Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Woodward–Hoffmann rules wikipedia , lookup
Homoaromaticity wikipedia , lookup
Molecular orbital wikipedia , lookup
Heat transfer physics wikipedia , lookup
Atomic orbital wikipedia , lookup
Physical organic chemistry wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Electron configuration wikipedia , lookup
Franck–Condon principle wikipedia , lookup
CFM centro de física de materiales Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU Electronic structure calculations: Methodology and applications to nanostructures Electron correlation methods in Quantum Chemistry 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 Outline • Brief introduction • Hartree- Fock • Basis sets • Configuration Interaction • Many-body perturbation theory • Coupled-cluster methods Electronic structure calculations: Methodology and applications to nanostructures Outline Post Hartree Fock • Brief introduction • Hartree- Fock • Basis sets • Configuration Interaction • Many-body perturbation theory • Coupled-cluster methods 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 Quantum chemistry is quantum mechanics applied to the electrons in atoms and molecules. Used to determine: 1.- Structure of the molecule (bond lengths, angles) 2.- Electronic energy (bond energies, enthalpies of formation, etc) 3.- Spectra (electronic, vibrational, rotational, etc) 4.- Electrical properties (dipole moment, polarizability) 5.- Molecular orbitals and derived properties such as effective charges, bond orders. 6.- Barriers to reaction and other rate properties. Electronic structure calculations: Methodology and applications to nanostructures Goal of quantum chemistry methods: Multi-electron atoms and molecules (r1,r2,...,rN ) H n (r1 , r2 ,..., rN ) En (r1 , r2 ,..., rN ) H ZI 1 1 2 i 2 i r iI i j rij iI Wave function of many electrons in an external potential (Borh-Oppenheimer) Finite system (no periodic boundary conditions) Electronic structure calculations: Methodology and applications to nanostructures Key quantity: The wave function • Fundamental object in quantum mechanics: wave function (r1,r2,...,rN ) • We want to find special wave functions such that H n (r1 , r2 ,..., rN ) En (r1 , r2 ,..., rN ) where H ZI 1 1 2 i 2 i r iI i j rij iI • 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 Variational principle 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 Quantum Chemistry: Learning a new language CIST, MP2, CC, CSF, TZV, ... Electronic structure calculations: Methodology and applications to nanostructures First (simple) approach: Hartree approximation As a first guess, one may try to write the many-electron wavefunction as a product of one-electron spin-orbitals ji(ri,si): 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 Spin-orbitals One-electron spin-orbitals ji(ri,si) 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 Hartree energy 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 Antisymmetric wave function: Slater determinant 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 Antisymmetric wave function: 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. Slater determinant Electronic structure calculations: Methodology and applications to nanostructures Variational principle on a Slater determinant: Hartree-Fock The minimization of the energy <|H|> 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 Variational principle on a Slater determinant: Hartree-Fock Coulomb term J Identical to Hartree Exchange term K New exchange term Again, this is a mean field self-consistent model. Electronic structure calculations: Methodology and applications to nanostructures Variational principle on a Slater determinant: Hartree-Fock Notice that adding the term i=j in the sums modifies nothing: It cancels out. Electronic structure calculations: Methodology and applications to nanostructures Hartree-Fock equations: Fock operator 2 2 x * x j x 2 2 2 j x e dx ' 1 Vext r e dx r r r r 2me x j j x We can define the following operators by their action on an orbital: J x j j x dx j x r r 2 K x j j x dx j j x and from them we define the Fock operator: 1 F 2 Vext r J x K x 2 so that the HF equations can be written as: Fj j x jj j x j* x j j x r r j x Electronic structure calculations: Methodology and applications to nanostructures Restricted Hartree-Fock (RHF) and Unrestricted Hartree-Fock (UHF) RHF: the spatial part of the one-electron spinorbitals ji(ri,si) is identical for spin-up and spindown (closed-shell) UHF: the spatial part of the one-electron spinorbitals ji(ri,si) 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 Accuracy of HF: Spin issues Slater determinants are always eigenfunctions of Sz. However, they are not necessarily eigenfunctions of S2. For the general case there are always linear combinations of determinants that are eigenfunctions of Sz and S2 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 Accuracy of Hartree-Fock 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 Accuracy of Hartree-Fock Hence, we have to improve over HF: How to do that? Electronic structure calculations: Methodology and applications to nanostructures Basis sets Orbitals are usually expanded in basis sets. Electronic structure calculations: Methodology and applications to nanostructures Basis sets Slater-type orbitals (STOs) n,l,m (r,,) = Nn,l,m, Yl,m (,) rn-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 Basis sets Cartesian Gaussian-type orbitals (GTOs) a,b,c (r,,) = N'a,b,c, xa yb zc exp(-r2), 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(-r2))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 Basis sets 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. tight Gaussian orbital with cusp at r = 0 loose Gaussian medium Gaussian r Electronic structure calculations: Methodology and applications to nanostructures Basis sets: what to do with these building bricks 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 Basis sets: what to do with these building bricks 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 Electron correlation 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 Electron correlation methods: Post Hartree-Fock 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 Configuration Interaction 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 Configuration Interaction 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 Configuration Interaction These determinants are often referred to as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc. Electronic structure calculations: Methodology and applications to nanostructures Configuration Interaction (CI) CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants. S CI=a0HF+aSS+aDD+…= aii Again we use the variational principle and look for the ai 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 Configuration Interaction (CI) S CI=a0HF+aSS+aDD+…= aii Structure of the CI matrix Brillouin’s theorem: Matrix elements between the HF reference determinant and singly excited states are zero. Electronic structure calculations: Methodology and applications to nanostructures Configuration Interaction (CI) S CI=a0HF+aSS+aDD+…= aii 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 Configuration Interaction (CI) S CI=a0HF+aSS+aDD+…= aii 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 Example: correlation in the H2 dissociation problem. Let’s try to illustrate how CI accounts for electron correlation taking as an example the dissociation of the hydrogen molecule H2 Take two 1s orbitals, one in each center of the molecule, A and B A B HF Electronic structure calculations: Methodology and applications to nanostructures Example: correlation in the H2 dissociation problem. The basis determinants for a full CI calculation are the following: Double Single 2+3 triplet SZ=0 Single 2-3 singlet Single : triplet SZ=1 Single : triplet SZ=-1 Electronic structure calculations: Methodology and applications to nanostructures Example: correlation in the H2 dissociation problem. The ground state 0 and the doubly excited 1 can be expanded in terms of the atomic orbitals: ionic covalent 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% H0H0. This is totally wrong!! Electron correlation is missing: electrons try to avoid each other! Electronic structure calculations: Methodology and applications to nanostructures Example: correlation in the H2 dissociation problem. We can solve that by using full CI. The full CI matrix can be shown to be: For 1Sg 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 a1=-a0 Electronic structure calculations: Methodology and applications to nanostructures Example: correlation in the H2 dissociation problem. 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. ionic covalent but now we have an additional triplet component Electronic structure calculations: Methodology and applications to nanostructures Example: correlation in the H2 dissociation problem. All this is conspicuous in the energy diagram: Electronic structure calculations: Methodology and applications to nanostructures Multi-Reference calculations 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 Active space in CI 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 Many-body perturbation theory 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 Møller-Plesset Perturbation Theory First, let us remember the Hartree-Fock equations: 2 2 x * x j x 2 2 2 j x e dx ' 1 Vext r e dx r r r r 2me x j j x These are the self-consistent equations that the single-particle wave functions should fulfill to obtain the minimum energy for a singledeterminant many-body wave function. We have the Fock operator, defined as: 1 F 2 Vext r J x K x 2 so that the HF equations can be written as: Fj j x jj j x Electronic structure calculations: Methodology and applications to nanostructures Møller-Plesset Perturbation Theory The exact Hamiltonian He is: 1 M 2 M 1 1 H e k Vext rk 2 k 1 2 k rk r k 1 We define an unperturbed Hamiltonian H0 as the sum of Fock operators: M H 0 Fk k 1 1 M 2 M H 0 k Vext rk J x k K x k 2 k 1 k 1 k , Electronic structure calculations: Methodology and applications to nanostructures Møller-Plesset Perturbation Theory The exact Hamiltonian He is: 1 M 2 M 1 1 H e k Vext rk 2 k 1 2 k rk r k 1 We define an unperturbed Hamiltonian H0 as the sum of Fock operators: M H 0 Fk k 1 1 M 2 M H 0 k Vext rk J x k K x k 2 k 1 k 1 k , In H0 we are summing up twice the electron-electron Coulomb interaction Electronic structure calculations: Methodology and applications to nanostructures Møller-Plesset Perturbation Theory The exact Hamiltonian He is: 1 M 2 M 1 1 H e k Vext rk 2 k 1 2 k rk r k 1 We define an unperturbed Hamiltonian H0 as the sum of Fock operators: M H 0 Fk k 1 1 M 2 M H 0 k Vext rk J x k K x k 2 k 1 k 1 k , In H0 the electron-electron interaction is considered in an average way. Electronic structure calculations: Methodology and applications to nanostructures Møller-Plesset Perturbation Theory The exact Hamiltonian He is: 1 M 2 M 1 1 H e k Vext rk 2 k 1 2 k rk r k 1 We define an unperturbed Hamiltonian H0 as the sum of Fock operators: M H 0 Fk k 1 1 M 2 M H 0 k Vext rk J x k K x k 2 k 1 k 1 k , The perturbation is therefore H’=He-H0: H 1 1 J x k K x k 2 k rk r k Not that small! Electronic structure calculations: Methodology and applications to nanostructures Møller-Plesset Perturbation Theory M H 0 Fk k 1 1 M 2 M H 0 k Vext rk J x k K x k 2 k 1 k 1 k , The unperturbed Hamiltonian H0 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 0HF. The unperturbed energy is E0=< 0HF | H0 | 0HF > ≠ EHF H 1 1 J x k K x k 2 k rk r k The energy adding the first-order correction is E1=< 0HF | H0+ H’ | 0HF > = EHF!! (it is the matrix element that we have varied to obtain the HF equations) Electronic structure calculations: Methodology and applications to nanostructures Møller-Plesset Perturbation Theory (MP) 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: 2 E ( MP2) i excited i 0 H 0 2 E 0 E i where the excited states (i)0. are eigenstates of the unperturbed Hamiltonian H0, 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 Convergence of MP perturbative series 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 Coupled-cluster methods 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 Coupled-cluster methods and the action of each Ti operator is to create the full set of i-excitations there is only one way to have a single excitation T1, but two ways to generate double excitations: a double excitation (T2) and two consecutive single excitations (T1 T1). etc. Electronic structure calculations: Methodology and applications to nanostructures Coupled-cluster methods and the action of each Ti operator is to create the full set of i-excitations there is only one way to have a single excitation T1, but two ways to generate double excitations: a double excitation (T2) and two consecutive single excitations (T1 T1). etc. 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 Coupled-cluster (CC) methods… in practice 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 M2N+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 Coupled-cluster (CC) methods… in practice 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 M2N+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. Electronic structure calculations: Methodology and applications to nanostructures Coupled-cluster methods Electronic structure calculations: Methodology and applications to nanostructures Coupled-cluster methods Electronic structure calculations: Methodology and applications to nanostructures In summary: • Quantum chemistry methods are very accurate but computationally expensive. • Key points: choice of basis set and methodology used to improve over Hartree-Fock. Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures Electronic structure calculations: Methodology and applications to nanostructures