Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September 20 1 / 24
Hartree-Fock limit In HF the electronic wave function is approximated by a single Slater determinant Not flexible enough to account for electron correlation therefore the Hartree-Fock limit is always above the exact energy Some electron correlation is already found in the electron exchange term HF wf Better description for the wavefunction needed for more accurate results exact wf Lecture 3. Electron correlation methods September 20 2 / 24
Electron correlation Some common nomenclature found in literature Fermi correlation arises from the Pauli antisymmetry of the wave function and is taken into account already at the single-determinant level. Example: The spatial part of wavefunction of two electrons with the same spin can be written as ψ(r 1,r 2 ) = 1 2 [φ 1 (r 1 )φ 2 (r 2 ) φ 1 (r 2 )φ 2 (r 1 )] (1) where r 1 and r 2 are the space coordinates of electrons 1 and 2, respectively. The wavefunction vanishes at the point where the two electron coincide. Around each electron there will be a hole in which there is less electron with a same spin: the Fermi hole. This exchange force is relatively localized. Lecture 3. Electron correlation methods September 20 3 / 24
Electron correlation Static correlation is the deviation from the exact energy caused by an attempt to represent a wavefunction by just one determinant when at least two are really needed Bond dissociation Excited states Near-degeneracy of electronic configurations (for example a singlet diradical CH 2 ). Dynamical correlation is associated with the instantaneous correlation among the electrons arising from their mutual Coulombic repulsion. Quite predictable, the major contribution is around 1 ev for each closed shell pair. Well accounted for by DFT functionals, perturbation theory, configuration interaction, and coupled cluster methods. Coulomb hole: The probability of finding two electrons at the same point in space is 0 as the repulsion becomes infinite. For the wavefunction approximated by Hartree-Fock method this requirement is not fulfilled. There is no explicit separation between dynamical and static correlations. Lecture 3. Electron correlation methods September 20 4 / 24
Configuration interaction (CI) Configuration interaction (CI) has the following characteristics: A post-hartreefock linear variational method. Solves the nonrelativistic Schrödinger equation within the BO approximation for a multi-electron system. Describes the linear combination of Slater determinants used for the wave function. Orbital occupation (for instance, 1s 2 2s 2 2p 1...) interaction means the mixing of different electronic configurations (states). In contrast to the HartreeFock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of configuration state functions (CSFs) built from spin orbitals, Ψ = C i Φ i = C 0 Φ 0 +C 1 Φ 1... (2) i=0 Lecture 3. Electron correlation methods September 20 5 / 24
Configuration interaction (CI) Coefficients from the wavefunction expansion are determined by a variational optimization respect to the electronic energy HC = E CI C (3) where H is the Hamiltonian matrix with matrix elements H ij = φ i H φ j (4) The construction of the CI wavefunction may be carried out by diagonalization of the Hamiltonian matrix, but in reality iterative techniques are used to extract eigenvalues and eigenfunctions (Newton s method). Lecture 3. Electron correlation methods September 20 6 / 24
Configuration interaction (CI) The first term in the CI-expansion is normally the Hartree Fock determinant The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree Fock determinant If only one spin orbital differs, we describe this as a single excitation determinant If two spin orbitals differ it is a double excitation determinant and so on This is used to limit the number of determinants in the expansion which is called the CI-space. The eigenvalues are the energies of the ground and some electronically excited states. By this it is possible to calculate energy differences (excitation energies) with CI-methods. Lecture 3. Electron correlation methods September 20 7 / 24
Example: CI calculation for Helium atom Lets us begin with two-configuration wavefunction expressed as a linear combination of hydrogenic wavefunctions having the form Ψ 1,2 = c 1 ψ 1 +c 2 ψ 2, (5) where ψ 1 arises from the configuration 1s 2 and ψ 2 arises from the configuration 1s2s. Specifically, the two wavefunctions are ψ 1 = 1 1s(1)α(1) 1s(1)β(1) 2 1s(2)α(2) 1s(2)β(2) = 1s 1s (6) ψ 2 = 1 2 ( 1s 2s + 1s2s ) (7) Lecture 3. Electron correlation methods September 20 8 / 24
Example: CI calculation for Helium atom Since both ψ 1 and ψ 2 describe the 1 S 0 states (S = L = 0), there will be no vanishing matrix elements of Ĥ. If we represent these matrix elements by H ij = ψ i Ĥ ψ j (i,j = 1 or 2), the secular determinant to be solved is H 11 E H 12 H 22 E = 0. (8) H 21 The diagonal matrix elements H 11 and H 22 are just the energies of single configurational calculations for the ground and exited states. Keeping in mind that the spin portions integrate out separately to unity ( α α = 1, α β = 0, etc.), we obtain for the H 11 H 11 = 1s(1)1s(2) Ĥ 1s(1)1s(2) = 2ǫ 1 +J 11 (9) where we represent the 1s orbital by subscript 1. The energy ǫ has the same form as the hydrogen atom energy ǫ n = Z2 2n 2 = 2 n 2 (10) Lecture 3. Electron correlation methods September 20 9 / 24
Example: CI calculation for Helium atom Similarly the result for H 22 is derived as H 22 = 1 2 1s 2s 1s2s Ĥ 1s 2s 1s2s (11) Which can be expanded to the form H 22 = 1 ( 1s 2s Ĥ 1s 2s ) 2 1s 2s Ĥ 1s2s + 1s2s Ĥ 1s2s ), 2 (12) which leads to an energy The off-diagonal matrix element becomes H 22 = ǫ 1 +ǫ 2 +J 12 +K 12 (13) H 12 = 1 2 ( 1s 1s Ĥ 1s 2s ) 1s 1s Ĥ 1s2s ) (14) Lecture 3. Electron correlation methods September 20 10 / 24
Example: CI calculation for Helium atom which is simplified because of vanishing spin integrals and thus one can obtain H 12 = 2 1s(1)1s(2) 1 r 12 1s(1)2s(2) (15) Combining the appropriate integrals, the matrix elements become H 11 = 2.75 (16) H 22 = 2.037 (17) H 12 = 0.253 (18) The roots of the quadratic formula (produced by secular determinant) are E = 2.831 and 1.956. The lower root represents the ground state whose experimental energy is 2.903. Note that this improves the single configurational result 2.75. The higher root represents the lowest excited state (experimental energy = 2.15). Lecture 3. Electron correlation methods September 20 11 / 24
Full CI The expansion to the full set of Slater determinants (SD) or CSFs by distributing all electrons among all orbitals is called full CI (FCI) expansion. FCI exactly solves the electronic Schrödinger equation within the space spanned by the one-particle basis set. In FCI, the number SDs increase very rapidly with the number of electrons and number of orbitals. For example, when distributing 10 electrons to 10 orbitals the number of SDs is 63504. This illustrates the intractability of the FCI for any but the smallest electronic systems. Practical solution: Truncation of the CI-expansion Truncating the CI-space is important to save computational time. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. These methods, CID and CISD, are in many standard programs. In the early days of CI-calculations, truncated expansion was constructed by selecting individual configurations. Today, such methods are impractical and even problematic because the relative importance of configurations changes across the potential energy surface. Lecture 3. Electron correlation methods September 20 12 / 24
Static and dynamical correlation CI-expansion truncation is handled differently between static or dynamical correlation. In the treatment of static correlation in addition to the dominant configurations, near degenerate configurations are chosen (referred as reference configurations). Active orbitals (typically valence orbitals) form complete active space (CAS) in which no restrictions are placed on the occupations. The inactive (core) and secondary orbitals are always doubly occupied or unoccupied. Dynamical correlation is subsequently treated by generating excitations from reference space. Unified framework for above treatments is called restricted active space (RAS). Lecture 3. Electron correlation methods September 20 13 / 24
Some problems (and solutions) of CI Excitation energies of truncated CI-methods are generally too high because the excited states are not that well correlated as the ground state is. The Davidson correction can be used to estimate a correction to the CISD energy to account for higher excitations. It allows one to estimate the value of the full configuration interaction energy from a limited configuration interaction expansion result, although more precisely it estimates the energy of configuration interaction up to quadruple excitations (CISDTQ) from the energy of configuration interaction up to double excitations (CISD). It uses the formula δe Q = (1 C 2 0)(E CISD E HF ) (19) where C 0 is the coefficient of the Hartree Fock wavefunction in the CISD-expansion Lecture 3. Electron correlation methods September 20 14 / 24
Size consistency CI-methods are not size-consistent and size-extensive Size-inconsistency means that the energy of two infinitely separated particles is not double the energy of the single particle. This property is of particular importance to obtain correctly behaving dissociation curves. Size-extensivity, on the other hand, refers to the correct (linear) scaling of a method with the number of electrons. The Davidson correction can be used. Quadratic configuration interaction (QCI) is an extension CI that corrects for size-consistency errors in the all singles and double excitation CI methods (CISD). This method is linked to coupled cluster (CC) theory. Accounts for important four-electron correlation effects by including quadruple excitations Lecture 3. Electron correlation methods September 20 15 / 24
Example: CI results for water Compared to CISD-method, the simpler and less computationally expensive MP2-method gives superior results when size of the system increases (MP2 is size extensive). For water monomer, MP2 recovers 94% of correlation energy which remains similar with increasing system (cc-pvtz basis). For stretched water monomer (bond length doubled) CISD recovers only 80.2% of the correlation energy. With the Davidson correction added, the error is reduced to 3%. When the number of monomers increases, the degradation in the performance is even more severe for the equilibrium geometry. For eight monomers, the CISD wavefunction recovers only half of the correlation energy and the Davidson correction remain more or less the constant. Lecture 3. Electron correlation methods September 20 16 / 24
Coupled Cluster method First observations: Coupled cluster (CC) method, especially The CCSD(T), has become the gold-standard of quantum chemistry. CC theory was poised to describe essentially all the quantities of interest in chemistry, and has now been shown numerically to offer the most predictive, widely applicable results in the field. The computatinal cost is very high. So, in practice, it is limited to relatively small systems. Some facts: Coupled cluster (CC) is a numerical technique used for describing many-body systems. The method was initially developed in the 1950s for studying nuclear physics phenomena, but became more frequently used when in 1966 it was formulate for electron correlation in atoms and molecules. It starts from the Hartree-Fock molecular orbital method and adds a correction term to take into account electron correlation. Lecture 3. Electron correlation methods September 20 17 / 24
Coupled Cluster method The wavefunction of the coupled-cluster theory is written in terms of exponential functions: Ψ = eˆt Φ 0 (20) where Φ 0 is a Slater determinant usually constructed from Hartree Fock molecular orbitals. ˆT is an excitation operator which, when acting on Φ0, produces a linear combination of excited Slater determinants. The exponential approach guarantees the size extensivity of the solution. However, it depends on the size consistency of the reference wave function. A drawback of the method is that it is not variational E[Φ] = Φ e ˆTHeˆT Φ Φ Φ (21) which for truncated cluster expansion becomes E[Φ] = Ω H Θ Φ Φ (22) where Ω and Θ are different functions Lecture 3. Electron correlation methods September 20 18 / 24
Coupled Cluster method The cluster operator is written in the form ˆT = ˆT 1 + ˆT 2 + ˆT 3 +... (23) where ˆT 1 is the operator of all single excitations, ˆT 2 is the operator of all double excitations and so forth. The exponential operator eˆt may be expanded into Taylor series: eˆt = 1+ ˆT + ˆT 2 2! +... = 1+ ˆT 1 + ˆT 2 + ˆT 1 2 2 + ˆT 1 ˆT2 + ˆT 2 2 2 +... (24) In practice the expansion of ˆT into individual excitation operators is terminated at the second or slightly higher level of excitation. Slater determinants excited more than n times contribute to the wave function because of the non-linear nature of the exponential function. Therefore, coupled cluster terminated at ˆT n usually recovers more correlation energy than CI with maximum n excitations. Lecture 3. Electron correlation methods September 20 19 / 24
Coupled Cluster method The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of ˆT. The abbreviations for coupled-cluster methods usually begin with the letters CC (for coupled cluster) followed by S - for single excitations (shortened to singles in coupled-cluster terminology) D - for double excitations (doubles) T - for triple excitations (triples) Q - for quadruple excitations (quadruples) Thus, the operator in CCSDT has the form ˆT = ˆT 1 + ˆT 2 + ˆT 3 (25) Terms in round brackets indicate that these terms are calculated based on perturbation theory. For example, a CCSD(T) approach simply means: It includes singles and doubles fully Triples are calculated with perturbation theory. Lecture 3. Electron correlation methods September 20 20 / 24
Example: Atomization energies Table 1: Calculated and experimental electronic atomization energies (kj/mol) Molecule HF CCSD Exp. F 2-155.3 128.0 163.4 H 2 350.8 458.1 458.0 HF 405.7 583.9 593.2 H 2 O 652.3 960.2 975.3 O 3-238.2 496.1 616.2 CO 2 1033.4 1573.6 1632.5 C 2 H 4 1793.9 2328.9 2359.8 CH 4 1374.1 1747.0 1759.3 CCSD calculations produce qualitatively correct results Eventhought CCSD is expensive method, it is unfortunately not very accurate Lecture 3. Electron correlation methods September 20 21 / 24
Example: Reaction enthalpies Table 2: Calculated and experimental electronic reaction enthalpies (kj/mol) Reaction HF CCSD Experiment CO + H 2 CH 2 O 2.7-23.4-21.8 H 2 O + F 2 HOF + HF -139.1-123.3-129.4 N 2 + 3H 2 2NH 3-147.1-173.1-165.4 C 2 H 2 + H 2 C 2 H 4-214.1-209.7-203.9 CO 2 + 4H 2 CH 4 + 2H 2 O -242.0-261.3-245.3 2CH 2 C 2 H 4-731.8-830.1-845.7 O 3 + 3H 2 3H 2 O -1142.7-1010.1-935.5 CCSD recovers majority of the electron correlation energy CCSD calculations do not achieve chemical accuracy (4 kj/mol) Lecture 3. Electron correlation methods September 20 22 / 24
Example: Water energies Table 3: Deviation of CI and CC energies from non-relativistic exact results (within B-O approximation) for H 2 O (mhartree = 2.6255 kj/mol) Method r e 1.5 r e 2 r e Hartree Fock 216.1 270.9 370.0 CID 13.7 34.5 84.8 CISD 12.9 30.4 75.6 CISDT 10.6 23.5 60.3 CISDTQ 0.40 1.55 6.29 CCD 5.01 15.9 40.2 CCSD 4.12 10.2 21.4 CCSDT 0.53 1.78-2.47 CCSDTQ 0.02 0.14-0.02 CI converges (too) slowly to exact energy CC has superior performance but show fluctuations Lecture 3. Electron correlation methods September 20 23 / 24
Historical facts The CC method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics. In 1966 Jiri Cek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. Kümmel comments: I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost gave up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Ji ri s work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then. Lecture 3. Electron correlation methods September 20 24 / 24