The treatment of atoms and molecules with a large number of electrons from HF or DFT techniques is still a challenge to quantum chemists due to the fact that computational time scales as Nm with N being the number of basis functions. The increase in CPU
power and the development of sophisticated algorithms over the last few decades has made it possible to calculate properties of lighter atoms and molecules with results at or near experimental accuracy. However, further approximations without significant loss of accuracy are needed for heavy or large molecules, as an all-electron treatment for these types of systems is still computationally time consuming. Such an approximation is called the pseudopotential approximation (PPA) or effective core potential (ECP) method originally introduced by H. Hellmann [64] as the combined approximation method [65]. In the PPA, the electrons are separated into core and valence spaces and applies to the basic idea that the valence electrons are primarily responsible for chemical bonding and processes. The PPA is introduced into the electronic Hamiltonian by [66]:
ˆ Hv =− 1 2 nv X i ∇2i + nv X i<j 1 rij + nv X i Nc X a · VP Pa (rai)− Qa rai ¸ + Nc X a<b QaQb rab (2.62) ˆ
Hv is the valence electron model Hamiltonian with nv valence electrons and Nc cores
(nuclei), andVP P is the corresponding pseudopotential for core-valence interactions. The
indices a and b run over all cores (nuclei) and i, j run over all valence electrons. Qa is
the charge of core a and is equivalent toZa−Nca, Nca being the number of core electrons
of atom a. For an all-electron treatment of a specific atom, Qa = Za. The last term in
Eq. 2.62 describes the Coulombic core-core repulsion. Additional corrections are needed if one goes beyond the frozen-core approximation.
Non-relativistic quantum chemical calculations on heavy atoms fail to reproduce an accurate treatment of basic properties such as ionization potentials, electron affinities, excitation energies, and static dipole polarizabilities; hence relativistic effects for these
systems cannot be neglected. Even though relativistic basis sets [67–70] as well as Dirac- Hartree-Fock (DHF) coupled cluster calculations including Breit interactions for many- electron heavy atomic systems are now available, a fully relativistic treatment for heavy elements is still a formidable task. There are two main approaches toward the treatment of large molecules with heavy elements. One of these methods is relativistic density dunctional theory (RDFT) [71] in which relativistic Hamiltonians are introduced into a DFT scheme. Current RDFT schemes suffer from the fact that they cannot be improved to high accuracy to describe certain interactions (for example Van der Waals) even with the development of more accurate functionals within the GGA [72].
The other alternative is to stay with relativistic ab-initio calculations and reduce the number of electrons in the Hamiltonian. Since the core electrons are not as important in describing most chemical properties as the valence electrons, one might consider a frozen core approximation and apply the PPA model to approach heavy elements. Such an approach is not trivial, as several fundamental issues need to be addressed before a PPA model is adopted. One concern with the PPA is that the electrons cannot be separated into core and valence electrons, as this violates the Pauli principle. Transferability among different molecules as well as fitting techniques for different pseudopotentials are of further concern. Relativistic perturbation operators act in the close vicinity of the nucleus, and it is not obvious how a relativistic scheme can be implemented into the PPA using nodeless orbitals. Lastly, core properties such as electric field gradients and magnetic shielding tensors cannot be obtained directly from pseudopotential schemes.
Given that there are several concerns associated with using a PPA description for heavy atoms, there are many advantages for using such an approximation. Atomic and molecular properties for elements with large relativistic effects have been calculated very accurately using the PPA. Removing the core and replacing them with nodeless pseu- doorbitals significantly reduces the number of basis functions for an atomic or molecular calculation. The amount of the BSSE is reduced for weakly interacting systems. Rela- tivistic effects can be included implicitly in the pseudopotential through the adjustment procedure to relativistic atomic calculations. This eliminates the need to carry out a fully relativistic treatment for molecules containing heavy elements.
2.10.1
Introducing Pseudopotentials into the Hartree-Fock Equa-
tions
Before introducing the PPA into the HF equations, appropriate core and valence spaces must be chosen. This task is not trivial, and several properties must be considered before selecting these corresponding spaces. First, the core should be chosen within a region where there is little overlap between the core and valence spaces. This can be achieved by comparing the outermost core orbital with the innermost valence orbital through the examination of orbital energies, overlap regions, r-expectation values, etc. Next, the static dipole polarizability of the core should be small. This defines a hard core where polarization effects from other electrons and nuclei are relatively small. The core density should not change significantly with the addition or removal of valence electrons. The valence electrons from other atoms should not penetrate into the core. Core penetration causes large errors associated with overlap between cores of different atoms. Given the above criteria, core spaces are typically chosen to be closed shell and contain the innermost
s,p, d, or f orbitals.
For a neutral atom with Nc core electrons and nv = Z −Nc valence electrons, the
Hartree-Fock core and valence electron densities can be written as:
ρc(r) = X c∈core Nc|φc(r)|2 (2.63) ρv(r) = X v∈valence nv|φv(r)|2 (2.64)
The HF equation for the valence electrons would be ˆ
F φv =ǫvφv (2.65)
hφv|φci= 0
For simplicity it is assumed that the valence space consists of a single electron, and the core space is closed-shell. An extension into systems with many electrons within the valence space is given in Ref. [73]. If a new valence orbital (χv) is created by mixing in
core orbitals, then we have:
χv = φv +
X
c
acvφc (2.66)
we obtain
hχv|Fˆ|χvi=ǫv +
X
c
a2cvǫc (2.67)
The linear combination in Eq. 2.67 can be chosen such that the radial part ofχv becomes
nodeless in the region (0,∞), which is called the pseudoorbital transformation. The aim is to construct a new Fock operator such that
ˆ
F′χx =ǫvχv (2.68)
where ǫv is identical to that in Eq. 2.66. The HF equation for χv thus becomes
ˆ F χv = ǫvφv+ X c acvǫcφc = ǫvχv +ǫv(φv−χv) + X c acvǫcφc = ǫvχv + X c acv(ǫc−ǫv)φc = ǫvχv + X c hχv|φci(ǫc−ǫv)φc (2.69)
The last term in Eq. 2.69 can be brought to the left hand side and after inserting Eq. 2.67 for the coefficients acv we obtain
[ ˆF + ˆPcore]χv = ˆFP Kχv =ǫvχv (2.70)
where ˆFP K stands for the Philips-Kleinman operator. ˆPcore is a specific core projection
operator such that
ˆ
Pcore=
X
c
(ǫv−ǫc)|φcihφc| (2.71)
The Fock operator can now be split into a sum of valence and core operators, with the core operator being absorbed into pseudopotential operator ( ˆVpp(r)) for the valence electron
at position r. Hence, the total Fock operator becomes ˆ F = ˆFvalence+ ˆVpp(r) = ˆFvalence− Nc r + X c (2 ˆJc−Kˆc) + ˆPcore (2.72)
Solving the HF equations for χv leads to
hχv|Fˆvalence+ ˆVpp|χvi
hχv|χvi
Equation 2.72 is considered the starting point for all pseudopotential approximations in which ˆVpp can take on a variety of different forms within different PPA schemes. There
are two major types of pseudopotentials, which will be discussed briefly below: Energy consistent pseudopotentials and Shape consistent pseudopotentials.
Within the pseudopotential approximation, ˆVpp is replaced by a local or non-local
potential which is bound from below by a linear combination of Gaussian type orbitals (GTO’s). One therefore chooses l-dependent semi-local pseudopotentials of the form [74–77] ˆ Vpp = ˆVpplocal(r) + Lmax X l=0 ˆ Vppl (r) l X m=−l |lmihlm| = NL X n=1 Anrkne−αnr 2 + Lmax X l=0 NSL X n=1 Blnrklne−βlnr 2 l X m=−l |lmihlm| (2.74)
where the sum over the m term is an l-dependent operator projecting onto the Hilbert subspace of angular momentum l, and k is an integer with a value k ≥ −2.
Energy consistent pseudopotentials (EC-PP or EC-ECP) are fitted such that the pseu- dopotential parametersAn,αn,Bln, andβln in Eq. 2.74 are adjusted to an atomic valence
spectrum by a least squares fit procedure.
X
i
wi(∆EiAE−∆EiP P)2 =min. (2.75)
where wi are weight factors, and ∆EiAE are ionization potentials, electron affinities, and
excitation energies for neutral or charged atoms. These reference data are obtained from numerical HF calculations [78].
The same parameters described above in Eq. 2.74 can also be adjusted such that the valence orbitals are reproduced to high accuracy above a certain cut-off radius (Rc).
This fitting process can be done by modifying all-electron orbitals such that they are nodeless in the core region below Rc and adjusting the pseudopotential parameters such
that they match these orbitals [79–81]. These pseudopotentials are called shape consistent pseudopotentials (SC-PP or SC-ECP). The primary drawback to this procedure is that a large number of Gaussians are required to obtain an accurate fit.
As mentioned previously, relativistic operators act in the vicinity of the nucleus. Pa- rameterization of the pseudopotentials to relativistic data therefore seems to successfully transfer both direct and indirect relativistic effects to the valence orbitals. The param- eterization procedure for modeling these effects into the PPA can come from a variety
of different data. Reference energies or orbitals can be taken from scalar relativistic, spin-orbit coupled, or fully relativistic all-electron calculations. The PPA can be intro- duced at the scalar (spin-orbit averaged) relativistic level (SRPP, ARPP, or AREP), at the two-component level (SOPP or REP), or derived from the 4-component Dirac equa- tion [82–85].