Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU centro de física de materiales CFM.

Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU centro de física de materiales CFM

Electronic structure calculations: Methodology and applications to nanostructures

Lectures on Quantum Chemistry: Tuesday March 17th: 9.45 --> 12.30 Theoretical background Wednesday March 18th : 9.45 --> 12.30 Practical exercise Electronic structure calculations: Methodology and applications to nanostructures

Electronic structure calculations: Methodology and applications to nanostructures Post Hartree Fock

Electronic structure calculations: Methodology and applications to nanostructures Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry—an aberration which is happily almost impossible–it would occasion a rapid and widespread degeneration of that science. Auguste Comte, 1830.

Electronic structure calculations: Methodology and applications to nanostructures In conclusion, I would like to emphasize my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the computing machine instead of the laboratory, for increasingly many facets of chemical information, is already at hand. There is only one obstacle, namely, that someone must pay for the computing time. Robert Mulliken. Nobel Prize address, 1966.

Electronic structure calculations: Methodology and applications to nanostructures Wave function of many electrons in an external potential (Borh-Oppenheimer) Finite system (no periodic boundary conditions)

Fundamental object in quantum mechanics: wave function We want to find special wave functions such that where This is a fundamentally many-body equation! A large variety of methods have been proposed and are being used to solve this problem. Electronic structure calculations: Methodology and applications to nanostructures

Electronic structure calculations: Methodology and applications to nanostructures This will be one of our main tools today. It states that the energy calculated from an approximation to the true wavefunction will always be greater than the true energy: Thus, the better the wavefunction, the lower the energy. At a minimum, the first derivative of the energy will be zero.

Electronic structure calculations: Methodology and applications to nanostructures CIST, MP2, CC, CSF, TZV,...

As a first guess, one may try to write the many-electron wavefunction as a product of one-electron spin-orbitals  i (r i,  i ): Electronic structure calculations: Methodology and applications to nanostructures An important feature of the Hartree description is that the probability of finding one electron at a particular point in space is independent of the probability of finding any other electron at that point in space. Thus, due to the independent particle model, the motion of the electrons in the Hartree approximation is uncorrelated.

Electronic structure calculations: Methodology and applications to nanostructures One-electron spin-orbitals  i (r i,  i ) are constructed as the product of a spatial orbital and a spin function. In general, they are molecular orbitals.

Electronic structure calculations: Methodology and applications to nanostructures Applying the variational principle to the above wave function, one can find the single-particle Hartree equations: self-consistent equations mean field

Electronic structure calculations: Methodology and applications to nanostructures The main problem with Hartree’s wave function is that it violates Pauli´s principle. The wave function of fermions must be antisymmetric and therefore two fermions cannot be in the same quantum state. Slater determinant

Electronic structure calculations: Methodology and applications to nanostructures Slater determinant -Exchanging any two rows of a determinant (exchanging two electrons) leads to a change in sign  antisymmetry. -Two electrons in the same quantum state  two identical rows  the determinant is zero.

Electronic structure calculations: Methodology and applications to nanostructures The minimization of the energy assuming that the wave function  is a Slater determinant leads to the Hartree-Fock approximation. The corresponding single-particle Hartree-Fock equations are the following:

Electronic structure calculations: Methodology and applications to nanostructures Identical to HartreeNew exchange term Again, this is a mean field self-consistent model. Coulomb term JExchange term K

Electronic structure calculations: Methodology and applications to nanostructures Notice that adding the term i=j in the sums modifies nothing: It cancels out.

Electronic structure calculations: Methodology and applications to nanostructures We can define the following operators by their action on an orbital: and from them we define the Fock operator: so that the HF equations can be written as:

Electronic structure calculations: Methodology and applications to nanostructures RHF: the spatial part of the one-electron spin- orbitals  i (r i,  i ) is identical for spin-up and spin- down (closed-shell) UHF: the spatial part of the one-electron spin- orbitals  i (r i,  i ) depend on the spin-orientation. Here, the wavefunction may be not a proper spin eigenfunction (spin contamination). The energy of a UHF wave function is always lower than (or equal to) the corresponding RHF wave function (there is more flexibility in the former).

Electronic structure calculations: Methodology and applications to nanostructures Slater determinants are always eigenfunctions of S z. However, they are not necessarily eigenfunctions of S 2. For the general case there are always linear combinations of determinants that are eigenfunctions of S z and S 2 at the same time. Such spin-adapted linear combination of determinants (configurations) are needed to describe open-shell systems.

Electronic structure calculations: Methodology and applications to nanostructures Hartree-Fock calculations often account for ~99% of the total energy of the system. The problem is that the remaining ~1% can determine the physical and chemical properties of the system.

Electronic structure calculations: Methodology and applications to nanostructures Hence, we have to improve over HF: How to do that?

Electronic structure calculations: Methodology and applications to nanostructures Orbitals are usually expanded in basis sets.

Electronic structure calculations: Methodology and applications to nanostructures Slater-type orbitals (STOs)  n,l,m (r, ,  ) = N n,l,m,  Y l,m ( ,  ) r n-1 e -  r are characterized by quantum numbers n, l, and m and exponents (which characterize the radial 'size' ) . Slater-type orbitals are similar to Hydrogenic orbitals in the regions close to the nuclei. Specifically, they have a non-zero slope near the nucleus on which they are located d/dr(exp(-  r)) r=0 = -  so they can have proper electron-nucleus cusps.

Electronic structure calculations: Methodology and applications to nanostructures Cartesian Gaussian-type orbitals (GTOs)  a,b,c (r, ,  ) = N' a,b,c,  x a y b z c exp(-  r 2 ), are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the orbital, and exponents  which govern the radial 'size’. GTOs have zero slope near r=0 because d/dr(exp(-  r 2 )) r=0 = 0. The Coulomb cusp at the origin is not properly described. But, computationally, multi-center integrals are much more efficiently obtained.

Electronic structure calculations: Methodology and applications to nanostructures To overcome the cusp weakness of GTO functions, it is common to combine two, three, or more GTOs, with combination coefficients that are fixed (and not treated as parameters), into new functions called contracted GTOs or CGTOs. However, it is not possible to correctly produce a cusp by combining any number of Gaussian functions because every Gaussian has a zero slope at r = 0 as shown here.

Electronic structure calculations: Methodology and applications to nanostructures Minimum basis set: the number of basis functions is equal to the number of core and valence electrons in the atom. Double zeta (DZ): there are twice as many basis functions as there are core and valence electrons. Triple zeta (TZ): there are three times as many basis functions as the number of core and valence electrons. Quadruple zeta (QZ), Pentuple Zeta (PZ or 5Z), etc. In any of them: split valence basis means that only the number of basis functions representing the valence electrons is increased. H C N HCN molecule: DZ basis allows for different bonding in different directions

Electronic structure calculations: Methodology and applications to nanostructures Polarization functions: a basis function with a higher component of angular momentum is added, p-functions to s-based orbitals, d-functions to p-based orbitals, etc. Double Polarization functions: basis functions with two higher components of angular momentum are added. For instance, double zeta with polarization (DZP), triple zeta plus double polarization (TZDP), etc. Polarization functions give angular flexibility in forming molecular orbitals between valence atomic orbitals. Polarization functions also allow for angular correlations in describing the correlated motions of electrons. H C N

Electronic structure calculations: Methodology and applications to nanostructures Hartree-Fock is an approximation: It replaces the instantaneous electron-electron repulsion by an average repulsion term. Strictly speaking, electron correlation energy is defined as the difference between the HF energy and the lowest possible energy that one can obtain within a given basis set. Physically, it corresponds to the fact that, on average, the electrons are further apart than the situation described by the (R)HF wave function. A clear example in RHF: electrons are paired in molecular orbitals and the spatial overlap between the orbitals of such pair-electrons is exactly one!

Electronic structure calculations: Methodology and applications to nanostructures To improve over Hartree-Fock and include electron-correlation, the easiest way is to start from the Hartree-Fock approximation and ADD new things. Different methodologies will be defined by the different ways to ‘add’ things to Hartree-Fock. Typically, they fall into two classes: Wavefunction expansion: The most common approaches are Configuration Interaction (CI) and Coupled-Cluster Methods (CC, CCSD). Perturbation theory: The most common approach is Møller-Plesset (MP2 or MP4).

Electronic structure calculations: Methodology and applications to nanostructures CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants.

Electronic structure calculations: Methodology and applications to nanostructures CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants. where we are adding new Slater determinants that are singly (  s ), doubly (  d ), triply (  t ), quadruply (  q ), etc. Excited relative to the original HF determinant. These determinants are often referred to as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc.

Electronic structure calculations: Methodology and applications to nanostructures These determinants are often referred to as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc.

Electronic structure calculations: Methodology and applications to nanostructures CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants.  CI =a 0  HF +a S  S +a D  D +…=  a i  i Again we use the variational principle and look for the a i coefficients that make minimal the wave function energy. Löwdin (1955): Complete CI gives exact wavefunction for the given atomic basis. For an infinite basis, it provides the exact solution. Orbitals are NOT reoptimized in CI!

Electronic structure calculations: Methodology and applications to nanostructures  CI =a 0  HF +a S  S +a D  D +…=  a i  i Brillouin’s theorem: Matrix elements between the HF reference determinant and singly excited states are zero. Structure of the CI matrix

Electronic structure calculations: Methodology and applications to nanostructures  CI =a 0  HF +a S  S +a D  D +…=  a i  i In order to develop a computationally tractable model, the number of excited determinants in the CI expansion must be reduced. Truncating the expansion at one (  s ) does not improve the HF result because of Brillouin’s theorem. The lowest CI level that improves over HF is CI with Doubles (CID). The number of singles is much lower than the number of Doubles. Therefore, including singles is not a big deal: CI with Singles and Doubles (CISD). Also with Triples: CISDT. Also with Quadruples (CISDTQ).

Electronic structure calculations: Methodology and applications to nanostructures  CI =a 0  HF +a S  S +a D  D +…=  a i  i The lowest CI level that improves over HF is CI with Doubles (CID). Weights of excited configuration in the Ne atom. Doubles have the highest weight!

Electronic structure calculations: Methodology and applications to nanostructures Let’s try to illustrate how CI accounts for electron correlation taking as an example the dissociation of the hydrogen molecule H 2 Take two 1s orbitals, one in each center of the molecule,  A and  B AA BB HF

Electronic structure calculations: Methodology and applications to nanostructures The basis determinants for a full CI calculation are the following: Double Single Single : triplet S Z =1 Single : triplet S Z =-1  2 +  3 triplet S Z =0  2 -  3 singlet

Electronic structure calculations: Methodology and applications to nanostructures The ground state  0 and the doubly excited  1 can be expanded in terms of the atomic orbitals: ioniccovalent Now, if we increase the bond length towards infinity, the HF wave function is still a mixture of ionic and covalent components and, in the dissociation limit will be 50% H + H - and 50% H 0 H 0. This is totally wrong!! Electron correlation is missing: electrons try to avoid each other!

Electronic structure calculations: Methodology and applications to nanostructures We can solve that by using full CI. The full CI matrix can be shown to be: For 1  g symmetry only these terms matter The variational parameters allow us to choose the best combination for each bonding distance. For instance, the ionic component disappears for a 1 =-a 0

Electronic structure calculations: Methodology and applications to nanostructures The problem can also be treated with a UHF wave function. Although the UHF wave function does not solve everything: spin contamination. We introduce a variational parameter c in the definition of the molecular orbitals. Now they are different for spin-up and spin-down. ioniccovalent but now we have an additional triplet component

Electronic structure calculations: Methodology and applications to nanostructures All this is conspicuous in the energy diagram:

Electronic structure calculations: Methodology and applications to nanostructures For almost degenerate levels it is crucial to optimize the orbitals as well: Multi-Reference Self-Consistent Field (MRSCF): a kind of CI in which the orbitals, as well as the coefficients, are optimized. Configurations included in MCSCF are defined by the active space. Multi-Reference Configuration Interaction (MRCI): A MRSCF function is chosen as reference. Singles, doubles, etc., are generated out of all the determinants that enter the MRSCF.

Electronic structure calculations: Methodology and applications to nanostructures Reduced CI methods Idea: Not all determinants are equally important. Ansatz: Only allow excitations from a subset of orbitals into a subset of virtual orbitals (active space). Allow only a maximal number of excitations.

Electronic structure calculations: Methodology and applications to nanostructures In perturbation theory, the Hamiltonian splits into: Perturbation, i.e., its effect should be small! Unperturbed Hamiltonian

Electronic structure calculations: Methodology and applications to nanostructures First, let us remember the Hartree-Fock equations: We have the Fock operator, defined as: These are the self-consistent equations that the single-particle wave functions should fulfill to obtain the minimum energy for a single- determinant many-body wave function. so that the HF equations can be written as:

Electronic structure calculations: Methodology and applications to nanostructures The exact Hamiltonian H e is: We define an unperturbed Hamiltonian H 0 as the sum of Fock operators:

Electronic structure calculations: Methodology and applications to nanostructures The exact Hamiltonian H e is: We define an unperturbed Hamiltonian H 0 as the sum of Fock operators: In H 0 we are summing up twice the electron-electron Coulomb interaction

Electronic structure calculations: Methodology and applications to nanostructures The exact Hamiltonian H e is: We define an unperturbed Hamiltonian H 0 as the sum of Fock operators: In H 0 the electron-electron interaction is considered in an average way.

Electronic structure calculations: Methodology and applications to nanostructures The exact Hamiltonian H e is: We define an unperturbed Hamiltonian H 0 as the sum of Fock operators: The perturbation is therefore H’=H e -H 0 : Not that small!

Electronic structure calculations: Methodology and applications to nanostructures The unperturbed Hamiltonian H 0 is a sum of one-electron operators without coupling. Therefore the solution will be the sum of all energies of the independent systems and the wavefunction will be the (antisymmetrized) product of the one electron wavefunctions (orbitals). In other words, the ground state wave function will be the Hartree-Fock wavefunction  0 HF. The unperturbed energy is E 0 = ≠ E HF The energy adding the first-order correction is E 1 = = E HF !! (it is the matrix element that we have varied to obtain the HF equations)

Electronic structure calculations: Methodology and applications to nanostructures Electron correlation corrections start at second order with this choice of the unperturbed Hamiltonian. MP2 means second-order in Moller-Plesset expansion. The perturbative correction is: where the excited states  (i) 0. are eigenstates of the unperturbed Hamiltonian H 0, i.e., they are determinants in which excitations have been created. Actually, doubly-excited (not single-excited) determinants are the first contribution! Following contributions to the perturbative expansion are MP3, MP4, etc.

Electronic structure calculations: Methodology and applications to nanostructures No smooth convergence, or no convergence at all is possible!! In practice, only low orders of the expansion can be included.

Electronic structure calculations: Methodology and applications to nanostructures The basic ansatz of coupled cluster theory is that the exact many-electron wavefunction  may be generated by the operation of an exponential operator on a single determinant.  0 is a single determinant wave function (usually, the Hartree-Fock wave function  0 =  HF is used) T is an excitation operator. The excitation operator can be written as a linear combination of single, double, triple, etc excitations, up to N fold excitations for an N electron system:

Electronic structure calculations: Methodology and applications to nanostructures etc. and the action of each T i operator is to create the full set of i-excitations there is only one way to have a single excitation T 1, but two ways to generate double excitations: a double excitation (T 2 ) and two consecutive single excitations (T 1 T 1 ).

Electronic structure calculations: Methodology and applications to nanostructures etc. and the action of each T i operator is to create the full set of i-excitations there is only one way to have a single excitation T 1, but two ways to generate double excitations: a double excitation (T 2 ) and two consecutive single excitations (T 1 T 1 ). To construct the coupled cluster wavefunction one must then determine the various amplitudes t through a system of coupled equations.

Electronic structure calculations: Methodology and applications to nanostructures In coupled cluster methods, for a given type of corrections (say, single excitations, for instance), all required terms are included. As a consequence, the method scales like M 2N+2 for M basis functions and N electrons  a very expensive scaling!! In practice, only corrections up to a given term are included: If higher-order terms are calculated in perturbation theory, they are indicated with Parentheses. For instance CCSD(T) means that the triples are obtained perturbatively.

Electronic structure calculations: Methodology and applications to nanostructures In coupled cluster methods, for a given type of corrections (say, single excitations, for instance), all required terms are included. As a consequence, the method scales like M 2N+2 for M basis functions and N electrons  a very expensive scaling!! In practice, only corrections up to a given term are included: Almost the only one used in practice.

Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU centro de física de materiales CFM.

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Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU centro de física de materiales CFM.

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