La Resistencia de Arica

A hybrid functional includes an amount of the exact exchange obtained in Hartree-Fock theory. The popular B3LYP xc-functional³³⁷ is an example of a semi-empirical hybrid functional containing exact exchange, LDA and GGA exchange (with the latter coming from the B88 functional^331,338), plus LDA and GGA correlation (with the latter coming from the LYP functional). The B3LYP functional is defined in 𝐸𝑞. 2.48. The parameters a ≈ 0.2, b ≈ 0.7 and c ≈ 0.8 are fit to experimental data.

𝐸_𝑥𝑐^{𝐵3𝐿𝑌𝑃} = (1 − 𝑎)𝐸_𝑥^{𝐿𝑆𝐷𝐴}+ 𝑎𝐸_𝑥^{𝑒𝑥𝑎𝑐𝑡}+ 𝑏Δ𝐸_𝑥^𝐵88+ (1 + 𝑐)𝐸_𝑐^{𝐿𝑆𝐷𝐴}+ 𝑐𝐸_𝑐^𝐿𝑌𝑃 (𝐸𝑞. 2.48)

75 The PBE0³³⁴ functional mixes exact exchange with exchange from the PBE functional, at an amount determined by perturbation theory, an example of a non-empirical hybrid functional.

The Random Phase Approximation (RPA) is the next level of improvement.

Information about virtual orbitals is included alongside information about occupied orbitals. Inclusion of unoccupied orbitals improves the treatment of dispersion interactions, a known weakness for most DFT methods.

2.8.5. Time-Dependent Density Functional Theory

Time-Dependent Density Functional Theory (TDDFT) is the study of the electron density associated with molecular or solid-state systems in time-dependent potential fields. Most importantly, TDDFT allows vertical excitation energies and oscillator strengths to be calculated and UV-Vis spectra to be simulated.

The formalism of TDDFT begins with the Runge-Gross theorem which is fully described in several published resources296,339,340, which states that given the starting conditions, the density of a system is related directly to the external time-evolving potential, for example, an electric field. The Kohn-Sham equations provide an approximation to the many-body, time-independent Schrodinger equation for a fictitious system of non-interacting particles, and the Runge-Gross theorem allows a generalisation of the Kohn-Sham equations to an approximation of the many-body time-dependent Schrodinger equation for a system of interacting particles. An external potential 𝑣_KS is defined to ensure that the density of the fictitious non-interacting system is the same as that of a given interacting system. Vertical excitation energies can then be calculated as dependent on the time-evolution of the potential.

2.8.6. Performing DFT Calculations

Single-point energy calculations are straightforward to understand: With the nuclear coordinates held fixed and an initial guess for the molecular orbitals coming from, for example, Extended Hückel Theory, the Kohn-Sham equations are solved iteratively to find the ground state energy of the system, i.e. the energy is minimised, as in Hartree-Fock theory.

76 Optimising a molecular geometry requires a stationary point on the potential energy surface to be found. At each step, the derivative of the energy with respect to nuclear position is calculated, and the nuclear coordinates adjusted according to the associated forces. The ground state energy is then minimised via the Kohn-Sham equations. This process is repeated until the energy gradients vanish to within a pre-defined tolerance.

The gradient can be "followed" down its steepest path to aid convergence because the derivative of the energy with respect to position will be largest far from an energetic minimum, and will reduce as one approaches this minimum.

Vibrational frequency analysis involves calculation of analytical or numerical second derivatives. This process can only be performed where the energy gradient is tolerably close to zero and is vital for ensuring that a minimum (rather than a saddle point or transition state) in the potential energy surface has been found. These calculations also allow the vibrational spectrum to be simulated, allowing comparison with experimental IR spectroscopic data in some cases, and for thermochemical energy corrections to be calculated.

In a numerical calculation, the molecular coordinates are distorted slightly along each of these modes, and a single point energy calculation is performed to ensure the geometry is a true minimum. In an analytical calculation, the second derivatives are calculated directly to give the vibrational modes. For a structure that is at a minimum in the potential energy surface, there should be no imaginary frequencies.

2.8.7. Problems with DFT

Density functional theory has been successful in many areas, with many thousands of papers published and this number growing rapidly. However, there are problems that have proved difficult to resolve. Dispersion interactions (e.g. Van der Waals) are poorly described by most functionals, although there are methods which have had some success³⁴¹. Self-interaction error is a major problem in DFT, present in all approximate exchange-correlation (xc-)functionals and can cause inaccuracies, particularly in systems with loosely bound electrons where the error may be larger than the binding energy, causing an electron to appear unbound^342,343. In the absence of the ideal functional, DFT relies on error cancelation, and there is no systematic way by which the quality of results can be improved. Parameterising functionals using

77 empirical data can improve results, but it is important to ensure that improvements apply to all systems, not just the data set that the functional has been fitted to.

Regarding f-block elements, DFT struggles to describe the strong static electron correlation that arises in systems where the ground state is well-described only with multiple near-degenerate determinants, for example, as a result of a partially filled f-shell, but good results have been achieved, particularly with closed shell systems40,43,44,58,77,94,97,221,241,244,254,344.