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Time-independent Schr¨odinger Equation

All ab initio methods which are presented in this chapter to perform quantum chemical calculations, aim at solving the stationary nonrelativistic Schr¨odinger equation [40]. This (time-independent) equation is given as ˆ (3.1) H|Ψ = E|Ψ.
In the previous equation the total energy, E, is calculated by applying the time-independent ˆ molecular Hamiltonian, H, on the wave function, |Ψ , describing the system’s properties. The molecular Hamiltonian can be described in terms of distinct parts as ˆˆ ˆˆ ˆ ˆ (3.2) H = Te + TN + VN e + VN N + Vee In the expression above one can notice that the spin contributions due to the electronic and nuclear particles of the system are neglected. The diﬀerent parts of the right hand 10 Chapter 3. Electronic Structure Methodology side of equation ( 3.2) are explicitly expressed as : ˆ = − h¯2 △i.

Born-Oppenheimer Approximation

Unfortunately, the Schr¨odinger equation, eq. 3.1, is not exactly solvable for most systems, except in the case of one-electron systems such as the hydrogen atom or the H+2 molecule. One way to overcome this inability is a separation between the nuclear terms and the electronic terms [41]. The most common way to do so is applying the Born-Oppenheimer approximation (BOA) [42] to equation 3.1. Since the mass diﬀerence between the electron and the nucleus is very important, for example in the case of the helium atom, the mass ratio of the nucleus to the electron is more than 3600, electrons are assumed to instantaneously adapt to the nuclear motion. In other words, the geometry of the nuclei can be considered as fixed when studying the electrons. Applied to the time-independent Schr¨odinger equation, the BOA leads to two distinct equation : the electronic Schr¨odinger equation and the nuclear Schr¨odinger equation which are introduced in the following.

Electronic Schr¨odinger Equation

Under the Born-Oppenheimer approximation (BOA) a separation between the motion of the electrons and the nuclei is done. He, the electronic Hamiltonian in Eq. 3.2, is the part of the total Hamiltonian which accounts for the electrons. In this electronic operator, ˆe, the kinetic energy term for the nuclei and the electrostatic nuclear interaction, are H dropped : ˆ = ˆ (3.9) He Te + VN e + Vee = − h¯2 △i − 1 ZAe2 + e2 (3.10) 2me i 4πǫ0 AiriA i j>i rij.
The purely electronic Schr¨odinger equation then becomes ˆ (3.11) He|Ψe(ri RA) = Ee(RA)|Ψe(ri RA).
In Eq. 3.11 |Ψe(ri RA) is the electronic wave function and explicitly depends on the elec-tron coordinates, ri, and parametrically on the nuclear coordinates, RA. The electronic energies, Ee(RA), also parametrically depend on the vector-coordinate RA.

Solving the Electronic Schr¨odinger Equation

The solutions of the electronic Schr¨odinger equation (Eq. 3.11) for diﬀerent geometries provide potential energy surfaces or curves. These curves will be used as pair potentials in DMC and dynamic calculations. In order to solve Eq. 3.11 it is important to appropriately represent the electronic wave functions. A simple and approximate way to represent a wave function is the use of Slater determinants.

Restricted Closed-Shell Hartree-Fock

When solving the Hartree-Fock (HF) equation, Eq. 3.27, one needs to evaluate the shape of the spin orbitals. For a closed-shell molecule, the pair of spin orbitals have the same spatial function : ψi(r)α(σ) ϕ2i(x) = i = 1 2 K (3.31) ψi(r)β(σ) Introducing this equation into the HF Eq. 3.27 results in two distinct HF equations, ˆ = εi|ψi(r1)α(σ1) (3.32) f (x1)|ψi(r1)α(σ1) ˆ = εi|ψi(r1)β(σ1) (3.33) f (x1)|ψi(r1)β(σ1) given in terms of their spin functions, α(σi) and β(σi). In order to get an expression exclusively in terms of the spatial orbitals, it is necessary to remove the spin function from the Fock operator. To do so, the spin orbital is replaced by its spatial orbital and spin functions. In a closed-shell system, the contributions from the α terms and those from β are equal. It is then suﬃcient to multiply Eq. 3.32 from the left by α∗(σi) and integrate over its spin ∗ ˆ |ψi(r1) = εk |ψk(r1) (3.34) α (σ1)f (x1)α(σ1)dσ1.

Basis Set Superposition Error

Using finite basis sets in calculations of potential energy surfaces involves the presence of basis set superposition errors. This phenomenon is due to the fact that for a given bound molecule AB, the atom A can be stabilised by the close presence of the basis functions of atom B and vice versa. The system is hence not only bound by any true interaction between A and B but also by this additional superposition eﬀect.
A possible and approximate correction of this eﬀect is obtained via the counterpoise method of Boys and Bernardi [56]. This method involves the calculation of the energy of each atom or fragment both with its basis functions, EA , EB , and with the basis functions of the entire system EA(B) , E(A)B . This counterpoise correction is respectively given for A and B by:
ΔEACP = EA(B) − EA (3.69).
ΔEBCP = E(A)B − EB (3.70).
The total counter poise correction to the interaction energy is the sum of the counterpoise corrections, ΔEACP + ΔEBCP . The counter poise corrected interaction energy is finally written as:
EABcorrected = EA + EB − EAB + ΔEACP + ΔEBCP (3.71).
Another way to accurately determine interaction energies can be achieved by the complete basis set extrapolation limit which is the subject of the next section.

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Binding energy definition

In the case of a diatomic molecule, the binding energy is defined as the diﬀerence between the molecular electronic energy and those of its components. This definition, of course, can be generalised to any molecule which contains more than two atomic components. For the AB molecule, we can formulate 1 its binding energy as Ebind = EAB − (EA + EB ) (3.72).
In the previous relation, Ebind is the binding potential energy between atom A and B, EA and EB their respective energies. Each of these energies can be decomposed into two contributions: a Hartree-Fock part and a correlation one.

Hartree-Fock energy

The Hartree-Fock part of the energy, EH F , is expected to evolve, when increasing the number of ζ , x as EH F (x) = EH∞F + αe−βx (3.73).
where EH∞F is the Hartree-Fock energy for an infinite basis, α and β are the fitting pa-rameters. The Dunning-Feller [53, 57, 58] (exponential form (Eq. 3.73)) has extensively been shown to better extrapolate the HF part of the binding energy [59] than any power law does.

Table of contents :

1 General introduction
2 Weak interactions
2.1 Introduction
2.2 Dispersion interactions
3 Electronic Structure Methodology
3.1 Time-independent Schr¨odinger Equation
3.1.1 Born-Oppenheimer Approximation
3.1.2 Electronic Schr¨odinger Equation
3.1.3 Nuclear Schr¨odinger Equation
3.2 The Variational Principle
3.3 Solving the Electronic Schr¨odinger Equation
3.3.1 Slater Determinant
3.3.2 Hartree-Fock Equations
3.3.3 Restricted Closed-Shell Hartree-Fock
3.3.4 Roothaan-Hall Equations
3.3.5 Self-consistent Field
3.4 Configuration Interaction (CI)
3.5 Perturbational theories
3.5.1 Møller-Plesset Perturbation Theory
3.6 Coupled Cluster Theory
3.7 Basis Sets
3.8 Basis Set Superposition Error
3.9 Complete Basis Set Extrapolation
3.9.1 Binding energy definition
3.9.2 Hartree-Fock energy
3.9.3 The correlation energy
4 The CaHe X1+ state
4.1 Introduction
4.2 Computational details
4.2.1 Influence of basis sets size
4.2.2 Bond-functions role
4.3 Comparison of methods
4.4 Determination of dispersion coefficients
4.5 Comparison with literature
4.6 Vibrational levels of the CaHe 1+ state
4.7 Conclusion
5 The MgHe 1+ state
5.1 Introduction
5.2 Computational details
5.3 Results and discussions
5.3.1 Basis set
5.3.2 Influence of core correlation effect
5.3.3 Influence of bond functions
5.3.4 PES characteristics : r0 and ǫ
5.3.5 Difference between basis and C-basis set
5.3.6 Basis set superposition error (BSSE)
5.4 Fit quality
5.5 Conventional CBS approximation
5.5.1 Fitting of the HF energies
5.5.2 Fitting of the correlation energies
5.6 Non conventional CBS approximation
5.7 Vibrational level of MgHe ground state
5.8 Conclusion
6 Introduction to quantum Monte Carlo methods
6.1 Variational Quantum Monte Carlo
6.1.1 Energy point calculation
6.1.2 VMC wave functions
6.2 Metropolis algorithm
6.3 Diffusion Monte Carlo
6.3.1 Why diffusion?
6.3.2 DMC method
6.3.3 Time evolution and Green’s function
6.3.4 Move acceptance
6.3.5 DMC wave function
6.3.6 DMC Energy Evaluation
6.4 Error analysis
6.4.1 Correlated samples
6.4.2 Correlation analysis
6.4.3 The DMC case
6.4.4 Statistical errors
6.4.5 Systematic errors
6.5 Calculation of main properties
6.5.1 Radial distribution
6.5.2 Pair correlation function
6.5.3 Two-dimensional histograms
6.6 Pseudo-codes
6.6.1 VMC
6.6.2 DMC
6.7 Conclusion
7 Doped helium nanodroplets
7.1 Introduction
7.2 4He nanodroplet properties
7.2.1 Superfluidity
7.2.2 Temperature of the droplets
7.3 Experimental aspects
7.3.1 Production of helium nanodroplets
7.3.2 Doping of droplets
7.4 Applications of helium nanodroplets
7.4.1 Helium Nanodroplet Isolation Spectroscopy
7.4.2 Other applications
8 DMC computational details
8.1 Introduction
8.2 Influence of the number of walkers
8.3 Influence of the time step
8.4 Influence of the duration of the simulation
8.5 Influence of the number of blocks
8.6 Influence of the random number seed
8.7 Trial wavefunction and parameters
8.8 Conclusion
9 DMC results for MgHen and CaHen clusters
9.1 Introduction
9.2 Ancilotto’s model
9.2.1 Principle
9.2.2 Limits of the model
9.2.3 The alkaline earth case
9.3 Pair potential of the He2, MgHe and CaHe
9.4 Energy calculation
9.4.1 Binding energy model
9.5 Comparison with literature
9.6 Ca and Mg positions on the droplets
9.6.1 Radial probability densities
9.6.2 Helium densities in cylinder coordinates
9.6.3 Structural relaxation of the MgHeN cluster
9.7 Pair density distributions
9.8 Adiabatic model for Mg solvation
9.8.1 Energy profile with a geometrical constraint
9.8.2 Evolution of the helium density
9.8.3 Rovibrational calculation in the constrained potential
9.9 Conclusion
10 Dynamics of Mg doped Helium Clusters
10.1 Introduction
10.2 Potential energy curves
10.2.1 Mg2 (X1+ g ) .
10.2.2 MgHe (X1+)
10.2.3 Effective He2 potential
10.3 Dynamic results
10.3.1 MgHe1998
10.3.2 Mg2He1997
10.4 Conclusion
11 General conclusions
A Electronic energies
B Position of Mg for several MgHe potentials