Methodological Background

The resolution of the KS equations gives access to the electronic density of the adiabatic ground state of the molecular system of interest. In other words, one finds by the SCF procedure a relaxed electronic density that corresponds to a minimum of the DFT energy. It is, however, possible that one desires to investi-gate properties of electronic densities that do not correspond to the electronic ground states. The purpose of constrained DFT is precisely to enable the user to impose a particular atomic charge or a spin density while solving the KS equations. Although artificial, this procedure can have considerable interest for the modeling of (bio-)chemical systems. We will provide various examples of applications in the second part of this section but we can already mention the question of electron transfer reactions within the context of the Marcus Theory [213];

this theory relies on the definition of phenomenological states that cannot be obtained by standard DFT (which provides adiabatic states). cDFT is a means to define such ad hoc diabatic states. As another example we remark that a strength of computational chemistry is to give access to systems or situations that are not amenable to direct experimental measurements. Chemical trends like the elec-tronic effects of substituents on a reactive energy profile can be investigated by various methods including cDFT or other methods by imposing artificial constraints on the chemical system of interest. As an example of the usefulness of other procedures we refer to [214] where cDFT is not used but the nuclear charge of nitrogen atoms is varied to modify its electronegativity and indirectly the energy cost for activating an alkyl C–H bond.

In the next paragraphs we explain how this objective is achieved with cDFT and how an efficient implementation within the context of ADFT can be devised.

Our aim is not to provide here a delineated mathematical derivation of the

Fig. 17 Potential energy and free energy profiles for the nucleotidyl addition reaction of a DNA polymerase.

Reproduced with permission from [211]

formalism, but rather to introduce the key aspects of the method so as to provide the reader with a view of the potential applications of cDFT. For more mathe-matical details, the reader is referred to recent reviews on the method [215,216].

The foundations of constrained DFT are to be found in the pioneering works of Gunnarsson and Lundqvist [217] and of Dederich et al. in the early 1980s [218].

Wesolowski, Muller, and Warshel also developed a related, though algorithmi-cally different, constrained DFT approach in the 1990s [219] with remarkable success for the modeling of condensed phase proton transfers, electron transfer reactions [220], and S_N2 reactions [221], in connection with the Empirical Valence Bond approach [222]. We will not, however, cover this methodology in this chapter and the reader is referred to the references for more details.

Continuing with the principal works on cDFT formalisms we finally mention that of Wu and Van Voohris in 2005 [223]. In 1976 Gunnarsson and Lundqvist showed that in addition to the true ground state it was possible to obtain relaxed electronic densities of the particular molecular symmetries. A few years later Dederich et al. proposed a generalization of this idea that used a Lagrange Multiplier to constrain the DFT energy:

e½r; lc ¼ minrmax_lc E½r þ lc

In this equation E is the nonconstrained DFT energy whose mathematical expression has been given in Sect.2.1,w(r) is a weight function that defines the constraining property, andNCis the set-point supplied by the user. For example to constrainNC electrons to occupy a volume O the weight function would equal 1 inside O and zero everywhere else. Bothw(r) and NCare thus user-defined terms.

We will come back later to the practical definition of w(r) within the LCAO framework. For the moment we focus on the Lagrange multiplierlcthat needs to be determined. To emphasize the role of this term in the formalism it is useful to write down the set of modified KS equations. These are obtained by differentiation of the cDFT energy with respect to the MO coefficients under the orthonorma-lization constraint.

Within the LCGTO formalism employed in deMon2k the elements of the constrained KS matrix are given by

Kmn  @eADFT It is apparent from these equations that the product lcW acts like a supple-mentary potential felt by the electrons. Its role is to drive the convergence of the

SCF procedure toward a relaxed density that fulfills the desired constraint. It must be noted, however, that lc is not known beforehand (contrary to W or NC) and must be determined. In the older applications of cDFT, it was customary to scan over possible values oflc to find the correct one. This was obviously a cumber-some strategy that probably restricted the application of cDFT to only a few fields of research. In addition, a critical point when solving the cDFT KS equations is related to the fact that the Coulomb and XC potentials on the one hand and the lcw potential on the other are inter-dependent. This makes the resolution of the constrained KS equations difficult. Wu and Van Voorhis proposed an efficient solution to these difficulties building on Optimized Potential Theory. By examin-ing the stationary conditions of the cDFT energy with respect tolc, these authors proved that, for agiven Coulomb and Exchange Correlation (XC) potential there is a uniquelc that leads to an electronic density fulfilling the desired constraint.

In addition, the correctlccorresponds to a maximum of the energy. The first and second derivatives of the energy with respect tolcare easily obtained from the KS orbital coefficients so that an automatic algorithm can now be implemented for searching the correctlc. In practice this optimization is conducted according to the combination of steepest descents and a Newton–Raphson algorithm for best efficiency [224]. Then to address the difficulty of the coupling between the Coulomb, XC potential, and the constraint, Wu and Van Voorhis proposed a dual-loop strategy whereby the determination of KS orbital energies is decoupled from the determination oflc. The algorithm is illustrated in Fig.18. At every SCF step lc is optimized keeping the Coulomb and XC potential constants (“inner loop” in the figure). Once the correct lc is determined the constrained density matrix is used to calculate the new Coulomb and XC potentials for the next SCF step. Our experience told us that the convergence of the inner loop is usually not problematic in terms of CPU time, especially if a guess forlcfrom the previous SCF iteration can be provided. At global convergence the SCF procedure provides a converged electronic density fulfilling the desired constraint. The most demanding task in ADFT/cDFT remains the numerical calculation of the XC matrix elements, but since these terms remain constant when converging the inner loop the necessity to optimize lc is not lethal for the computational efficiency of the ADFT framework. In other words, cDFT is compatible with ADFT and its implementation with the min–max algorithm [225]. The calculation of the contribution of the constraining term to the energy gradients with respect to the nuclear positions has also been implemented in deMon2k, allowing geometry optimization, frequency analysis, and BOMD with cDFT [226]. However, we identified a computational bottleneck in the calculation that becomes problematic for molecular systems of a few tens of atoms (ca. 70–80 atoms). This is due to the involvement of the orbital density in the constraining term and therefore to the calculation of products of GTOs in the numerical evaluation of the energy derivative terms. We are pursuing our efforts to derive a fully cADFT formalism where the constraint would apply on the auxiliary function density instead of on the orbital density.