5.13 Methods Based on “Constrained” DFT

Under certain circumstances it is desirable to apply constraints to the electron density during an SCF calculation. For example, in a transition metal complex it may be desirable to constrain the net spin density on a particular metal atom to integrate to a value consistent with the $M_{S}$ value expected from ligand field theory. Similarly, in a donor/acceptor complex one might be interested in constraining the total density on the acceptor group so that the formal charge on the acceptor is either neutral or negatively charged, depending as the molecule is in its neutral or its charge-transfer state. In these situations, one is interested in controlling the average value of some observable, $O(\mathbf{r})$, to take on a given value, $N$:

 $\int\rho(\mathbf{r})\,O(\mathbf{r})\;d\mathbf{r}=N$ (5.60)

There are of course many states that satisfy such a constraint, but in practice one is usually looking for the lowest energy such state. To solve the resulting constrained minimization problem, one introduces a Lagrange multiplier, $V$, and solves for the stationary point of

 $V[\rho,V]=E[\rho]-V\left(\int\rho(\mathbf{r})\,O(\mathbf{r})\;d\mathbf{r}-N\right)$ (5.61)

where $E[\rho]$ is the energy of the system described using density functional theory (DFT). At convergence, the functional $W$ gives the density, $\rho$, that satisfies the constraint exactly (i.e., it has exactly the prescribed number of electrons on the acceptor or spins on the metal center) but has the lowest energy possible. The resulting self-consistent procedure can be efficiently solved by ensuring at every SCF step the constraint is satisfied exactly. The Q-Chem implementation of these equations closely parallels those in Ref. 1070.

The first step in any constrained DFT calculation is the specification of the constraint operator, $O(\mathbf{r})$. Within Q-Chem, the user is free to specify any constraint operator that consists of a linear combination of the Becke’s atomic partitioning functions:52

 $O(\mathbf{r})=\sum_{A}^{\rm atoms}\sum_{\sigma=\alpha,\beta}C^{\sigma}_{A}\,w_% {A}(\mathbf{r})$ (5.62)

Here the summation runs over the atoms in the system and over electron spins. The weight function $w_{A}$ is designed to be $\approx 1$ near the nucleus of atom $A$ and rapidly fall to zero near the nucleus of any other atom in the system.52 As discussed in Section 10.2.1, the default Becke partitioning scheme in Q-Chem leads to highly unphysical results when using the atom-centered quadrature to count electrons. It is highly recommended to set BECKE_SHIFT to one of the two atomic cell shifting schemes in order to obtain physically meaningful results, particularly if the constrained atoms are in close contact with the unconstrained space.

The user-specified coefficients $C^{\sigma}_{A}$ are input using a $cdft input section having the following format. $cdft
CONSTRAINT_VALUE_X
COEFFICIENT1_X       FIRST_ATOM1_X       LAST_ATOM1_X    TYPE1_X
COEFFICIENT2_X       FIRST_ATOM2_X       LAST_ATOM2_X    TYPE2_X
...
CONSTRAINT_VALUE_Y
COEFFICIENT1_Y       FIRST_ATOM1_Y       LAST_ATOM1_Y    TYPE1_Y
COEFFICIENT2_Y       FIRST_ATOM2_Y       LAST_ATOM2_Y    TYPE2_Y
...
$end  Here, each CONSTRAINT_VALUE is a real number that specifies the desired average value ($N$) of the ensuing linear combination of atomic partition functions. Each COEFFICIENT specifies the coefficient of a partition function or group of partition functions in the constraint operator $O$. For each coefficient, all the atoms between the integers FIRST_ATOM and LAST_ATOM contribute with the specified weight in the constraint operator. Finally, TYPE specifies the type of constraint being applied—either “CHARGE” or “SPIN”. For a CHARGE constraint the spin up and spin down densities contribute equally ($C_{A}^{\alpha}=C_{A}^{\beta}=C_{A}$) yielding the total number of electrons on the atom A. For a SPIN constraint, the spin up and spin down densities contribute with opposite sign ($C_{A}^{\alpha}-C_{A}^{\beta}=C_{A}$) resulting in a measure of the net spin on the atom A. Each separate CONSTRAINT_VALUE creates a new operator whose average is to be constrained—for instance, the example above includes several independent constraints: X, Y, $\ldots$. Q-Chem can handle an arbitrary number of constraints and will minimize the energy subject to all of these constraints simultaneously. If an atom is not included in a particular operator, then the coefficient of that atoms partition function is set to zero for that operator. The TYPE specification is optional, and the default is to perform a charge constraint. Further, note that any charge constraint is on the net atomic charge. That is, the constraint is on the difference between the average number of electrons on the atom and the nuclear charge. Thus, to constrain CO to be negative, the constraint value would be 1 and not 15. Note: Charge constraint in$cdft specifies the number of excess electrons on a fragment, not the total charge, i.e., the value 1.0 means charge $=-1$, whereas charge constraint of $-$1.0 corresponds to the total $+$1 charge.

The choice of which atoms to include in different constraint regions is left entirely to the user and in practice must be based somewhat on chemical intuition. Thus, for example, in an electron transfer reaction the user must specify which atoms are in the “donor” and which are in the “acceptor”. In practice, the most stable choice is typically to make the constrained region as large as physically possible. Thus, for the example of electron transfer again, it is best to assign every atom in the molecule to one or the other group (donor or acceptor), recognizing that it makes no sense to assign any atoms to both groups. On the other end of the spectrum, constraining the formal charge on a single atom is highly discouraged. The problem is that while our chemical intuition tells us that the lithium atom in LiF should have a formal charge of +1, in practice the quantum mechanical charge is much closer to +0.5 than +1. Only when the fragments are far enough apart do our intuitive pictures of formal charge actually become quantitative.

Note that the atomic populations that Q-Chem prints out are Mulliken populations, not the Becke weight populations. As a result, the printed populations will not generally add up to the specified constrained values, even though the constraint is exactly satisfied. You can print Becke populations to confirm that the computed states have the desired charge/spin character.

Finally, we note that SCF convergence is typically more challenging in constrained DFT calculations as compared to their unconstrained counterparts. This effect arises because applying the constraint typically leads to a broken symmetry, diradical-like state. As SCF convergence for these cases is known to be difficult even for unconstrained states, it is perhaps not surprising that there are additional convergence difficulties in this case. See Section 4.5 on SCF convergence algorithms for ideas on how to improve convergence for constrained calculations. Also, CDFT is more sensitive to grid size than ground-state DFT, so sometimes increasing the integration grid to SG-2 or SG-3 (see Section 5.5.2), instead of the default SG-1 grid, improves the convergence.

Note:  1. To improve convergence, use the fewest possible constraints. For example, if your system consists of two fragments, specify the constrains for one of them only. The overall charge and multiplicity will force the “unconstrained” fragment to attain the right charge and multiplicity. 2. The direct minimization methods are not available for constrained calculations. Hence, some combination of DIIS and RCA must be used to obtain convergence. Further, it is often necessary to break symmetry in the initial guess (using SCF_GUESS_MIX) to ensure that the lowest energy solution is obtained.

Analytic gradients are available for constrained DFT calculations.1071 Second derivatives are only available by finite difference of analytic gradients. For details on how to apply constrained DFT to compute magnetic exchange couplings, see Ref. 833. For details on using constrained DFT to compute electron transfer parameters, see Ref. 1072, 1073.