THE UNIVERSITY OF AKRONDEPARTMENT OF CHEMISTRYCOMPUTATIONAL SPECTROSCOPY3150:713DAVID PERRYExcited States and Electronic SpectroscopyEdward C. LimMarch 27, 2008A Quick Review of the Wavefunction-based ab initio Methods1. Single reference methods(use single determinant SCF wavefunction as a starting point)Hartree-Fock (HF) or SCFConfiguration interaction (CI), CIS, CISD, etc.Møller-Plesset (MP) perturbation theory, MP2, MP3, etc.Coupled-Cluster (CC), CCD, CCSD, CCSDT, etc.Instead of including al configuration to a particular order as in MP theory, eachexcited configuration is included to infinite order via an exponential excitationoperator;€ ΦCC= eˆ T Φ0 ≡ 1 + T + ˆ T 22! + ˆ T 33! + … ⎡ ⎣ ⎢ ⎤ ⎦ ⎥Φ0, 2. Multireference method(Use a linear combination of CSFs, € ψ = CiΦii∑ and varies not only Cis but also thecoefficients of the orbitals that are used to construct the determinants. CSF stands for“configuration state function”.)Multiconfigurational SCF (MCSCF)The most common of these is the Complete Active Space SCF (CASSCF)method. As for any full CI expansion, the CASSCF becomes unmanageably largeeven for small active spaces.Multireference configuration interactions, MRCI (Combines the MCSCF and conventional CI)CASPT2 (implementation of MP2 on CASSCF reference)CASPT2//CASSF is a very popular, albeit very expensive, method.2Coupled Cluster Methods(1)(2)where the cluster operator T is given by(3)The € Ti operator acting on a HF reference wavefunction generates all ith excited Slater determinant, viz.(4)Combining eqs. (2) and (3), the exponential operator eT may be written as(5)The first term generates the reference HF and the second all singly excited states. The firstparenthesis generates all doubly excited states, the second all triply excited states, and so on.Using eq. (1), the Schrödinger equation becomes(6)Multiplying from the left by € Φ0* and integrating, we have(7)As the Hamiltonian operator contains only one- and two-electron operators, we have3(8)When HF orbitals are used to construct the Slater determinant, the first matrix elements are zero (Brillouins theorem), and the second matrix elements are simply two-electron integrals over MOs(10)Eq. (10) is exact and contains all possible excited determinants, (as in full CI). For practical reasons, the cluster operator (T) must be truncated at some excitation level.Inclusion of T1 only does not give any improvement over HF, as the matrix element between the HF and singly excited states are zero.1. The lowest level of approximation is therefore, T = T2. This is referred as Coupled Cluster Doubles (CCD).2. T = T1 + T2 gives Coupled Cluster Singles and Doubles (CCSD).3. T = T1 + T2 + T3 yields CCSDT.CCSD is the only generally applicable coupled cluster method. For CCSD,(15)The CCSD energy and amplitude can be derived by multiplying eq. (6) with a singly exciteddeterminant and integrating(16)4Recently developed CC2 (Coupled Cluster approximate double) is derived from CCSDby including only the double contribution arising from the lowest order in perturbation theory(where perturbation is defined as in MP theory). CC3 is an approximation to CCSDT.Both in terms of computational cost and accuracy,€ HF << CC2 < CCSD < CC3 < CCSDTEOM-CC theoryIn the EOM-CC (equation of motion CC) approach to excited states, the excited-state(x) wave function is given by simple parameterization of the ground-state (g) wave function:€ Ψx = R Ψg ,(1)where R is a linear excitation operator, given by€ R = R0 + R1 + R2 + R3 + … ,(2)€ Rn = 1n!2 rijk…abck…abc…∑ ,(3)with i, j, k, … representing occupied orbitals and a, b, c … representing unoccupied orbitals.The ground-state wave function is given by the CC approximation.€ Ψg = eTΦ0 ,(4)where € ψg is a single Slater determinant. Substitution of eq. (1) and (4) into the Schrödingerequation yields€ HR eT Φ0 = ER eT Φ0 .(5)As R and T are excitation operators that commute eq. (5) can be written as€ e–THeTR Φ0 = ER Φ0 ,(6)5or€ H R Φ0 = ER Φ0(7)where € H is the similarity transformed Hamiltonian. i.e.,€ H = e–T H eT .MMCC (Method of Moments Coupled Cluster)Standard CC methods, such as CCSD, CCSD(T), and EOMCC (equation of motionCC), work well for non-degenerate systems and excited states dominated by single excitationfrom the ground state. For degenerate or quasi-degenerate system and excited states dominatedby double excitation from the ground state, biradicals, and other open-shell systems,renormalized coupled-cluster methods have been developed by Piecuch and co-workers. TheseCC methods, termed as CR-CCSD(T), CR-CCSD(TQ) and CR-EOM-CCSD(T), and sizeextensive formulations known as CR-CC(2,3), which are all derived from the method ofmoments of CC equations, are available in the GAMESS package. These are single-referenceapproaches that eliminate the failure of standard CC methods.The basic idea of the MMCC theory is that of the non-interative, state-specific, energycorrections€ δK(A) ≡ EK – EK(A)which, when added to the energies of the ground (K=0) and excited (K>0) states obtained in thestandard CC or EOM-CC calculations (termed method A), recover the exact (full CI) energiesEK. The main purpose of all MMCC calculations is to estimate € δK(A), such that the resultingMMCC energies€ EKMMCC = EK(A) + δK(A)are close to the corresponding exact energies EK.61. Ground-State MMCC TheoryIn the single-reference CC theory, the ground-state wave function € ψ0 of an N-electron system is given by€ ψ0 = eTΦ0(1)If A represents the standard single-reference CC approximation, the cluster operator is€ T(A) = Tnn =1mA∑(2)where Tn, n=1, 2, … mA are the many-body component of T(A).In all standard CC approximation, the cluster operator T(A) is obtained by solvingthe nonlinear algebraic equations€ Q(A)H(A)Φ0 = 0(3)where€ H(A) = e–T(A)HeT(A)(4)and Q(A) is the sum of projection operators onto the n-tuply excited configurations relativeto reference € Φ. By analyzing the relationship between multiple solutions of thenonlinear equations, representing different CC approximations (CCSD, CCSD(T), etc.),Piecuch and Kowalski arrived at an expression for the non-iterative correction € δ0(A) € δ0(A ) = E0 — E0(A)(5)7where