpotential
H.1 1 // 0 ! this helps SCF start-up
H.2 0 // 1 ! but is here not necessary
end
end input
eor
rm TAPE21 logfile
This should be a fair approximation (in the lda model) to the bonding energy of H2 with respect to the unrestricted H atoms. The difference between the bonding energies of this and the previous run should be very close to the energy of the not-self-consistent unrestricted H-atom with respect to the restricted basic atom (calculation #2).
Excited state
$ADFBIN/adf <<eor
title H2 excited
ATOMS
H 0 .0 0.375
H 0 .0 -0.375
end
fragments
H t21H
end
fragoccupations
H
s 1 // 0
subend
end
unrestricted
charge 0 2
occupations
sigma.g 1 // 0
sigma.u 1 // 0
end
end input
eor
Finally the calculation of an excited state, with respect to unrestricted fragments. The excitation energy is obtained by comparing the energy with the energy of the ground state calculation. This difference compares reasonably, but not accurately, to the difference in one-electron ground state energies of the involved orbitals (Koopman’s theorem).
Note that excitation energies can also be calculated with Time-Dependent DFT, using the RESPONSE module of ADF. See related sample runs.