where:
Hel({ri}; {Rfix}) = Tel({ri}) + Vnucl−el({ri}; {Rfix}) + Vel−el({ri}) (1.9) while the nuclear part of the dynamics is described by the relation:
Hnucl(R) · χnucl(R) = E · χnucl(R) (1.10) with:
Hnucl(R) = Tnucl(R) + Vnucl−nucl(R) + Eel(R) (1.11) where the electronic energy Eel(R) is the ground state energy of the electron system when the nuclei are in the configuration R.
Through out the next sections (1.2 - 1.4) we will be concerned only with the electronic part. For simplicity and brevity we will use in these sections Rydberg atomic units. Within this unit system:
~= 1, me= 1
2, e2 = 2, c = 2
α = 274.074 .
The energy is then expressed in Rydberg and the length scale is the Bohr radius:
1Ry ≃ 13.6058 eV, aB = 0.529177 ˚A .
With these simplifications the relations (1.4), (1.6) and (1.7) become:
Tel = −
In spite of the BO splitting the many body problem remains a formidable task. Thus, further approximations need to be done in order to accurately and efficiently perform the total-energy calculations. These approximations includes the density functional theory to model the electron interactions, the pseudopotential theory to model the electron-ion interactelectron-ions and the supercell approach to cope with large periodic or aperiodic atomic systems [3].
1.2 The density functional theory (DFT)
The DFT comes into play when the electron-electron interactions (the Vel−elterm from the electronic Hamiltonian) have to be included into our calculations. This term brings with it the Coulombic long range coupling between the position of one electron and the positions of all other electrons. These couplings - which does not allow one to deal with single electron equations - can not be neglected: the electron-electron interaction is Coulombic
and as such it has an infinite range of action whose strength is particularly large for small electronic distances.
One way out of this problem would be the replacement of the exact electron-electron repulsion term Vel−el with an average field Vav due to the presence of all the other elec-trons, except the considered one. Together with wavefunctions of proper symmetry (i.e.
wavefunctions which fulfill the Pauli Exclusion Principle) this approach allows one to split the many-body electronic problem (which we can not solve) into N independent one-electron equations (which we can solve) and takes also into account a large part of the electronic Coulomb interaction. The results obtained by this method (also known as the Hartree-Fock[HF] method [4]) are good but they still do not reproduce accurately the ground state energy for most systems. This is due to the fact that the Hartree-Fock method does not consider the so-called correlation energies, contributions which come from the fact that electrons with different spin states tend to avoid one another due to the Coulombic repulsion. This neglection leads to an unappropiate description of the two-electron (correlated) density and thus to a higher total energy for the simulated sys-tems. Corrections to the total HF energy, in order to incorporate these contributions, can be brought either by using a perturbatory approach (like in Moller-Plesset Perturbation Theory [5]) or by using a mixture between ground and excited configurations (like in Configuration Interaction or Coupled Cluster methods [6]). While successfull from the point of view of the accuracy provided, these extensions are exceptionally demanding in terms of computer performance (CPU time, memory, hard disk needs). Thus for all but the smallest molecules, these methods are basically impractical.
Is then there any hope to capture these correlation effects and still end with a com-putationally efficient method ? The answer is yes, but not with a wavefunction based method, rather with a density functional based method.
1.2.1 The Hohenberg-Kohn theorems
Unlike HF and post-HF methods which are wavefunction based, DFT comes with a new philosophy: why not use instead of the demanding wavefunctions - which are difficult to calculate, store or interpret - something more feasible, like the charge density. This substi-tution solves largely our problems because of the simplicity brought by the manipulation of a 3-dim object instead of a 3N-dim object. (This is readily clear. Imagine, for example, that you want to analyze a relatively common and simple system - the benzene molecule C6H6. With 42 electrons this would be an object in the 126-dim space if a wavefunction method is used and a 3-dim space object if a density based method is used. You can easily make and analyze a 3-dim plot, but you will run into difficulties even to make a picture of a 126-dim object! The storing space is also 10123 larger when using a wavefunction description than when using a density based description.)
The DFT is based on two remarkable theorems by Hohenberg and Kohn (HK) [7].
The first of these states that: the external potential Vext acting on a system of electrons is determined (up to an additive constant) by the ground state electronic charge density ρ(r) alone.
The external potential from the first HK theorem can be, but it is not restricted to, Vnucl−el. Its dependence on the charge density means that also the Hamiltonian, the wavefunctions and any observable (as the total energy) will depend on the charge density, i.e. they will be functionals of ρ(r).
1.2 The density functional theory (DFT) 15 Following these, one can write the ground state total energy as:
E[ρ(r)] = Z
ρ(r)·Vext(r)·dr+F [ρ(r)] = Z
ρ(r)·Vnucl−el(r)·dr+(Tel+Vel−el)[ρ(r)] (1.15) where F [ρ(r)] is an unknown, but universal (i.e. independent on the external potential) functional of the electron density.
The second HK theorem states that: for any density distribution ρ′ 6= ρ (where ρ is the correct ground state density) the energy E[ρ′] will be always higher than the true ground state energy E[ρ], i.e.
E[ρ′] > E[ρ], f or ρ′ 6= ρ (1.16) In this framework, the determination of the ground state of a system is reduced to the minimization of E[ρ] under the constraint that:
Z
ρ(r) · dr = N (1.17)
i.e. to the solution of the Euler-Lagrange equation:
∂
∂ρ{E[ρ] − µ · [ Z
ρ(r) · dr − N]} = 0 (1.18)
where the Lagrange multiplier µ (the chemical potential) is introduced in order to impose the condition from (1.17).
An equation similar to (1.18) is encountered also in the Fock or post Hartree-Fock theories. However, the evaluation of (1.18) is much easier now due to the simpler variable dependence - on ρ(r) and not on Φel(r1, r2, r3, ..., rN).
1.2.2 The Kohn-Sham equations
While the HK second theorem states that the ground state of a system can be reached through a minimization of E with respect to ρ(r), it provides, however, no prescription on how to actually construct the E[ρ] functional. An essential step towards turning DFT into a practical tool for simulations is a scheme, proposed by Kohn and Sham [8], which offers an approximate way for getting E[ρ]. Prior to this scheme an analytic form -density dependent - was only known for two components of E[ρ], namely for the energy of interaction between nuclei and electrons:
Enucl−el[ρ(r)] = Z
ρ(r) · Vnucl−el(r) · dr (1.19) and for the classical (Hartree) electron-electron repulsion term (a part of Vel−el)
EH[ρ(r)] =
Z Z ρ(r) · ρ(r′)
|r − r′| · dr · dr′ (1.20) These make however only a small fraction of the total electronic energy of a system, which reads:
Eel= EH+ Enucl−el+ Tel+ (Vel−el− EH) (1.21)
Important contributions like the kinetic energy (which for Coulombic systems equals, according to the virial theorem, the absolute value of the total energy |Eel|) or certain correlation effects (the Vel−el − EH term) had received so far no prescription in terms of the electron density.
Kohn-Sham dealt with this problem by considering an ”associated” system of non-interacting particles - with kinetic energy Ts and single particle potential vs - and whose ground state density ρs(r) matches exactly the ground state density ρ(r) of the interacting system. From the ”associated” one-particle Schr¨odinger equation:
[−∇2(r) + vs(r)] · φs,i(r) = ǫi· φs,i(r) (1.22) one obtains the single particle orbitals φs,i(r). These are ρ(r) dependent, i.e. φs,i(r) = φs,i([ρ], r), as the potential which has been used in (1.22) has been chosen such that ρs(r) = ρ(r).
Using these single particle orbitals, as obtained from (1.22), one can write the non-interacting kinetic energy Ts as:
Ts[ρ] = XN
i=1
< φs,i([ρ], r)| − ∇2i|φs,i([ρ], r) >
The difference between Tel and Ts is added at the last term from (1.21), which we will name from now on the exchange and correlation energy:
Exc[ρ] = (Tel[ρ] − Ts[ρ]) + (Vel−el[ρ] − EH[ρ]) (1.23) Using (1.23) the total electronic energy from (1.21) can be re-written as:
Eel = EH+ Enucl−el+ Ts+ Exc Through a variational principle, applied to the above relation, one is lead to the Euler-Lagrange equations: The exchange and correlation potential Vxc[ρ] from (1.26) is obtained as the functional derivative of Exc[ρ]:
Vxc[ρ] = δExc[ρ]
δρ(r) (1.27)
A closer look at (1.25) reveals that this is nothing else than the equation which one obtains when applying the density functional theory to a system of non-interacting electrons, moving in an external potential vs = veff. This means that for a given veff, the density ρ(r) that satisfies the Euler-Lagrange equation (1.25) can be obtained by solving the N one-electron equations:
{−∇2+ veff([ρ], r))} · φs,i = ǫs,i· φs,i (1.28)
1.2 The density functional theory (DFT) 17 and calculating the ground state electron density by occupying the N one-electron orbitals with the lowest eigenvalues:
ρ(r) = XN
i=1
|φs,i(r)|2 (1.29)
Equations (1.26), (1.27), (1.28), (1.29) are typically referred as the Kohn-Sham equa-tions.
These equations have to be iterated to self-consistency (SC) because the effective potential depends on the density which should be generated from it.
Within the SC procedure, graphically depicted in Fig. 1.1 (the inner loop; the outer loop, which concerns the optimization of the atomic coordinates in order to find the minimum energy configuration of the system considered, is described in Chapter 1.5), one starts typically with an initial guess for the density ρ(r) and constructs veff([ρ], r) according to (1.26) and (1.27). From (1.28) and (1.29) a new density ρnew(r) is obtained.
If this one matches ρ(r) the SC procedure ends, while if not a mixture between the old and the new density is used as input density to generate a new veff([ρ], r). The SC procedure ends when the old and the new density are equal (up to a chosen numerical accuracy) or, in other words, when the potential generated from the ρ(r) density is able to generate back the same ρ(r) density.
Once the self-consistent charge density is obtained it can be used to calculate any ground-state observable, like the ground state total electron energy (which can be obtained from the density via the relation (1.24)).
1.2.3 The exchange and correlation functionals
So far the DFT of Hohenberg, Kohn and Sham is exact, i.e. no approximations have been made till now. However, not all the terms in equation (1.24) have been clarified: We still don’t have an explicit form for the Exc[ρ] functional, which contains all the many-body effects of the problem under work. An explicit expression for this contribution is needed in order to calculate the Vxc[ρ] and solve the Kohn-Sham equations.
So far, no exact analytic formulation for the electron density dependence of the XC-energy has been found. Practical computations rely, therefore, on approximated forms of Exc[ρ].
The oldest and probably the simplest approximation for the exchange and correla-tion energy funccorrela-tional is called the local density approximacorrela-tion (LDA) [8]. In LDA the exchange-correlation energy of an electronic system (i.e. Exc[ρ]) is constructed by assum-ing that the exchange-correlation energy per electron at a point r (i.e. ǫxc(r)) of the system considered is equal to the exchange-correlation energy per electron in a homoge-neous electron gas which has the same density as the considered system at the point r.
Thus:
This approximation yields an explicit expression because the exchange energy of a homogeneous electron gas can be obtained exactly (analytically), while the correlation
φi − ∆2
(r) (r)φi (r)
(r)
RI = ∆calculate forceFI −(+el EEnucl−nucl) +s]T[+EXC[]ρρ][ρ(r)=[]+EHel Eρ[]ρEnucl−elcompute electronic energy
move the atomsminimum reached(F)<0.1mRy/a.u.?max yesno yes no cycle SC
+==H[]eff vεi φi {} RI
I eff v
Σi 2 mixing:
mix = [
(r)i φ ρ
ρ;] newr)(ρρnewr)(= ρ
newr)(= (r)?ρρ
Figure1.1:RunningschemeforaDFTbasedcode
1.3 The Bloch theorem and the supercell approach 19