There are some situations in which a system is not well-described by a DFT calculation on a single configuration. For example, transition states are known to be poorly described by most functionals, with the computed barrier being too low. We can, in particular, identify homolytic dissociation of diatomic species as situations where static correlation becomes extremely important. Existing DFT functionals have proved to be very effective in capturing dynamic correlation, but frequently exhibit difficulties in the presence of strong static correlation. Configuration Interaction, well known in wave function methods, is a multi-reference method that is quite well-suited for capturing static correlation; the CDFT-CI technique allows for CI calculations on top of DFT calculations, harnessing both static and dynamic correlation methods.
Constrained DFT is used to compute densities (and Kohn-Sham wave functions) for two or more diabatic-like states; these states are then used to build a CI matrix. Diagonalizing this matrix yields energies for the ground and excited states within the configuration space. The coefficients of the initial diabatic states are printed, to show the characteristics of the resultant states.
Since DFT only gives converged densities, not actual wave functions, computing the off-diagonal coupling elements is not completely straightforward, as the physical meaning of the Kohn-Sham wave function is not entirely clear. We can, however, perform the following manipulation:1073
where the converged states are assumed to be the ground state of with eigenvalue ). This manipulation eliminates the two-electron integrals from the expression, and experience has shown that the use of Slater determinants of Kohn-Sham orbitals is a reasonable approximation for the quantities and .
Note that since these constrained states are eigenfunctions of different Hamiltonians (due to different constraining potentials), they are not orthogonal states, and we must set up our CI matrix as a generalized eigenvalue problem. Symmetric orthogonalization is used by default, though the overlap matrix and Hamiltonian in non-orthogonal basis are also printed at higher print levels so that other orthogonalization schemes can be used after-the-fact. In a limited number of cases, it is possible to find an orthogonal basis for the CDFT-CI Hamiltonian, where a physical interpretation can be assigned to the orthogonal states. In such cases, the matrix representation of the Becke weight operator is diagonalized, and the (orthogonal) eigenstates can be characterized.1068 This matrix is printed as the “CDFT-CI Population Matrix” at increased print levels.
In order to perform a CDFT-CI calculation, the interacting states must be defined, which is accomplished using a $cdft input section in a fashion similar to the specification of CDFT states:
$cdft STATE_1_CONSTRAINT_VALUE_X COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X ... STATE_1_CONSTRAINT_VALUE_Y COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y ... --- STATE_2_CONSTRAINT_VALUE_X COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X ... STATE_2_CONSTRAINT_VALUE_Y COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y ... $end
Each state is specified with the CONSTRAINT_VALUE and the corresponding weights on sets of atoms whose average value should be the constraint value. Different states are separated by a single line containing three or more dash characters.
If it is desired to use an unconstrained state as one of the interacting configurations, charge and spin constraints of zero may be applied to the atom range from 0 to 0.
Note: It is mandatory to specify a spin constraint corresponding to every charge constraint (and it must be immediately following that charge constraint in the input deck), for reasons described below.
In addition to the $cdft input section of the input file, a CDFT-CI
calculation must also set the CDFTCI flag to TRUE for the calculation
to run. Note, however, that the CDFT flag is used internally by
CDFT-CI, and should not be set in the input deck. The variable
CDFTCI_PRINT may also be set manually to control the level of output.
The default is 0, which will print the energies and weights (in the diabatic
basis) of the CDFT-CI states. Setting it to 1 or above will also print the
CDFT-CI overlap matrix, the CDFT-CI Hamiltonian matrix before the change of
basis, and the CDFT-CI Population matrix. Setting it to 2 or above will also
print the eigenvectors and eigenvalues of the CDFT-CI Population matrix.
Setting it to 3 will produce more output that is only useful during application
For convenience, if CDFTCI_PRINT is not set in the input file, it will be set to the value of SCF_PRINT.
As mentioned in the previous section, there is a disparity between our chemical intuition of what charges should be and the actual quantum-mechanical charge. The example was given of LiF, where our intuition gives the lithium atom a formal charge of ; we might similarly imagine performing a CDFT-CI calculation on , with two ionic states and two spin-constrained states. However, this would result in attempting to force both electrons of onto the same nucleus, and this calculation is impossible to converge (since by the nature of the Becke weight operators, there will be some non-zero amount of the density that gets proportioned onto the other atom, at moderate internuclear separations). To remedy problems such as this, we have adopted a mechanism by which to convert the formal charges of our chemical intuition into reasonable quantum-mechanical charge constraints. We use the formalism of “promolecule” densities, wherein the molecule is divided into fragments (based on the partitioning of constraint operators), and a DFT calculation is performed on these fragments, completely isolated from each other.1068 (This step is why both spin and charge constraints are required, so that the correct partitioning of electrons for each fragment may be made.) The resulting promolecule densities, converged for the separate fragments, are then added together, and the value of the various weight operators as applied to this new density, is used as a constraint for the actual CDFT calculations on the interacting states. The promolecule density method compensates for the effect of nearby atoms on the actual density that will be constrained.
The comments about SCF convergence for CDFT calculations also apply to the calculations used for CDFT-CI, with the addition that if the SCF converges but CDFT does not, it may be necessary to use a denser integration grid or reduce the value of CDFT_THRESH.
Analytic gradients are not available. Many of the CDFT-related rem variables are also applicable to CDFT-CI calculations. For details on using CDFT-CI to calculate reaction barrier heights, see Ref. 1069.