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COUPLEDCLUSTER BASED METHODS FOR EXCITED STATE ENERGIES AND GRADIENTS By STEVEN RAY GWALTNEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997 ACKNOWLEDGMENTS Soli Deo Gloria, to God alone be the glory. I thank God for creating a universe so full of wonder and mystery and for creating me and allowing me to explore it. I also thank my wife, Charity, for her constant support and encouragement, even when it meant that she had to do more than her fair share. I love you and will always remember how much you sacrificed for me and for this degree. I want to thank my parents for never letting me settle for being less than I could be, and I thank them for pushing me when I was growing up and then letting me go far away when it was time. I would like to thank my research advisor, Prof. Rodney Bartlett, from whom I have learned so much. I would also like to thank Prof. Ernest Davidson. Who knows if I would be doing quantum chemistry if he had not allowed me to do undergraduate research under him, even though I did not know what I was doing. Thanks go out to all the members of the Quantum Theory Project, to the members of Dr. Bartlett's group, and especially to my office mates through the years. The environment of cooperation here is wonderful. Thanks to Prof. Michael Zerner for help understanding the free base porphin problem. Finally, special thanks go to Prof. Marcel Nooijen. I learned much in our discussions. The work presented here has been funded by the National Science Foundation through a Graduate Research Fellowship and by the United States Air Force Office of Scientific Research through grant number F496209510130 and through AASERT grant number F4962095I0421. Computer time for the porphin calculations was provided by the CEWES Depart ment of Defense Major Shared Resource Center. The CEWES computer center is grate fully acknowledged for their help in getting these calculations done. The EOMCCSD porphin calculations were performed in conjunction with Dr. Howard Prichard at Cray Research/SGI. TABLE OF CONTENTS ACKNOWLEDGMENTS ........... ...................... ii ABSTRACT .... ..................................... vi CHAPTERS 1. AN OVERVIEW OF CURRENT METHODS FOR EXCITED STATES .... 1 CoupledCluster M ethods ....................... ........ 1 Other Current Excited State Methods ............... ....... 3 Gradients for Excited State Methods ......................... 8 2. PARTITIONED EQUATIONOFMOTION COUPLEDCLUSTER METHODS FOR EXCITED STATE ENERGIES ........................... 9 Introduction .................... .................. 9 Theory .................... .. ....................13 Partitioned EOMCC for Excited States ..... .............. 13 MBPT[2] Ground State ............................. 15 Calculations ................... ................... 17 Be Atom ................... .................. 17 Example M olecules ............................... 18 Conclusions ................... ................... 21 3. GRADIENTS FOR THE PARTITIONED EQUATIONOFMOTION MBPT(2) METHOD FOR EXCITED STATES .................. ........ 28 Introduction ................... ................... 28 Theory ..................... .................... 30 EOMCC Gradients ............................... 30 PEOMMBPT(2) Gradients ...........................41 Application to Diatomic Molecules ................ .. .... ..46 Vertical and Adiabatic Excitation Energies .................. 48 Bond Lengths and Vibrational Frequencies .... .......... 50 Application to Polyatomic Molecules ............. .... ...... 51 The S1 State of Ammonia ............... ............ 51 Transbent and Vinylidenic Isomers on the S1 Surface of C2H2 53 Simple Carbonyls ................................ 54 Conclusions ................... ................... 55 iv 4. THE SIMILARITY TRANSFORMED EQUATIONOFMOTION COUPLEDCLUSTER METHOD .......................... 69 Introduction .... .......... ................. ....... 69 The STEOMCC Method ........... Discussion About STEOMCC . 5. THE SPECTRUM OF FREE BASE PORPHIN Introduction ................... Computational Details ............... STEOMCC ................ Basis Set and Geometry ....... Results . . . Ionized and Electron Attached States .. Excited States ............... Discussion .................... Conclusions ................... S73 . 78 . 83 . 83 . 85 . 85 . 86 S88 . 88 . 89 .91 .94 6. GRADIENTS FOR THE SIMILARITY TRANSFORMED EQUATIONOFMOTION COUPLEDCLUSTER METHOD .......... 110 Theory ..................... .................. 110 A applications .. ... . .. .. .. .. 120 7. A FINAL WORD ................... ............... 127 BIBLIOGRAPHY ......... ............................. 129 BIOGRAPHICAL SKETCH ....... ....................... 139 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COUPLEDCLUSTER BASED METHODS FOR EXCITED STATE ENERGIES AND GRADIENTS By Steven Ray Gwaltney December 1997 Chairman: Rodney J. Bartlett Major Department: Chemistry Coupledcluster based methods have become widely used for calculating energies and properties of molecular ground states. In this dissertation, I present a new method, the partitioned equationofmotion coupledcluster method based on a second order many body perturbation theory ground state (PEOMMBPT(2)) for calculating energies of molecular electronic excited states. I also describe how to calculate gradients for two excited state methods, PEOMMBPT(2) and the similarity transformed equation ofmotion coupledcluster singles and doubles (STEOMCCSD) method. These new methods are then applied to several example molecules. CHAPTER 1 AN OVERVIEW OF CURRENT METHODS FOR EXCITED STATES CoupledCluster Methods Coupledcluster theory'14 provides a routinely applicable method for calculating energies and properties of molecules in the lowest state of a given spin and symmetry. By adding triple,15, 16 quadruple,17, 18 and higher excitations, it is possible to converge on the exact answer within a given basis set. The perturbative inclusion of triple excitation effects1921 is now routine, meaning that energies and properties can approach near chemical accuracy for many molecules. Unfortunately, these methods do not, in general, work for electronic excited states. The primary problem with describing excited states with normal coupledcluster (CC) theory is its inherent single reference nature. The wavefunction for the state can be expressed as a linear combination of determinants, with the reference determinant having a coefficient of one. Although in principle coupledcluster theory can be made exact, the cost of such a calculation would increase in cost at the same rate as full configuration interaction (CI), i.e. as the size of the system factorial.22 Therefore, the set of excitations used in the calculation are truncated. In such a truncated CC, a second determinant with a coefficient of more than 0.2 to 0.3 can cause the accuracy of the computed results to drop significantly. However, for an openshell singlet excited state, the second determinant must have a coefficient of one. Coupledcluster theory has been extended in order to be able to describe such multireference states. Multireference coupledcluster theories can be divided into two 2 main categories. They are the Hilbert space approaches2331 and the Fock spaces approaches.3245 A very simplified view of Hilbert space multireference coupledcluster theory is that each determinant in the active space gets its own set of T amplitudes, and a small effective Hamiltonian is diagonalized over the space of these wavefunctions. In the Fock space coupledcluster method, a single exponential operator is used for all determinants in the active space, but this exponential contains operators that change the number of electrons in the system. Hilbert space coupledcluster theory is more suited to studying one or a few states, while Fock space coupledcluster theory is more suited to studying energy differences between many states and to calculating spectra. An alternative method for calculating excitation energies within a coupledcluster framework is to follow the response of the system to an external time dependent perturbation. This was introduced by Monkhorst,46 and has been developed into the coupledcluster linear response (CCLR) method.4754 The same equations can also be derived from an equationsofmotion55 framework based on a coupledcluster ground state. This gives the equationofmotion coupledcluster (EOMCC) method.5659 In EOMCC (or CCLR) the ground state coupledcluster operator is used to perform a similarity transformation of the Hamiltonian, which is then diagonalized over a space of excited determinants. EOMCC and CCLR are identical for excited state energies, but the approximations made by the two methods differ for oscillator strengths and polarizabilities. Typically, the ground state operator and the space of excited determinants is truncated to include only single and double excitations. This gives the EOMCCSD58 59 (or CCSDLR)53 54 method. Models have also been developed which partially include the effects of triple excitations.6063 3 Recently, Nooijen and Bartlett have developed a new method for calculating excited state energies and properties. It is the similarity transformed equationofmotion coupled cluster (STEOMCC) method.6466 For singly excited states, STEOMCC is closely related to the Fock space coupledcluster method,66 but conceptually they are very different. In STEOMCC, ionization potential EOMCCSD67 and electron attachment EOMCCSD68 calculations are done. Information about the states with one fewer and one more electron is then used to perform a second similarity transformation on the Hamiltonian69 such that the terms which directly couple single and double excitations are set to zero. A detailed description of STEOMCC will be given in Chapter 4. Other Current Excited State Methods An attempt to catalog all ab initio methods currently used for calculating excited states would be futile and will not be attempted. Instead, several of the methods which are used most often will be reviewed. But first it will be useful to take a more general view of excited states and break down how the various methods describe excited states. In this view, the excited state energy is divided into several components. They are the zeroth order description of the wavefunction, the orbital relaxation, and the dynamic correlation in the excited state. Properly describing the zeroth order description simply means being able to include all of the determinants which play a dominant role in the excited state. For example, this means being able to give the two determinants in an openshell singlet the same weight. The orbital relaxation refers to how much the orbitals are allowed to rearrange themselves in response to the excitation. 4 The dynamic correlation for the excited state is further broken down into the dynamic correlation for the ground state and the differential dynamic correlation between the ground and excited states. The reason for this division of the dynamic correlation is that, especially for the single reference theories, first a ground state is calculated and then the excited states are built upon it. The justification for this division is that most electrons do not take part in the excitation and therefore still interact with each other in the same way in the excited state as in the ground state. These definitions are not unique, especially when considering orbital relaxation versus differential dynamic correlation. The coupledcluster viewpoint will be taken here. The key to the coupledcluster viewpoint is Thouless' theorem,70 which states that any two nonorthogonal determinants can be written as )i) =eT J), (11) where Ti is a single excitation operator. Therefore, any set of single excitations or any product of single excitations will be considered orbital relaxation, and dynamic correlation will be reserved for "connected" double and higher excitations.13 Table 11 provides a convenient representation for how the selected methods break down into the four terms. A "+" means that the method treats this term adequately. Two pluses mean that the method treats the term very well. A "" means that the term is included, but poorly. Finally, a "0" means that the term is completely neglected. The question mark means that how well the term in question is treated varies based on specifics of the calculation. The first method to be considered is probably the simplest of all ab initio excited state methods. It is the TammDancoff approximation,71, 72 also known as the configuration 5 Table 11: An overview of excited state methods. h o r o ground state differential zeroth order orbital da dynamic dynamic description relaxation correlation correlation correlation CIS + 0 0 RPA + + 0 0 MCSCF ++ (?) ++ CASPT2 ++ (?) ++ + + CIS(D) + + + EOMCCSD ++ + ++ + STEOM SD + + ++ ++ (?) CCSD MRCI ++ (?) ++ (?) ++ (?) ++ (?) interaction singles (CIS)7375 method. The names will be used interchangeably. In CIS the excited state is determined from diagonalizing a Hamiltonian matrix made up of single excitations with respect to the HartreeFock reference determinant. Since, by Brillouin's theorem,22 single excitations cannot couple with the HartreeFock determinant, this diagonalization leaves the ground state unchanged and gives excited states. Because all possible single excitations are included, any excited state which corresponds to a single electron being promoted can be described. However, the orbital relaxation is extremely limited and the method contains no dynamic correlation. The next method is the random phase approximation (RPA).76, 77 Although RPA can be derived several ways, the one most useful here is to consider RPA as the response of a wavefunction written as12, 13 1 ) = (1 + Ti + T2) TI2o). Clearly, then, RPA also contains no dynamic correlation, but it does contain more orbital relaxation than CIS. The next method to be considered is multiconfigurational SCF (MCSCF).78, 79 In MCSCF, a set of configurations is chosen. Then the coefficient for each configuration 6 is variationally optimized while simultaneously optimizing the orbitals from which the configurations are built. Since the procedure involves variationally choosing the optimal orbitals, orbital relaxation can be treated quite well. Because of the flexibility to choose any configurations wanted, any excited state, no matter its excitation level, can be described. Thus the zeroth order description can also be very good if the configurations are chosen properly. Because of the multideterminantal nature of the final wavefunction, some dynamic correlation is recovered, but normally the amount is small. A special way of choosing the configurations is to choose a set of orbitals and to create all possible configurations with a given number of electrons distributed among that set of orbitals. This gives rise to the complete active space SCF (CASSCF)80 variant of MCSCF. The CASSCF wavefunction can be used as the zeroth order wavefunction in a multireference perturbation theory expansion. The most popular of the methods derived this way is the CASPT2 method.81, 82 Since it is based on a CASSCF wavefunction, the arguments above about the zeroth order description and the orbital relaxation still apply. The perturbation theory then allows a treatment of the dynamic correlation. Overall, this method works well, but because of issues involving properly choosing the active space and a potential intruder state problem, it is sometimes difficult to get consistently good results. Starting with the CIS wavefunction being used as the zeroth order wavefunction in a perturbation expansion, it is possible to derive the CIS(D) method.83 CIS(D) shares CIS's weakness in term of orbital relaxation, but it does allow for a second order perturbation description of both the ground state dynamic correlation and the differential dynamic correlation. 7 The next method to be considered is EOMCCSD,58 59 discussed above. Since the excited state wavefunction consists of both single and double excitations, it can have a proper zeroth order description of not only singly excited states, but also doubly excited states and states which are mixtures of singles and doubles. On the other hand, since the reference state determines the orbitals, they are fixed, and the excited state wavefunction has only limited flexibility to relax them through the other singles and through the doubles acting as a single times a single. This linear excited state also limits to what extent differential dynamic correlation is included. More is probably included than for any other method considered so far, but in EOMCCSD, unlike CASPT2, the differential dynamic correlation needs to compensate for the orbital relaxation. The ground state dynamic correlation is very well described through the ground state coupledcluster calculation, and therefore for states without large orbital relaxation and differential correlation, EOM CCSD works well. In STEOMCCSD,64, 66 the use of a singles only excited state operator limits the zeroth order description to those kinds of states describable in CIS. The orbital relaxation is also somewhat limited, but some orbital relaxation comes indirectly from the con sideration of the ionized and electronattached states. The ground state coupledcluster calculation again describes the ground state dynamic correlation very well. The differen tial dynamic correlation is handled through the second similarity transformation, which is built from an active space of ionized and electronattached states. How well this works in practice for a large variety of molecules is not clear yet, but the amount of differential correlation included will depend heavily on the completeness of the active space.66 8 Finally, there is multireference CI (MRCI).84 Because of the flexibility offered by a MRCI calculation, it is possible to achieve extremely good accuracy by properly accounting for all four parts of the excited state energy. But MRCI has two very significant drawbacks. It often takes an experienced practitioner to properly build the trial wavefunction, and the cost of the method is such that it can only be applied to very small systems. It is not size extensive, either, and its lack of size extensivity will cause the error in the calculation to grow as the molecule grows. Gradients for Excited State Methods One of the most important advances in quantum chemistry was Pulay's derivation of gradients for SCF wavefunctions.85 This allowed for the convenient computation of the derivative of the energy with respect to a general perturbation. By taking the perturbation to be moving an atom, his discovery made practical the exploration of ground state potential energy surfaces.86 People could readily find minima and transition states and calculate vibrational frequencies. Pople et al.87 extended gradient technology to correlated methods with the development of gradients for MBPT(2). For a review of current analytical gradient techniques, see Ref. 88. In terms of excited states, gradients were first introduced for the MCSCF method.86 Although not in general use, gradients are also available for MRCI wavefunctions.89 Gradients for CIS were first developed in 1992.75 Then came gradients for EOMCCSD in 1993,9092 followed by CIS(D) gradients.93 For a general overview of gradients for coupledcluster based methods for excited states, see Ref. 94. CHAPTER 2 PARTITIONED EQUATIONOFMOTION COUPLEDCLUSTER METHODS FOR EXCITED STATE ENERGIES Introduction The equationofmotion coupledcluster (EOMCC) method5759 is a conceptually single reference, generally applicable, unambiguous approach for the description of excited,5759 electronattached68 or ionized states.67 All follow from simple consideration of the Schrodinger equation for two states, a reference state ITO) (not necessarily the ground state), and an excited (electronattached or ionized) state I',,). Considering H to be in second quantization, where the number of particles is irrelevant, the Schr6dinger equations for the two states are HI o) = Eol o), (21) and HIT,) = EI4,). (22) By choosing to represent the excited state eigenfunction as T.) = 7Z[lo), (23) it is readily obtained that [H, Rn]o}) = R, Io) (24) for w, = E, Eo, from subtracting Eq. (21) from Eq. (22) after left multiplication by 7K. 10 The choice of 7Z defines the particular EOM with i, j, k, ... indicating occupied orbital indices and operators, while a, b, c, ... are unoccupied orbitals and operators. Also, p, q, r, refer to orbitals and operators of either occupation. For electronic excited states, EE = r() + E r ({ai + + E r (s){ a ibtj+ (25) i,a i for electron attachment IZEA a E a(){a} E rKb(){atjb} ' (26) a a,b,j and for ionization RIP = f) {ri){} + 2J E .(\)fi j + f ., (27) i ij,a Coupledcluster theory is introduced by choosing Wo) = eTIO) (28) where T is the usual excitation operator, and 10) represents some independent particle model reference. Since [7,, T] = 0 for any of the above choices, we can commute the operators to give [H, J]O) = (HZ)710) = wj,,Z10), (29) where H = eTHeT, (210) and (HR 1,) indicates the open, connected terms that remain in the commutator. Two operators being connected means that they share at least one index. In this way, all of 11 the ground state CC information is contained in H, now generalized to have three and fourbody terms.58 Note, H is also nonHermitian, necessitating that both its left C, and right R, eigenvectors, which form a biorthogonal set, LiR.j = 6ij, be considered for a treatment of properties. In matrix form, (HR,)c = Ruw. (211) LH = Lw,. Consequently, EOMCC reduces to a CIlike equation that provides the relevant excitation energies directly. Besides the obvious approximations, such as T = Ti + T2, and R, being limited to single and double excitations, which defines EOMCCSD, and various triple excitation extensions, EOMCCSD(T),61 EOMCCSDT1,61, EOMCCSD(T),63 and EOMCCSDT 3,63 one can conceive of many other approximations to the basic EOMCC structure. Instead of Eq. (28), for the reference state a perturbation approximation might be chosen, such as m))= (1 + T(1)+ T(1)T(1) + (2) +. ..) (212) S2 (212) where one truncates at a particular order, m. (See Ref. 95 for the coupledcluster, non HartreeFock definition of the various orders.) Alternatively, rather than retaining the full perturbation approximation to Io(m)), we could truncate H itself to some order.96, 97 It is possible to conceive of perturbative approximations to R,, too. The latter are, perhaps, most easily viewed from the partitioning approach to perturbation theory.98 That is, Eq. (212) can be partitioned into the spaces P and Q, where P represents the principal 12 configuration space (of dimension p) and Q (of dimension q) represents its orthogonal complement. Then it is well known that we can consider an effective Hamiltonian, Hpp(w ) = Hpp + HpQ(wK HQQ)Hp, (213) whose eigenvectors are solely defined in the P space Hppcp = cpwu (214) for the first several eigenvalues, w,. Expanding the inverse in Eq. (213) provides a series of perturbative approximations to Hpp or to the eigenvectors cp in Eq. (214).98 With P chosen to be the space of single excitations, and Q that of double excitations, such partitioned EOMCCSD results, first presented in 198957 have been shown to retain most of the accuracy of the full EOMCCSD method. Other approximations can be made. To introduce triple and quadruple excitations, selective double excitations could be retained in P and at least diagonal approximations could be made for the triple and quadruple excitation blocks of HQQ. In this chapter such partitioned and perturbationbased approximations to the full EOMCCSD method will be reconsidered. It will be demonstrated that very good accuracy may be obtained within a much less expensive computational structure. In particular, whereas the full EOMCCSD is proportional to noNAirt oc n6, for n basis functions, a n cNc rt cx n procedure will be developed that shows promise for large molecules. 13 Theory Partitioned EOMCC for Excited States In a typical EOMCCSD calculation, the excitation energy and excited state properties are calculated via diagonalization of the nonsymmetric matrix,58 () [ ss HSD (215) ()= [HDS HDD (215) where Hss stands for the singlessingles block of the matrix, etc. The major step in an iterative diagonalization99 of the matrix is multiplying the matrix by a trial vector C. In the EE case, the equations for this multiplication are as follows:58' 60 [IssC]a = FacCF FmiCi + WamieC, (216) e m cm [HSDC]? = FmC, + i WamifC WmniCa, (217) em me f rnne [HDSC/]j =P(ab) E WmaijCb, + P(ij) E WabejCf m C + P(ab) Wbn fe ) (218) f \me / P(Cj) (nW Iie n me abriaa 1 [HDDC]j =P(ab) Fb P(j) FmC + b WafC' e m ef + WmnijCn + P(ab)P(ij) E WbmjeC mn emn 2 rP(abE) EWnc f C ea (219) S\mnec + P(ij)E EWnfC, )III n \wef 14 The permutation operator P(qr) is defined as P(qr) (...q ... r...) = ...q.. .r...) (...r...q...) (220) Fpq and Wpqrs are presented elsewhere.58 In the partitioned scheme, HDD is replaced by Ho1n, where H0 is the usual M0ller Plesset unperturbed Hamiltonian from manybody perturbation theory.13 In other words, the unfolded matrix analogous to Eq. (213) to be diagonalized is approximated by IT = /SS HSD \ (H= F HoR ) (221) \HDS HODD/ and Eq. (219) becomes [HODDC]b = P(ab) E fE ,Cje P(ij) E ,,m a, (222) C m Here, fpq is the Fock operator. In the case when the reference function is composed of canonical or semicanonical HartreeFock orbitals, HooD becomes diagonal with differences of orbital energies on the diagonal. From examining the equations, it is clear that the method is formally an iterative n5 method, and the results are invariant to rotations among occupied or unoccupied orbitals. This partitioning scheme differs from that presented in Ref. 57 in two ways. Geertsen et al.57 included the full H elements on the diagonal instead of just including Ho elements in the doublesdoubles block, thus destroying the orbital invariance. Also, Geertsen et al.57 did not include the three body terms (the last two terms in Eq. (218)) in their work. After a matrix in the form of Eq. (213) has been diagonalized, the eigenvectors consisting of single excitations and double excitations are built. Therefore, the same techniques used to calculate the properties of a full EOMCCSD wavefunction58 can 15 be used to calculate properties of the partitioned EOMCCSD (i.e. PEOMCCSD) wavefunction. Since H is not Hermitian, its left and right hand eigenvalues are the same, but the eigenvectors differ.58 The equations for the left hand side are similar to those presented above. MBPT[2] Ground State The first logical perturbative approximation to H =eTHeT = (HeT)c (223) = Fpq {p q} + 1 Wpqrs{ptqtsr} + higher order terms (22 pq pqrs is obtained by keeping only terms through second order. Such an approximation defines an EOMMBPT[2] method for excited states (i.e. an EOMCC calculation based on a MBPT[2] ground state instead of a coupledcluster ground state).97 If we insist upon a method correct through secondorder for the nonHartreeFock case, the explicit equations for H(2 are Fai = 0, (224) Fab = fab + tafmb t (mallfb) ta (mnllbe), (225) m frm, emn Fj = fij + tfie + t,(imije) Z te(im ef), (226) e em cfm Fia = fia + te (iml ae), (227) em Wabij = 0, (228) Wijkl = (ijkl) + P(kl) ft(ike) + ~{(ij Ief), (229) e oef 16 Wacd = (ab lcd) P(ab) t (amllcd) + t+n (mnllcd), (230) m mn Waibc = (aillbc) ta(millbc), (231) Wijka (jkllia) + E t (jkllea), (232) C' ijab = (ijlab), (233) Waijb = (ailjb) + t t(ailleb) t (milljb) tjr(milleb), (234) e m em 1 Wabci =(abllci) P(ab) t ( + tImfe + E t (abllce) P(ab) E trn(mbllci), m e m Wijk =(ia P(ij) T k(im\je) + yt(iaef) en ef (236) + Etf + (imjk) P(ij) E t(ia I ek). e m e In all of these equations tq and tO refer to their values through first order. That is, tl1] = 1 and t[lab (abllij) i ~ , ij ~ ,+j6ah' Since approximating the excited state with a partitioned EOMCC calculation and ap proximating the ground state with a MBPT(2) calculation are independent approximations, we can obviously combine them into a partitioned EOMMBPT(2) (PEOMMBPT(2)) method. When combined, the H elements Wijkl, Wijab, and the most numerous, Wabcd, are not needed. Therefore, in the HartreeFock case, the (abllcd) integrals never need to be calculated. The (abllcd) integrals would contribute to the Wabci H element multiplied by a T11, but in the HartreeFock case all TI ]'s are zero. A few n6 terms still remain in the calculation of the H elements, but these terms only need to be calculated once. In most calculations the cost of the iterative n5 step in the excited state calculation will dominate over the n6 step involved in calculating the H elements. 17 Calculations Be Atom Table 21 lists energies calculated using the four methods described above (EOM CCSD, PEOMCCSD, EOMMBPT(2), and PEOMMBPT(2)) for the first few singlet excited states in beryllium. The basis set from Ref. 100 was used in the calculations. From the mean absolute errors, it would appear that only the EOMCCSD calculation is able to reproduce the full CI excitation energies (which are exact within the basis set used). However, if the 11D state is excluded from the determination of the mean absolute error, the errors become 0.010 eV for EOMCCSD, 0.151 eV for PEOMCCSD, 0.431 eV for EOMMBPT(2), and 0.300 eV for PEOMMBPT(2). With an approximate excitation level (AEL) value of 1.60, the I'D state is dominated by double excitation character. The AEL is a measure of the number of electrons excited in an excitation.58 A value of 1.00 is a pure single excitation, while 2.00 corresponds to a pure double excitation. Since the doublesdoubles block of the EOM wavefunction is approximated in this partitioned scheme, it is reasonable to expect any state with appreciable double excitation character will be poorly described by this partitioned EOM calculation. Even with the I'D excluded, the calculations based on a MBPT(2) ground state are still very poor. This poor behavior can be ascribed to the inadequacy of the MBPT(2) wavefunction in describing the ground state of Be. In a CCSD calculation the largest T2 amplitude is 0.065, while the largest TT1 amplitude in the MBPT(2) calculation is only 0.034. Also, the MBPT(2) calculation only obtains 73% of the correlation energy recovered by the CCSD calculation. Clearly, MBPT(2) is not adequate to describe the 18 ground state, and any excited state calculation based on that MBPT(2) ground state will suffer accordingly. In Table 22 the first few triplet excited states of beryllium starting from the singlet ground state are presented. Since all of the calculated states are single excitations, the EOMCCSD method does exceedingly well, while the PEOMCCSD also does a good job of describing the states. Example Molecules While comparisons with full CI calculations are useful, it is also informative to look at the performance of the methods for more common molecules. These calculations will also provide an opportunity to compare the current methods with other single reference methods used today. Tables 23 to 26 present calculations on four molecules: formaldehyde, acetaldehyde, ethylene, and butadiene. The calculations are performed at the MP2/631G* geometries given in Refs. 101103. A 6311(2+,2+)G** basis setl01 is used for formaldehyde and ethylene, with a 6311(2+)G* basis set102 being used for acetaldehyde and butadiene. All electrons are correlated except in butadiene, where the first four core orbitals are dropped. All of the methods presented, except for the CISMP2,75 can be viewed as approxi mations to the full EOMCCSD method. TDA (or CIS)104 is the crudest of the methods in that the excited state is given as a linear combination of single excitations out of a single reference ground state. This method includes no dynamic correlation. CIS(D)83 provides a noniterative n5 perturbation correction to the CIS energy. CISMP2 also provides a correction to the CIS energy, but the method is not sizeconsistent and scales as n6.83 The partitioned methods provide an iterative n5 excited state method based on either a 19 n5 ground state (PEOMMBPT(2)) or an iterative n6 ground state (PEOMCCSD). The full EOMCCSD method is an iterative n6 method. Except for two differences, our assignment of states agrees with the assignments of Wiberg et al. and with the corrected assignments of HeadGordon et al.,105. The states have been ordered based on their experimental excitation energies, or, where not available, their EOMCCSD excitation energies, instead of their CIS excitation energies. Also, HeadGordon etal.83 had incorrectly assigned the 'A1 EOMCCSD state at 9.27 eV in formaldehyde to the valence 41A, state. Based upon the state's properties, including an 'Al EOMCCSD state has been reassigned to the Rydberg 3'A, state. The 4'A, state (the 7r*+r state), with an of 10.00 eV. Considering the basis sets and the relatively inexpensive methods used here, these calculations are not meant to be definitive. For previous results on these molecules see the references in Refs. 101103. Instead, the mean absolute errors of the methods compared to experiment and compared to EOMCCSD will be used to access the quality of the methods. The mean absolute error for each of the methods with respect to the experimental values given is 0.66 eV for CIS, 0.41 eV for CISMP2, 0.32 eV for CIS(D), 0.17 eV for PEOMMBPT(2), 0.20 eV for PEOMCCSD, and 0.14 eV for the full EOM CCSD method. Compared to the more complete EOMCCSD method, the mean absolute errors of the various methods are 0.70 eV for CIS, 0.39 eV for CISMP2, 0.35 for CIS(D), 0.12 eV for PEOMMBPT(2), and 0.16 eV for PEOMCCSD. 20 As should be expected from its simple nature, TDA performs poorly. The average errors for CISMP2 and CIS(D) are similar, although, as previously noted,83 the CISMP2 energies are more erratic. The PEOMMBPT(2) and PEOMCCSD energies are also similar, seldom differing by 0.1 eV. This suggests that for these cases where MBPT(2) is able to well represent the ground state, PEOMMBPT(2) should be able to well describe singly excited states. A balance argument between the ground and excited state would also tend to favor the PEOMMBPT(2) method, since the partitioning, as discussed above, can be viewed as a secondorder in H perturbation expansion for the excited state. Looking only at the valence states, as designated by Wiberg et al.,101103 the mean absolute errors from the EOMCCSD energies are 0.45 eV for CIS, 0.68 eV for CISMP2, 0.33 eV for CIS(D), 0.23 for PEOMMBPT(2), and 0.38 eV for PEOMCCSD. This is compared to the Rydberg states, where the mean absolute errors from the EOMCCSD energies are 0.76 eV for CIS, 0.32 eV for CISMP2, 0.35 eV for CIS(D), 0.10 for P EOMMBPT(2), and 0.11 eV for PEOMCCSD. It appears that the partitioned methods on average do not describe valence states as well as they describe Rydberg states. Since orbital relaxation is often greater for valence states, it is possible the reduced relaxation available by approximating the doubles vector might cause larger errors for valence states. These calculations also give a good chance to measure the time savings of the partitioned method. For example, for butadiene the full EOMCCSD calculation took 25905 seconds on an IBM RS/6000 model 590 to calculate eighteen excited states (both right and left hand sides). To calculate the same states with the PEOMCCSD method took 3998 seconds. These times only include the time for the excited states. PEOM 21 MBPT(2) also saves time in the calculation of the ground state and in the formation of H, as well as saves disk space. Conclusions In this chapter a formalism is presented and results are given for partitioned methods based on the equationofmotion coupledcluster theory, where the ground state can be described by either a CCSD or an MBPT(2) wavefunction. The partitioned methods provide an iterative n5 method (plus an n6 step for forming H elements) for excited states. When the ground state of the system is well described by an MBPT(2) wavefunction, the PEOMMBPT(2) method provides an inexpensive way to accurately calculate the energies and properties of singly excited states. For systems less well described by a MBPT(2) wavefunction, the PEOMCCSD method is a generally accurate, but more economical, alternative to a full EOMCCSD calculation. For vertical excitation energies, PEOMMBPT(2) is a superior n5 method to CIS(D). Table 21: Be singlet excitation energies (in eV) EOM CCSD 5.323 6.773 7.139 7.468 8.055 8.084 8.309 8.548 8.583d 8.700 PEOM CCSDb 5.666 6.985 10.579 7.653 7.892 8.203 8.432 8.548 8.694 8.792 0.485 (0.151)e a)Ref. 60. The AEL is for the EOMCCSD wavefunction. b)Because of a different implementation of the partitioning, these numbers are slightly different than those in Ref. 57. C)Ref. 100. d)Ref. 60. e)Average error without 11D double excited state. state liP 2'S I'D 21P 2'D 31S 31P 3'D 4'S 4'p AELa 1.07 1.06 1.60 1.06 1.21 1.04 1.05 1.09 1.04 1.04 EOM MBPT(2) 4.869 6.329 6.682 7.023 7.618 7.648 7.873 8.115 8.267 0.428 (0.431)e PEOM MBPT(2) 5.228 6.576 10.170 7.224 7.477 7.788 8.011 8.132 8.278 8.374 full CIc 5.318 6.765 7.089 7.462 8.034 8.076 8.302 8.536 8.600 8.693 mean abs. error 0.014 (0.010)e 0.578 (0.300)e Table 22: Be EOM CCSD 2.729 6.447 7.301 7.748 7.991 8.278 8.456 8.58d 8.70d 0.008 23 triplet excitation energies (in eV) PEOM CCSDb 2.819 6.583 7.424 7.866 8.089 8.366 8.539 8.647 8.766 0.104 EOM MBPT(2) 2.271 5.997 6.868 7.316 7.557 8.025 0.436 PEOM MBPT(2) 2.366 6.140 7.006 7.448 7.668 7.949 8.121 8.228 8.348 a)The AEL is for the EOMCCSD wavefunction. b)See footnote b, Table 21. c)Ref. 100. d)Ref. 59. AELa 1.01 1.04 1.04 1.04 1.04 1.03 1.03 state 13P 23S 23P 13D 33S 43P 23D 43S 53P full CI 2.733 6.444 7.295 7.741 7.985 8.272 8.449 8.560 8.686 mean abs. error 0.321  24 Table 23: Formaldehyde (energies in eV) CIS PEOM PEOM EOM CISa MP2a CIS(D)b MBPT(2)c CCSDC CCSD Exp.a 11A2(V) 4.48 4.58 3.98 4.31 4.41 3.95b 4.07 11B2(R) 8.63 6.85 6.44 7.10 7.10 7.06b 7.11 21B2(R) 9.36 7.66 7.26 7.97 7.95 7.89b 7.97 2'A1(R) 9.66 8.47 8.12 8.02 8.01 8.00b 8.14 21A2(R) 9.78 7.83 7.50 8.25 8.23 8.23b 8.37 31B2(R) 10.61 8.46 8.21 8.99 8.97 9.07b 8.88 I'BI(V) 9.66 9.97 9.37 9.61 9.68 9.26b 31Ai(R) 10.88 8.75 8.52 9.24 9.21 9.27c 41B2(R) 10.86 8.94 8.63 9.39 9.38 9.40b 4'1A(V) 9.45 9.19 8.80 10.08 10.24 10.00c a)Ref. 102. b)Ref. 83. C)Present work. 25 Table 24: Acetaldehyde (energies in eV) CIS PEOM PEOM EOM CISa MP2a CIS(D)b MBPT(2)C CCSDc CCSD Exp.a I'A"(V) 4.89 5.27 4.28 4.65 4.71 4.26b 4.28 21A'(R) 8.51 6.71 6.13 6.88 6.84 6.78b 6.82 3'A'(R) 9.22 7.57 7.04 7.57 7.52 7.49b 7.46 2'A"(R) 9.37 7.37 6.90 7.70 7.64 7.64b 41A'(R) 9.30 8.00 7.42 7.77 7.70 7.68b 7.75 51A'(R) 10.19 8.09 7.70 8.42 8.35 8.39b 8.43 61A'(R) 10.26 8.08 7.70 8.53 8.47 8.51b 8.69 3'A"(R) 10.31 8.10 7.74 8.58 8.51 8.57b 4'A"(V) 9.78 10.34 9.34 9.61 9.65 9.23c 71A'(V) 9.73 9.07 8.50 9.58 10.07 9.44C a)Ref. 102. b)Ref. 83. C)Present work. Table 25: Ethylene (energies in eV) CIS PEOM PEOM EOM CISa MP2a CIS(D)b MBPT(2)c CCSDc CCSD Exp.a llB3u(R) 7.13 7.52 7.21 7.45 7.51 7.31b 7.11 11BluV) 7.74 8.39 8.04 8.20 8.36 8.14b 7.60 l1Big(R) 7.71 8.14 7.84 8.08 8.14 7.96b 7.80 11B2g(R) 7.86 8.12 7.86 8.12 8.18 7.99b 8.01 21Ag(R) 8.09 8.42 8.18 8.43 8.48 8.34b 8.29 2'B3u(R) 8.63 8.92 8.69 8.95 9.00 8.86b 8.62 I'Au(R) 8.77 9.00 8.80 9.07 9.12 9.01b 3'B3u(R) 8.93 9.14 8.96 9.23 9.28 9.18b 2'B1u(R) 9.09 9.38 9.18 9.42 9.51 9.39C 9.33 21Big(R) 9.09 9.31 9.12 9.42 9.48 9.38c 9.34 a)Ref. 101. b)Ref. 83. c)Present work. Table 26: Butadiene (energies in eV) CIS PEOM PEOM EOM CISa MP2a CIS(D)b MBPT(2)c CCSDc CCSD Exp.a l'Bu(V) 6.21 7.00 6.29 6.52 6.63 6.42b 5.91 1'Bg(R) 6.11 6.73 6.11 6.40 6.39 6.20b 6.22 11Au(R) 6.45 7.03 6.44 6.72 6.73 6.53b 21Au(R) 6.61 7.11 6.55 6.85 6.84 6.67b 6.66 21Bu(R) 6.99 7.58 7.03 7.29 7.34 7.17b 7.07 21Bg(R) 7.22 7.66 7.17 7.47 7.47 7.31b 7.36 2'Ag(R) 7.19 7.74 7.19 7.43 7.44 7.10b 7.4 31Bg(R) 7.25 7.74 7.24 7.54 7.53 7.39b 7.62 41Bg(R) 7.39 7.87 7.40 7.70 7.69 7.55b 7.72 3'Ag(R) 7.45 7.88 7.44 7.74 7.73 7.61b 31Bu(R) 8.05 8.40 8.01 8.32 8.31 8.21c 8.00 3'Au(R) 7.78 6.75 7.73 8.07 8.07 7.92c 8.18 41Au(R) 7.92 7.66 7.86 8.19 8.18 8.06c 8.21 a)Ref. 103. b)Ref. 83. C)Present work. CHAPTER 3 GRADIENTS FOR THE PARTITIONED EQUATIONOF MOTION MBPT(2) METHOD FOR EXCITED STATES Introduction The equationofmotion coupledcluster method with single and double excitations for excitation energies (EOMCCSD)58 59 provides an accurate method for calculating the energy and properties of many excited states of molecules. With the development of gradients for EOMCCSD9092 it has become possible to study the excited state potential energy surfaces, just as the development of coupledcluster gradients for ground state methods106109 made routine the study of ground state potential energy surfaces. More recently, Stanton and Gauss have developed second derivatives for EOMCCSD110 and gradients for ionization potential EOMCCSD (IPEOMCCSD).67 The key to all coupled cluster gradients has been the recognition of how the interchange theorem of Dalgarno and Stewart,"1 initially used by Adamowicz et al.106 and Bartlett107 for coupledcluster gradients and sometimes known in quantum chemistry as the Zvector method,112 could be used to avoid computing the response of the ground state T amplitudes to each perturbation. One of the primary problems with the EOMCCSD method is its cost. An EOM CCSD gradient calculation involves four iterative steps which scale as nccNvirt, where nocc stands for the number of orbitals occupied in the reference determinant, and Nvirt stands for the number of orbitals unoccupied in the reference determinant These iterative no~N4irt steps are calculating the ground state T amplitudes, calculating the 29 right hand eigenvector 7, calculating the left hand eigenvector C, and inverting H as part of calculating the Z vector. In the philosophy of replacing ground state cluster amplitudes with secondorder perturbation approaches,113, 96 Stanton and Gauss97 developed an EOMCC method based on a H that was expanded through second order. For calculations based on a Hartree Fock reference, expanding H through second order and replacing the CCSD amplitudes with the MBPT(2) amplitudes in the EOMCC equations are functionally equivalent. This truncation of H reduced the cost of calculating the T amplitudes to an noniterative nocc(noc + Nvirt)4 step and reduced the cost of inverting H to n,,.ocir.97 However, the iterative norcNirt steps for calculating R and L still remained. Since this did not significantly reduce the cost, the method was not very useful in practice. In order to further reduce the cost and to create a more practical method geared toward providing a robust secondorderlike treatment of excited states, in analogy with the robust MBPT[2] for ground states,114 we proposed partitioning away the doubles doubles block of H. It is in the doublesdoubles block of H that all of the iterative N.ccNirt steps in calculating R and arise. In practice this means replacing H in the doublesdoubles block with Ho, which is diagonal in the HartreeFock case. Thus, the iterative nccNirt steps involved in calculating R and are replaced with iterative ccNirt steps. The overall effect is that the cost is significantly reduced, with little loss in accuracy in vertical excitation energies.114 Now, however, the noniterative noccNvirt step involved in calculating H elements can also become significant. The point at which the cost of calculating the one n' NaN step in H becomes dominant over the cost of calculating the three iterative n Cirt steps 30 in the excited state can be estimated. The two will have approximately equal cost when the number of virtual orbitals is three times the number of iterations. Since it typically takes around fifteen iterations to solve for each R and each L, this would imply that the cost of calculating H would become significant when the number of virtual orbitals is more than about ninety per excited state being calculated. In practice, some of the terms not included in this estimate also contribute significantly to the cost, increasing the breakeven point substantially. Also, when calculating vertical excitation energies, multiple excited states can be calculated while forming H only once. Theory EOMCC Gradients Before we beginning with an overview of EOMCC theory and EOMCC gradients, it will be helpful to define some notation. Given a set of spin orbitals and a reference determinant 10), the labels i,j, k,. represent spin orbitals occupied in 10), the labels a, b, c, .represent spin orbitals unoccupied in 10), and the labels p, q, r, represent spin orbitals whose occupation is not specified. The reference determinant is usually, but not necessarily, the SCF determinant. The space of all possible determinants with n electrons formed from the spin orbitals is represented by jh). The space Ih) is then divided into Ih) = Ip) D Iq). Here, 1p) represents the space spanned by the operators C and R, where it is assumed that T, L, and R all have the same rank. Finally, Ip) is divided into lp) = 10) Ig). For EOMCCSD g) consists of all determinants singly and doubly excited with respect to 10). This is the same notation used by Stanton.90 The ground state coupledcluster equations can be written as (01H 0) = Ecc, (31) and (gIHrO) = 0. (32) In pure CC methods (e.g. CCSD or CCSDT), H can be written as S= eTHe = (He (33) where the subscript c indicates that the terms are connected; they share at least one index in common. T = tai + {a tibtj} +... (34) i,a '., a,b is an excitation operator which accounts for the ground state correlation. In the iterative triples methods such as the CCSDTn methods,15 or in manybody perturbation theory, H does not have such a concise form. Therefore, Eqs. (31) and (32) will be taken as the defining equations of H. It is now possible to write the EOMCC equations58 as (0oJHlp) = E(Ollp), (35) and (plHR I0) = E(pIIIO). (36) Here, E is the total energy for the excited state, Si= lo + {i'ajtb} + ... (37) a,i a,b ij is a left eigenvector of Ht, and S= ro + Z ai} + I EI afibtj} +... (38) ia ,,j .,b 32 is a right eigenvector of H. It is also possible to include the equationofmotion in the 7Z equation. Eq. (36) then becomes59 (p [H, Z] 10) = w(plR0), (39) where w = E ECC (310) is the excitation energy. It can be seen that in EOMCC the excited states are the eigenvectors of H, with the corresponding eigenvalues being the energies. Since H is formed from a similarity transformation, it is not Hermitian, but its eigenvectors do form a biorthogonal set. Choosing the norm of the excited state vectors to be one gives that (01l7Zk0) = 6jk (311) for all states j and k. Therefore, from either Eq. (35) or (36) we can get an expression for the excited state energy E = (01 T H 0). (312) When Stanton first derived the equations for EOMCC gradients,90 he followed the same development as that of Refs. 106, 107, 109 by taking the derivative of Eq. (312) with respect to a general external perturbation. In the process, he had to introduce the perturbation independent quantity zeta (Z) to account for the effect of the excited state perturbation on the ground state T amplitudes. Since L and R are eigenvectors for the excited state, they are variationally optimum and therefore stationary with respect to the 33 perturbation. On the other hand, T is never stationary (even for the ground state), and another operator must be included to account for its response. Szalay94 then used the approach of a general functional analogous to the A functional in ground state coupledcluster theory14, 116 by introducing Z to force T to be stationary with respect to the external perturbation. Following this approach, here is introduced an EOMCC functional which has the property that all quantities are stationary, and therefore the functional will satisfy the generalized HellmannFeynman theorem.117, 118 Such a functional is F = (OICH1Z0) + (01ZH 0) + E(1 (01L 10)). (313) By construction Z = ('ita + iaC ajb + (314) a,i a,b is a pure deexcitation operator of rank equal to T. Since the last two terms of Eq. (313) do not contribute to the value of the functional, the value of the functional is just the energy. By taking the derivative of each of the quantities on the right hand side of Eq. (313) and setting them equal to zero, the EOMCC gradient equations will be recovered. For example, OF DE = 0 = 1 (01OR 0) (315) is a restatement of Eq. (311), the normalization condition for L and R. Taking the derivative of the functional with respect to R (C) gives Eq. (35) (Eq. (36)), which is the equation for C (R). Taking the derivative of the functional with respect to Z gives Eq. (32), which is the ground state T equation. 34 However, taking the derivative of the functional with respect to T is much more complicated. Since the form of H, and therefore the equations for T, vary between different CC methods, it should not be surprising that the equations for Z also vary between different EOMCC methods. Here will be assumed a pure EOMCC method (e.g. EOMCCSD, EOMCCSDT, ...). When the derivative with respect to one of the operators is taken, L, R, T, or Z, what is really being done is separately taking the derivative with respect to every coefficient in the operator. For example, , really means 9 for all t amplitudes in T. This still leaves the excitation or deexcitation part of the operator. So when jr is taken as a part of getting 4, what results is 8H = TeTHeT + eTHeTQT OT (316) = [fH,QT]. Here T ti + {atibtj} ... (317) i'a '>j a,b is the pure excitation operator without any coefficients. Therefore, OF 0T = 0 = (ol0[H, QT]n IO) + (OIZ[H, QT] 0). (318) After significant simplification, Eq. (318) reduces to (0ZIg) = (01Lrq)(qjlZg) ((gl(Eccl H) g)), (319) where 1 stands for a unit matrix of rank Ig). This is exactly equivalent to the equation for Z in Ref. 90. Finally, since the functional satisfies the generalized HellmannFeynman theorem, for any perturbation X OF OE ox= = (OlHxZIO) + (OlZHrXlO). (320) 35 Here, Hf represents the derivative of the bare Hamiltonian elements with respect to the perturbation X within the similarity transformed Hamiltonian H. For the pure EOMCC methods this means Hx = eT He. T(321) Eq. (320) is the same as Eq. (36) of Ref. 90. Although the form of the functional was simply postulated, the fact that it produces equations (and therefore density matrices) identical to those produced through straightforward derivation of the energy expression90 proves that the functional is valid. For a discussion of the properties of the density matrices resulting from these equations, see Ref. 119. It is also possible to derive an energy expression from Eq. (39). Projecting on the left by (0Lp) gives the expression E = (OI[H, R] I0) + (I0H O), (322) where the second term, Ecc, is added in to give a total energy instead of just the excitation energy. The functional which corresponds to this energy expression is F = (0[H, ]] 10) + (0oHI0) + (OIZrI0) + w(1 (0IrL10)). (323) Superficially, Eqs. (313) and (323) appear similar. In fact, Eq. (322) can be straight forwardly derived from Eq. (312), and both functionals have the same value, the total energy E. However, these energy functionals contain several important differences. The first is that w, the excitation energy, appears on the right instead of E. More importantly, though, the equations for and value of Z have changed. This will seen later, but first it is important to see that the EOMCC equations are still contained in the new functional. 36 As before, the derivative of each of the quantities on the right hand side of the functional will be taken and set to zero. First, OF = 0 = 1 (070) (324) again gives the normalization condition for C and R. OF S= 0 = (glHI0) (325) again gives the coupledcluster equations. Next, taking the derivative of the functional with respect to L gives OF OL = 0 = (pl [H, 7] 10) w(pIR.0), (326) which is equivalent to Eq. (39). Expanding the commutator gives w(PlIZ0) = (plrIZ0) (plRHI0). (327) Inserting a resolution of the identity Ih)(hl into the last term gives w(pnlZIO) = (pIHRIO) (pjlZq)(qlH0O) (pIR7g)(glHj0) (pIRo0)(0IHIO) (328) = (plHRIO) (pR0IO)Ecc, or (Ecc + w)(plZIO) = (pIHR0O). (329) The third term in the first equality of Eq. (328) disappears because of the solution of the coupledcluster equations. The second term in the first equality of Eq. (328) disappears because R, being an excitation operator, can never decrease the excitation level. Thus 37 it cannot have any nonvanishing terms between (p and 1q). Note that Eq. (329) is equivalent to Eq. (36). Taking the derivative of the functional with respect to R gives OF S= = 0 = I(O[f a] 10) w(oILOc R1). (330) In analogy to QT above, R is the excitation part of the operator R without any coefficients. The difference is that fT acting on 10) gives 1g), while Q7 acting on 0) gives p), since Qf contains a constant part and QT does not. Rewriting Eq. (330) gives w(0oCp) = (0lCHft o0) (01Loc rH0). (331) Inserting a resolution of the identity into the last term gives wo(01lP) = (0oIHIp) (OILnzO0)(OIHI0) (01LS lp)(plHo0) (OIl0 CQq)(qlR0) (332) = (0oLCHp) (OIClp)Ecc, or (w + Ecc)(0O Ip) = (0CHflp). (333) This is simply Eq. (35). The third term in the top equality of Eq. (332) disappears because of the solution of the coupledcluster equations. The last term in top equality of Eq. (332) disappears because Qn acting on q) gives back 1q), and cannot give nonvanishing terms between (01 and q). For the Zeta equations a pure EOMCC method will again be assumed. Taking the derivative of the functional with respect to T gives OF T = =(0[[r, r],7Z] i0) + (01 [H, QT] 0) + (OIZ[H ,T] 10) = (OI[H, RT]) 10) (01CZ[H, QT] I0) + (0[, QT] 10) (334) + (01Z [r, T] 10). 38 Notice that the first and last terms are the same as Eq. (318). Expanding the commutators and inserting resolutions of the identity gives 0 =(0o CHq)(qlTRZlO) + (0IHlp)(plTRZI0) (OITjq)(qlHRI0) (0L7TIp)(pIHR10) (I07Zq)(q[H, rT] 0) (0jlZg)(gl [f, T] 0) (I7ZI0)(01 [f, T] 0) + (01 [H, T] 10) (335) + (0Zlq)(ql rT0j) + (0Zg)(glHnTI0) + (0IZl0)(0HfTI0) (0 q)(q (IZTlqqlI (OIZnTrg)(gljH0) (0IZQrTO)(0Ho10). The third, fifth, and twelfth term disappear for the same reason that the (OILQ q) disappeared above. Since the norm is chosen to be one, the seventh and eighth terms cancel. The ninth and eleventh terms disappear since Z cannot connect (01 to either 10) or Iq). The thirteenth term disappears because of the solution of the coupledcluster equations. Eqs. (35) and (36) can be used to simplify the second and fourth terms. Doing this gives 0 =(0oJH q)(qjlZjg) + E(OILIp)(p)ORTZI) E(0)ITIp)(p RI0) (01LRg)(g Hf2T 0) + (Oj0C(g)(g sT q)(qjHI0) (336) + (0[7Zlg)(glrTIg)(glHI0) + (OjLR7g)(glTIO0)(OjHI0) + (OIZIg)(glHlg) (0IZIg)Ecc. The fact that R and QT, both being excitation operators, commute has been used. Since R acting on 10) only connects with (pl and since acting on (01 only connects with Ip), the second and third terms cancel. The fifth term disappears since QT cannot connect (gl to Iq). The sixth term disappears because of the solution of the coupledcluster equations. Finally, the Zeta equation becomes (OjZjg)(gj(Eccl H) )g) =(0LfIq)(qj7Zg) (337) + (0 7Z g)(g (Ecc1 R) Ig), or (0OIZg) = (ol0rq)(qRl g)((gl(Ecc1 f) g))1 + (0olRg). (338) The first term in Eq. (338) is the same as Eq. (319). However, the second term is new. The expression for the energy derivative corresponding to this functional is OE O = (OIL[Hx, R] 10) + (olI0XI) + (OIZHXI0). (339) In fact, this equation and Eq. (320) produce the same density matrices. In the first term of Eq. (320) Hx and RZ are not required to be connected, as they are in Eq. (339). Those disconnected term can be divided up into two categories. In the first, C is connected to only R. When all such terms are added up, the norm can be factored out, leaving the second term of Eq. (339). The other category is when C is connected to Hx. These terms lead to the extra term in Eq. (338). It is important to note that the two different functionals have different equations for Z, but since they give the same density matrices, either may be used. Therefore, the operator Z is not unique, but only has meaning within the context of its functional. To better understand the differences between the functionals, let us examine how the ground state fits into each. In EOMCC theory, the ground state is just an eigenstate of H with the special property that R = 1, and C = 1 + A, where A, a pure deexcitation operator, is the usual A from coupledcluster theory.107 Substituting these values into the first functional gives F = (01(1 + A)HIO), (340) 40 the usual ground state functional. In this functional Z for the ground state is zero. This can be seen from Eq. (319), where the constant operator 1 cannot connect (ql and 1g). These give for the energy derivative, Eq. (320), 9E = (01(1 + A)HX 0), (341) again the usual coupledcluster expression. For the second functional the picture is somewhat more complicated. Substituting the appropriate values for R and C into Eq. (338) gives that (0IZg) = (0(1 + A)l1g) = (0Ag). In other words, Z for the ground state in the second functional is A. But in this case, R for the ground state, a constant operator, gives zero for the commutators in the first terms of Eqs. (323) and (339). So the usual coupledcluster expressions, Eqs. (340) and (341), are still obtained. The above equations derived from the second functional have a small computational advantage over those derived from the first functional. Specifically, the second Zeta equation suggests a simpler way to calculate some of the terms in the density matrix equations. Stanton and Gauss92 suggest combining Z and roL into a composite operator, and using this new operator in a ground state coupledcluster gradient code to calculate many of the excited state density matrix contributions. The approach presented here suggests including all of (0C7Ig) into the composite operator, reducing even further the number of terms which would need to be calculated separately. In practice, this should make very little difference in terms of how long a calculation takes. However, it should make programming somewhat easier. 41 It is possible to derive a third Z equation. Starting with the first line of Eq. (334) and not breaking the commutators gives (0Zig) = ((0[[H., ], T] 10)+ (0 [H,s,] 10))((gl(Eccl Hr)g))l. (342) In the first term H must be connected to both R and QrT, while in the second term H is connected to just Tr, but Z and L cannot appear. In diagrammatic language saying that H is connected to OT is equivalent to saying that at least one of the lines from H must be dangling from the bottom. Since the second and third Z equations were derived starting from the same place, they must be equivalent. For EOMCCSD the third Z equation has two more diagrams than the first Z equation. None of the diagrams where the two differ would change the overall computational scaling of the method. PEOMMBPT(2) Gradients The defining equations for PEOMMBPT(2) are identical to Eqs. (34) and (35), but with a modified H. Detailed equations for H, Z, and have been given previously.114 For (nonpartitioned) EOMMBPT(2) the functional is97 F=(0 c [(Ho] + H1+ H1 + ]T[11R) 0) + 0 H0o + H1 + (H[']11 Tl) 0) (343) + (0 Z[(H[] Eo)T[11 + HXI] o) + E(1(0 R o)). The c means that the operators are connected; they share at least one index. Here, the usual manybody perturbation theory partitioning of the Hamiltonian is assumed. H[n] 42 is the nth order term of the Hamiltionian, and TIn] is the nth order T amplitude from manybody perturbation theory. For PEOMMBPT(2) the functional becomes F =( (1 + 2)[(H[o+] + H[11 + H[l]T[1I) I] 0) + (0 [(H[] + H[11 + HI1]T[l])72] 0) + (0 C2(H[oZ2) 0) (344) + (0 H[ol + H11l + (Hl]T[1I) 0) O + (0 Z[(H[] E[])T[1] + H['I] 0) + E(1 (0 I 0)). This form of the Zeta operator comes from the defining equation for the T[11 amplitudes, which is (g (H[] E[])T1l + H 1 0) = 0. (345) For the rest of the paper it will be assumed that the reference determinant is the HartreeFock solution. In this case, H[ol = EZ {iti+ EZ{aa}, (346) i a and H1 = H H[o S (pq rs) ptq sr (347) pqrs Also, T[I= _1[ a ibtj } (348) nfh where ab1l] (ab ij) tia= (349) bl z + tj a  b, 43 Since T111 is now a double excitation operator, (0 1 [HIITL1' 2] 0) must vanish, and the functional is slightly simpler. Another consequence of T['] being a pure double excitation is that Z consists only of double deexcitation operators. Taking derivatives of the (simplified) Eq. (344) gives the equations for the gradients. It can be shown that taking derivatives with respect to the various R and C operators recovers the energy expression. Again, by construction, taking the derivative with respect to Z recovers the T[1] equation. Finally, taking the derivative with respect to T[11 gives OF 0 =(o0 I (H[llR,,) 0)+(0 HI[l] d) + (0 Z (H Elo]) d). Here, Qd = 1 C {afibtj} is the pure doubles excitation operator. The fact that R i,j,a,b commutes with Qd has been used. Solving for Z gives (o0 Z d) ={(o I(H[ ']Z d)c 0) (0 H[1 d)} x((d (Eil H[l) d))(351) Previously, solving for Z involved inverting the full H, but now only the inverse of (d (E[o] H[]) d is needed, which, in the HartreeFock case, is just the denominator from MBPT(2). It is now convenient to introduce a new operator, E, which is the inhomogeneous part of the Z equation. For PEOMMBPT(2) ai (352) =(0 1L(H['I zl d) 0o) (0 H111 d). Computationally, to solve for Z, E must first be calculated and then the denominator applied. 44 Now the derivative can be rewritten as OE OF OHi (l 2[O ] OH[1] ____ 9E aF i90 all['] a98HI O 0 ( + + + + rT[1] 0 9Oxx QX ax S [(H[o] H9Il 9T9H11 ) \ C O X , X + x T[1 R2] 0/ +(0 o 2 (HOIz2) 0) (353) \ 9 + 5 x 10 +(o H[o] 9 H[' + TH[1 T+I1) 0 Z T x ) 0. This expression only contains quantities which are independent of the perturbation plus derivatives of the Hamiltonian elements. Thus, the derivative can be recast as one and twoparticle effective density matrices times derivatives of integrals; E Ofq q + pq( rs) (354) Ox = x rs ax Explicit spin orbital equations for the effective density matrix elements in terms of R, L, T[1], and Z are P, = I ;i je 1 / imtef S= 2 1E e j. + ij, (355) e m,e,f P= lbm + 2 e "ae (356) b m,n,a SImn ftea lmnracf. V7 m aca p=  l ef i mn 2 lrfn cmi + c rim, (357) m,n,e,f m,n,e,f m,e Pa = 0, (358) 1ij Pka (ijr)P (kl)ik, (359) p = 4 1 r (360) ff a k ea m r (8 + e E f tjk S= + ) 42lr a + 8 1a e nm,e,f m,e,f 8 k() Yme, f m,e,f Pb 1 P_(ij)P_(ab)lra, Pb 4E 1 e64 + (ab) m l,"i m n m,n,e mn b .ea ba "c ro tinn ar f tan eTba m,,ec mn m,n,e. pi = IE ,m a Pbc mia m m P2 i _(ij) E\ lmretab P_ ) r "'a Teb ab ' e i mj 4 2 e m rnj m,e m,e 1 1 + P(ij)P(ab) I, m raet + 4e ri I 4 4 , m,e pa = 0 cd 0, pb = 0. Pzj The explicit spin orbital equation for 5 is iJb = P(ab) E lmr(ij Ieb) P_(ij) m ,e + P(ij)P(ab) lr (rnjIeb) m,c P (ij)P(ab) e e(m ) m,n,e + P(ij)P(ab) E 'r' (fj leb) m,e, r + P(ab) 1 i'r e(fmllbe) m,e,f  P(ij) r (jnjjne) 1 r'(fmllba) m,n,e m,ef)  (ien) + (ijjllab). m,n,e (361) (362) (363) (364) (365) (366) (367) m1rm (mj lab) me (368) 46 In these equations P.(pq) stands for [1 P(pq)], where P(pq) means permute p and q. The equations for p and for ( are subsets of the corresponding equations for EOM MBPT(2) gradients,97 and the PEOMMBPT(2) gradients have been implemented in the ACES II program system* starting from Stanton and Gauss's EOMMBPT(2) gradient code.97 One final note should be made about the cost of the PEOMMBPT(2) gradients. Previously it was argued that the iterative n Ncctirt steps should dominate the time over the noniterative noccNirt step for normal vertical excitation energy calculations. This was partially because the cost of the no cANirt step can be amortized over multiple excited states. However, the gradient for only one excited state can be calculated at a time. Therefore, this amortization cannot occur. More importantly, there are now many more noccNirt steps, which, when added together will also contribute significantly to the cost of the calculation. Therefore, PEOMMBPT(2) gradients is most accurately considered a noniterative n2 N 4irt method. Application to Diatomic Molecules In order to assess the performance of the CIS(D)83 method for excited state potential energy surfaces, Stanton et al.93 studied singlet excited states for six diatomic molecules. * The ACES H program is a product of the Quantum Theory Project, University of Florida. Authors: J. F. Stanton, J. Gauss, J. D. Watts, M. Nooijen, N. Oliphant, S. A. Perera, P. G. Szalay, W. J. Lauderdale, S. R. Gwaltney, S. Beck, A. Balkovi, D. E. Bernholdt, K.K. Baeck, H. Sekino, P. Rozyczko, C. Huber, and R. J. Bartlett. Integrals packages included are VMOL (J. Alml6f and P. R. Taylor), VPROPS (P. R. Taylor), and a modified version of the ABACUS integral derivative package (T. U. Helgaker, H. J. Aa. Jensen, J. Olsen, P. Jorgensen, and P. R. Taylor). 47 In order to compare this new work with the methods presented there, those systems have also been studied here. The six diatomics are CO, N2, C2, H2, BH, and BF. Except for N2, the studied states are the lowest singlet excited state of the molecule. For N2 the 11'll state was studied. The basis sets used were 631G*,120 augccpVDZ,121 and augccpVTZ.121t From attempting to match the reported numbers, it would appear that for the 631G* and augccpVDZ basis sets all six Cartesian d functions were used previously, and for the augccpVTZ basis set only the spherical d and f functions were used. The same was done here. These states have been carefully studied with very accurate calculations. However, the point of this work is to compare the current method with other single reference excited state methods. Therefore, an attempt to review the volumous literature on these molecules will not be made. Instead the three inexpensive methods (CIS, CIS(D), and P EOMMBPT(2)) will be compared to the more expensive and more complete EOMCCSD and to experiment.122 All of the other methods can be viewed as various approximations to EOMCCSD. For valence excited states there are frequently important contributions from triples to consider.6063 t Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 1.0, as developed and distributed by the Molecular Sciences Laboratory which is part of the Pacific Northwest Laboratory, P. O. Box 999, Richland, Washington 99352, USA, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy under contract DEAC0676RLO 1830. Contact David Feller, Karen Schuchardt, or Don Jones for further information. 48 Results for the diatomic molecules are presented in Tables 31 to 35. There are some small differences between the numbers reported here and the values given in Ref. 93. Several values given in the previous paper were in error. Those errors have been corrected here. Vertical and Adiabatic Excitation Energies The vertical excitation energies are given for the six molecules, for the four methods, and for the three basis sets in Table 31. The adiabatic excitation energies are given in Table 32. The ground state minimum geometries used are listed in Table 33, with the excited state minimum geometries in Table 34. The appropriate ground state minimum for CIS is the SCF minimum. The appropriate ground state minimum for CIS(D) and P EOMMBPT(2) is the MBPT(2) minimum. The appropriate minimum for EOMCCSD is the CCSD minimum. As has been noted before,93 the CIS answers are too poor to be reliable CIS has several errors of greater than 1 eV, and for C2 CIS even predicts the wrong sign for the excitation energy. The mean absolute deviation with respect to experiment for the adiabatic excitation energies with the augccpVTZ basis is 0.724 eV. These failings are directly attributable to the method's complete lack of dynamic correlation. CIS(D) improves on the CIS in all cases, except those four where the CIS is accidentally already in good agreement. In all cases, the CIS(D) energies are within 0.2 eV of the EOMCCSD energies, and in all of the cases with the best basis set, the error with respect to experiment is less than 0.3 eV. The mean absolute deviation with respect to experiment for the adiabatic excitation energies with the augccpVTZ basis is 49 0.112 eV. This is partially due to the choice of excited states considered and is actually better accuracy than was initially reported for the method.83 The PEOMMBPT(2) method, on the other hand, actually performs worse than originally reported."14 The vertical excitation energies are slightly worse than the CIS(D) energies. The adiabatic energies are slightly better to somewhat worse than the CIS(D) energies, with their mean absolute deviation with respect to experiment for the augcc pVTZ basis being 0.279 eV. The fact that the PEOMMBPT(2) energies are worse than the CIS(D) energies seems to be caused by the choice of states and by a curious feature of CIS(D). In the previous study,114 it was noted that PEOMMBPT(2) performed better on Rydberg states than on valence states. This can be understood, since Rydberg excitations essentially involve pulling an electron out of the valence region and putting it in a very diffuse orbital, while valence excitations put the electron back into the valence space in a different arrange ment. Thus valence excitations should involve more orbital relaxation and differential correlation compared to Rydberg excitations. Therefore, it is reasonable that a method as simple as PEOMMBPT(2) could have more trouble accurately describing valence excitations than Rydberg excitations. EOMCCSD is also more accurate for Rydberg states than valence states, where it takes triples to partially correct the problem.123 CIS(D), on the other hand, performs better for valence states than for Rydberg states.114 This is curious, since the method does not allow the states to relax in the presence of electron correlation. All of the states studied here are valence states, which is probably why the CIS(D) energies are better than the PEOMMBPT(2) energies. Also, these states are well separated from other states of the same symmetry, meaning 50 that mixing of states, which the zeroth order wavefunction, the CIS, would have trouble handling, does not occur in these problems. The EOMCCSD augccpVTZ results agree quite well with experiment, which is not unexpected for such relatively simple states. The mean absolute deviation with respect to experiment for the adiabatic excitation energies is 0.106 eV. Actually, there are not significant differences between the basis sets, except for H2, where 631G* does not contain any polarization functions. Bond Lengths and Vibrational Frequencies Excited state equilibrium bond lengths are presented in Table 34 and excited state vibrational frequencies are presented in Table 35. Certain trends hold true. CIS bond lengths are always too short and vibrational frequencies are all too high. This is completely analogous to the situation for HartreeFock for the ground state. However, like HartreeFock, the bond lengths and vibrational frequencies are reasonable. The CIS(D) and PEOMMBPT(2) bond lengths and vibrational frequencies are quite similar. And, as should be expected, the EOMCCSD is superior to the other methods. For CIS, the mean absolute deviation in the bond length is 0.028 A, and the mean absolute deviation in the frequency is 188 cm1. For comparison, the mean absolute deviation in the ground state bond lengths for the SCF was 0.015 A. The mean absolute deviation in the bond length drops to 0.015 A for CIS(D), with the mean absolute deviation in the vibrational frequencies being 108 cm1. This compares to a mean absolute deviation of 0.011 A for the PEOMMBPT(2) bond lengths, 106 cm1 for the PEOMMBPT(2) vibrational frequencies, and 0.009 A for the MBPT(2) ground state bond lengths. The EOMCCSD bond lengths have a mean absolute error of 0.008A, and the vibrational 51 frequencies have a mean absolute error of 67 cm1. The ground state CCSD has a mean absolute deviation of 0.004 A for the bond length. All comparisons were made between the augccpVTZ results and the experimental results. In general, the ground state methods were slightly better than the excited state methods, and EOMCCSD was better than PEOMMBPT(2), which was slightly better than CIS(D). Application to Polyatomic Molecules The PEOMMBPT(2) method has also been applied to some prototypical polyatomic molecules. Because of the much more complicated chemistry possible, polyatomics can provide a much more demanding test of a method. The S1 State of Ammonia The Si state of ammonia has been studied extensively.124 125 As a methods test, one of its attractive features is its predissociative nature. The character of the state changes from Rydberg like near its D3h minimum to valence like near the C2v transition state.126 Therefore for a method to properly describe the barrier height, it must equally treat the valence and Rydberg parts of the potential energy surface. Studying the Si state of ammonia will also provide an opportunity to compare the PEOMMBPT(2) with the (nonpartitioned) EOMMBPT(2) method97 in order to assess the effect of the partitioning apart from the effect of replacing the CCSD ground state with the MBPT(2) ground state. The basis set used for this study was the "A" basis set of Ref. 127. This basis set contains 65 contracted functions and includes multiple diffuse functions at each atom. It is flexible enough to give a reasonable description of the entire area of interest of the potential energy surface. Table 36 gives the geometry and vibrational spectrum for the 52 D3h minimum. It is believed that the NH bond distance is 1.055 0.008 A.128 The EOMCCSD bond distance falls within these error bars. However, a larger basis set would tend to shorten it.129 The CIS bond length is once again too short and the frequencies are too large. The CIS(D) PEOMMBPT(2), and EOMMBPT(2) bond lengths are all similar and slightly shorter than the EOMCCSD bond length. For the vibrational frequencies, the CIS(D) results are slightly more erratic compared to the EOMCCSD than the are PEOMMBPT(2) and EOMMBPT(2), which are similar. The infrared intensities show some interesting patterns. In every case CIS(D) substantially overestimates the correlation correction to the intensity. The PEOM MBPT(2) is only partially able to correct that behavior. Most of the error in the PEOM MBPT(2) can be traced to the very approximate treatment of the doublesdoubles block, since the EOMMBPT(2) intensities are much closer to the EOMCCSD intensities. The intensities provide a very sensitive measure of the quality of the wavefunction, and these large errors for the PEOMMBPT(2) and especially the CIS(D) methods suggest that their descriptions of the wavefunction are not as good as the bond lengths and vibrational frequencies suggest. The structure and vibrational spectrum of the transition state are presented in Table 37. The CIS(D) bond length for the hydrogen being extracted is much too long. This bond length for the PEOMMBPT(2) method is also too long, although shorter than for CIS(D). CIS actually manages to get this bond length right compared to EOM CCSD. These elongated bonds for CIS(D) and PEOMMBPT(2) cause w6 to be too small93 with the CIS(D) actually predicting that the transition state is not C2v. Stanton et al.93 suggested that the long bond length was caused by the tendency of CIS(D) to 53 underestimate the energy of Rydberg states,83 thus causing a "late" transition state. For PEOMMBPT(2) the problem is the opposite. It tends to overestimate the valence states. However, that will also lead to a "late" transition state and a longer CH bond. Once again, CIS(D) and PEOMMBPT(2) have problems with the intensities, with 12, 13, and 15 having significant errors with respect to EOMCCSD and EOMMBPT(2). On the other hand, the CIS(D) and PEOMMBPT(2) dipole moments are pretty reasonable. The barrier heights, though, are much too large, with the CIS(D) barrier height being much worse than the PEOMMBPT(2) barrier height. Even the EOMCCSD barrier height is well above the experimental barrier height, estimated to be about 2100 cm'.128 Transbent and Vinylidenic Isomers on the Si Surface of C2H2 The S1 surface of acetylene provides a nice test for excited state methods. The C2h transbent minimum for the S1 state was described in the 1950's.130132 However, in a recent theoretical paper, it was suggested that the global minimum was a C2v vinyli denic structure with both hydrogens bonded to one of the carbons.133 The vinylidenic structure was calculated to be 13 kcal/mol lower than the transbent acetylenic structure. This prediction that the vinylidenic state was lower was confirmed with multireference CI, approximate coupledpair functional,134 and averaged quadratic coupledcluster'35 calculations. On the other hand, both CIS and CIS(D) predict the wrong ordering of the isomers.93 The structures, vibrational frequencies, and energies of the two isomers are presented in Tables 38 and 39. The calculations were performed with a TZ2P basis set.136 Not including zero point energy corrections, CIS predicts the acetylenic structure to be more stable by 2.1 kcal/mol. The CIS(D) predicts the acetylenic structure to be more stable 54 by 0.9 kcal/mol. The PEOMMBPT(2) also has the wrong order, with the acetylenic structure being 2.9 kcal/mol lower in energy than the vinylidenic structure. Simple Carbonyls As the simplest of the carbonyls, formaldehyde's spectrum has been studied extensively.137' 138 The first excited state, n+7r*, has two distinctive geometrical fea tures. The CO bond lengthens and the molecule becomes pyramidal. CIS severely underestimates the bond lengthening,102 while CIS(D) severely overestimates the bond lengthening.93 Also, CIS(D) predicts the molecule to be almost flat.93 Table 310 presents the PEOMMBPT(2) results. Other than predicting a CO bond that is 0.02 A too long, the PEOMMBPT(2) geometry agrees well with experiment, especially considering the relatively poor basis set used (631G*,120 the same that was used in Ref. 93). The frequencies are also reasonable, except for w4. The dipole is somewhat too large, but is still in better agreement with experiment than the CIS(D) dipole. Table 311 presents a comparison of the formaldehyde, acetaldehyde, and acetone geometries, all with the 631G* basis set.120 Hydrogen "a" is a hydrogen attached to the carbonyl carbon. Hydrogen "b" is the "in plane" hydrogen. Hydrogen "c" is the hydrogen pointing into the pyramid. Finally, hydrogen "d" is the hydrogen pointing away from the pyramid. Since acetaldehyde and acetone are pyramidal in their first excited state like formaldehyde is, there is no true in plane hydrogen. However, in both cases there is a hydrogen with a dihedral angle only about five degrees out of planarity. The difference with the ground state, however, is that the in plane hydrogenn points away from the oxygen instead of being hydrogen bound to it.139 This can be attributed to the 55 removal of one electron from the in plane lone pair on oxygen, letting the steric effects dominate over the weakened hydrogen bonding. Zuckermann et al.140 carefully measured the spectrum of acetone around the origin of the n+7r* band. Their analysis led to the positive analysis of three vibrational bands, with uncertainty about two others. Their assignments, along with the calculated vibrational frequencies, are presented in Table 312. Zuckermann et al. could not definitely distinguish between the first possibility and the second possibility; though they preferred the first. Determining which is correct depends upon the frequency of v19. The computed frequency of 368.4 cm1 is well below the 465.4 cm1 for the second possibility. On the other hand the PEOMMBPT(2) is low on all of the other frequencies except for v12, considering that the calculated frequencies are harmonic frequencies as opposed to the measured fundamental frequencies. Also, in acetaldehyde the PEOM MBPT(2) gets very good agreement between the calculated vibrational frequency (368.4 cm1) and the measured frequency (370 cm')139 for the corresponding mode. The next calculated mode in acetone is at 811.1 cm1. Hence, the calculated spectrum supports the second possibility over the first possibility. It would take better calculations, for example an EOMCCSD calculation with a better basis set and possibly with triples, to confirm which one it is. Conclusions An alternative and simpler derivation for EOMCCSD gradients based upon an excited state EOMCC functional is presented. Gradients for the PEOMMBPT(2) method have been derived and their cost discussed. Then, by studying a series of 56 diatomics and prototypical polyatomics the performance of PEOMMBPT(2) for low lying valence potential energy surfaces was studied versus other single reference methods for excited states for which gradients have been derived. The performance of CIS, CIS(D), and EOMCCSD has been discussed previously93, and so they will only briefly be discussed here. The CIS energies are too poor to be even qualitatively reliable. However, the CIS geometries are normally quite reasonable. CIS(D) has the opposite behavior. Its energies for valence states often has errors of 0.3 eV or less, but its geometries sometime fail dramatically. Even worse, there seems to be no indication of when the CIS(D) geometries will fail. EOMCCSD performed quite well in all of the tests presented here, both for the energies and for the geometries. Overall, the performance of the PEOMMBPT(2) method was mixed. The energies were reasonable but were not as good as had been reported previously.114 The geometries were also reasonable but were noticeably worse than the EOMCCSD geometries. Over all, in these tests PEOMMBPT(2) performed very similarly to CIS(D) when CIS(D) did not fail and was qualitatively correct for those cases for which CIS(D) did fail. The discrepancy between the PEOMMBPT(2) and the EOMCCSD geometries is attributable primarily to the partitioning of the doublesdoubles block in PEOM MBPT(2), since the (nonpartitioned) EOMMBPT(2) performed similarly to the EOM CCSD for ammonia. However, it is this partitioning that makes the method attractive, since it is the partitioning that reduces the cost from an iterative n2c rt to an iterative nocNirt plus a noniterative n Nirt 57 Table 31: Vertical excitation energies (in eV) for the excited singlet states CISa CIS(D)a PEOMMBPT(2) EOMCCSDa CISa CIS(D)a PEOMMBPT(2) EOMCCSDa CISa CIS(D)a PEOMMBPT(2) EOMCCSDa CISa CIS(D)a PEOMMBPT(2) EOMCCSDa CISa CIS(D)a PEOMMBPT(2) EOMCCSDa CISa CIS(D)a PEOMMBPT(2) EOMCCSDa a)Ref. 93. 631G* 15.354 15.341 15.349 15.257 3.029 2.991 2.991 3.117 9.385 8.911 9.196 8.833 10.378 9.444 9.883 9.425 6.888 6.865 6.935 6.898 1.180 1.310 1.423 1.526 augccpVDZ 12.639 12.574 12.574 12.484 2.849 2.864 2.875 2.972 9.272 8.768 9.047 8.638 10.360 9.389 9.805 9.339 6.559 6.481 6.560 6.482 1.258 1.151 1.261 1.283 augccpVTZ 12.714 12.832 12.835 12.717 2.852 2.810 2.824 2.914 9.330 8.767 9.076 8.666 10.550 9.532 9.980 9.514 6.601 6.455 6.545 6.454 1.229 1.222 1.328 1.308  ~I ~I Table 32: Adiabatic excitation energies (in eV) for the excited singlet states 631* augcc augcc Expt.a pVDZ pVTZ H2 CISb 12.765 11.269 11.352 CIS(D)b 12.980 11.290 11.429 PEOMMBPT(2) 12.943 11.274 11.408 EOMCCSDb 13.110 11.228 11.353 11.3694 BH CISb 3.024 2.845 2.849 CIS(D)b 2.989 2.861 2.807 PEOMMBPT(2) 2.989 2.872 2.821 EOMCCSDb 3.117 2.972 2.913 2.8685 CO CISb 8.784 8.739 8.802 CIS(D)b 8.269 8.219 8.246 PEOMMBPT(2) 8.566 8.527 8.580 EOMCCSDb 8.330 8.229 8.256 8.0684 N2 CISb 9.413 9.432 9.582 CIS(D)b 8.802 8.775 8.864 PEOMMBPT(2) 9.264 9.231 9.346 EOMCCSDb 8.760 8.714 8.839 8.5900 BF CISb 6.806 6.519 6.563 CIS(D)b 6.760 6.428 6.404 PEOMMBPT(2) 6.822 6.507 6.495 EOMCCSDb 6.811 6.444 6.417 6.3427 C2 CISb 1.275 1.350 1.319 CIS(D)b 1.143 0.997 1.075 PEOMMBPT(2) 1.293 1.138 1.208 EOMCCSDb 1.301 1.066 1.103 1.0404 a)Ref. 122. b)Ref. 93. Table 33: Equilibrium distances (in A) for the ground states augcc augcc pVDZ pVTZ H2 SCFb 0.7300 0.7481 0.7345 MBPT(2)b 0.7375 0.7549 0.7374 CCSDb 0.7462 0.7617 0.7431 0.7414 BH SCFb 1.225 1.233 1.221 MBPT(2)b 1.233 1.241 1.216 CCSDb 1.244 1.249 1.220 1.2324 CO SCFb 1.114 1.110 1.104 MBPT(2)b 1.150 1.147 1.134 CCSDb 1.141 1.138 1.124 1.1283 N2 SCFb 1.078 1.078 1.067 MBPT(2)b 1.130 1.131 1.110 CCSDb 1.113 1.113 1.093 1.0977 BF SCFb 1.260 1.270 1.249 MBPT(2)b 1.279 1.294 1.264 CCSDb 1.281 1.296 1.263 1.2626 C2 SCFb 1.245 1.253 1.241 MBPT(2)b 1.264 1.276 1.254 CCSDb 1.252 1.265 1.241 1.2425 a)Ref. 122. b)Ref. 93. 60 Table 34: Equilibrium distances (in A) for the excited singlet states 631G* augcc augcc Expt.a pVDZ pVTZ H2 CISb 1.544 1.239 1.239 CIS(D)b 1.599 1.256 1.273 PEOMMBPT(2) 1.626 1.273 1.295 EOMCCSDb 1.616 1.267 1.283 1.2928 BH CISb 1.204 1.214 1.204 CIS(D)b 1.219 1.223 1.199 PEOMMBPT(2) 1.216 1.223 1.199 EOMCCSDb 1.241 1.242 1.211 1.2186 CO CISb 1.228 1.220 1.213 CIS(D)b 1.295 1.286 1.263 PEOMMBPT(2) 1.293 1.282 1.259 EOMCCSDb 1.252 1.242 1.224 1.2353 N2 CISb 1.200 1.200 1.192 CIS(D)b 1.250 1.249 1.231 PEOMMBPT(2) 1.245 1.244 1.227 EOMCCSDb 1.221 1.220 1.202 1.2203 BF CISb 1.316 1.312 1.287 CIS(D)b 1.350 1.349 1.312 PEOMMBPT(2) 1.353 1.349 1.312 EOMCCSDb 1.345 1.342 1.304 1.3038 C2 CISb 1.293 1.301 1.289 CIS(D)b 1.333 1.346 1.320 PEOMMBPT(2) 1.325 1.337 1.312 EOMCCSDb 1.334 1.347 1.318 1.3184 a)Ref. 122. b)Ref. 93. Table 35: Harmonic vibrational frequencies (in cm') for the excited singlet states 631* augcc augccpt.a pVDZ pVTZ H2 CISb 1495 1610 1589 CIS(D)b 1438 1485 1422 PEOMMBPT(2) 1407 1410 1337 EOMCCSDb 1428 1439 1368 1358.09 BH CISb 2576 2536 2544 CIS(D)b 2426 2441 2517 PEOMMBPT(2) 2445 2443 2515 EOMCCSDb 2180 2243 2372 2251.0 CO CISb 1646 1615 1633 CIS(D)b 1282 1238 1323 PEOMMBPT(2) 1281 1244 1330 EOMCCSDb 1559 1517 1592 1518.2 N2 CISb 1939 1909 1897 CIS(D)b 1612 1583 1615 PEOMMBPT(2) 1638 1612 1635 EOMCCSDb 1856 1825 1854 1694.21 BF CISb 1339 1273 1367 CIS(D)b 1173 1097 1239 PEOMMBPT(2) 1160 1095 1241 EOMCCSDb 1199 1130 1279 1264.9 C2 CISb 1830 1797 1794 CIS(D)b 1618 1577 1631 PEOMMBPT(2) 1674 1635 1687 EOMCCSDb 1605 1563 1630 1608.35 a)Ref. 122. b)Ref. 93. Table 36: Geometries (in A), harmonic vibrational frequencies (in cm1), infrared intensities (in km/mol), and energies (in Hartrees) for the D3h equilibrium geometry of the S1 state of NH3 CIS(D)a 1.0441 2814.0 736.5 3277.2 1378.6 57.6 1538.5 981.9 35 56.240 3 PEOM MBPT(2) 1.0420 2971.4 767.2 3185.5 1370.6 48.3 3620.2 464.4 EOM MBPT(2)b 1.0487 3180.8 769.6 3021.0 1331.1 14.7 4622.1 293.0 376 56.225 152 56.234 687 EOM CCSDb 1.0512 2993.1 741.2 2997.5 1335.2 9.7 4447.1 376.0 56.246 539 CISa rNH wI(al') w1(ai') w2(a2") w3(e') w4(e') 12 13 I4 Energy 1.0213 3180.8 842.4 3356.7 1517.9 0.1 5803.2 19.4 55.968 6 a)Ref. 93. b)Ref. 97. Table 37: Geometries (in A and degrees), harmonic vibrational frequencies (in cm1), infrared intensities (in km/mol), dipole moments (in Debyes), energies (in Hartrees), and barrier heights (in cm1) for the C2v predissociative transition state of the Si state of NH3. The symmetry unique hydrogen is denoted by an asterisk. CISa 1.3497 1.0106 123.43 3538.5 1568.5 1446.6i 1025.3 3737.2 498.8 293.7 5.4 463.7 50.2 2.0 73.0 3.750 55.952 723 3492 CIS(D)a 1.4371 1.0421 126.78 3133.7 1539.1 1510.3i 1276.0 3313.9 104.8i 640.4 0.3 5.5 88.4 6.7 66.5 2.951 56.214 426 rNH* rNH 0(H*NH) wl(al) w;2(al) w3(al) w4(bl) w5(b2) W6(b2) I1 12 13 14 15 16 /t' Energy Barrier Height PEOM MBPT(2) 1.4108 1.0370 125.82 3203.6 1525.9 1543.2i 1191.1 3384.0 129.9 590.5 0.0 112.1 68.0 3.7 103.4 2.985 56.205 848 4237 EOM MBPT(2)b 1.3219 1.0423 124.51 3069.0 1478.5 1977.3i 1042.0 3274.9 396.5 1081.2 48.9 2301.1 43.9 58.1 158.0 2.631 56.223 512 2452 EOM CCSDb 1.3421 1.0441 124.27 3051.6 1456.0 1897.3i 1023.1 3255.2 419.6 1010.6 41.0 1921.3 69.9 56.5 167.5 2.727 56.234 405 2663 5695 a)Ref. 93. b)Ref. 97. Table 38: Geometries (in A and degrees), harmonic vibrational frequencies (in cm'), infrared intensities (in km/mol), and energies (in Hartrees) for the transbent isomer of the S1 state of acetylene PEOM CISa CIS(D)a EOMCCSDb MBPT(2) rCH 1.0788 1.0889 1.0890 1.0907 rcc 1.3521 1.3724 1.3717 1.3575 0(HCC) 124.55 122.06 121.86 123.64 w1(ag) 3293.5 3144.7 3141.7 3107.7 w2(ag) 1545.1 1423.6 1436.4 1471.4 w3(ag) 1162.4 1108.2 1114.1 1106.6 w4(au) 1230.8 687.2 907.8 614.6 w5(bu) 3280.3 3131.6 3126.2 3083.7 w6(bu) 805.7 882.4 887.5 745.8 14 2.8 111.1 40.7 100.1 15 2.5 3.6 3.9 6.9 16 342.4 281.6 295.6 405.9 Energy 76.681 20 76.969 01 76.966 77 76.981 76 a)Ref. 93. b)Ref. 133 Table 39: Geometries (in A and degrees), harmonic vibrational frequencies (in cm1), infrared intensities (in km/mol), and energies (in Hartrees) for the C2v vinylidenic isomer of the S1 state of acetylene PEOM CISa CIS(D)a ) EOMCCSDb MBPT(2) rcH 1.0847 1.0881 1.0879 1.0890 rcc 1.3877 1.4225 1.4174 1.4289 O(HCC) 122.94 122.15 122.38 122.28 wl(al) 3173.5 3086.1 3089.2 3060.1 w2(al) 1618.8 1464.2 1476.7 1486.5 w3(al) 1381.5 1225.7 1246.5 1213.1 w4(b2) 3235.5 3172.0 3171.0 3138.9 ws(b2) 1015.5 894.6 903.9 927.1 w6(a2) 977.1 702.3 708.6 763.2 I1 17.1 0.1 0.2 3.2 12 1.3 29.1 24.5 14.4 13 0.3 57.0 41.7 31.3 14 9.7 2.2 2.1 7.0 15 4.7 3.9 3.5 5.3 16 1.6 1.1 0.4 2.1 Dipole 3.817 2.926 3.029 2.875 moment Energy 76.677 85 76.967 63 76.962 08 77.003 07 a)Ref. 93. b)Ref. 133 Table 310: Geometries (in A and degrees), harmonic vibrational frequencies (in cm ), infrared intensities (in km/mol), dipole moments (in Debyes), and energies (in Hartrees) for the equilibrium geometry of the A 'A" state of formaldehyde. Experimental frequencies are fundamentals. PEOM EOM CISa CIS(D)b PEOM EOM Expt.a cIS CS(D)b MBPT(2) CCSDb Expt rco 1.258 1.384 1.340 1.324 1.321 rcH 1.085 1.088 1.093 1.096 1.097 0(OCH) 117.3 117.8 117.0 115.8 118.0 7(HOCH) 148.4 171.5 153.0 145.9 148.3 wi (a') 3209.5 3154.1 3102.3 3072.9 2847 w2(a') 1437.9 1420.5 1412.7 1407.9 1290 w3(a') 1660.8 895.7 1187.2 1248.5 1173 w4(a') 561.6 172.6 564.9 719.8 683 w5(a") 3299.6 3295.8 3223.6 3189.5 2968 w6(a") 993.5 914.0 932.9 934.3 898 11 29.6 2.6 0.7 2.8 12 52.0 6.3 0.8 3.5 13 2.1 109.6 66.8 93.7 14 108.2 38.3 36.0 30.8 15 0.0 1.6 2.9 2.8 16 3.4 3.6 3.4 4.0 Dipole 1.507 2.374 2.094 1.776 1.56(7)c) moment Energy 113.669 47 114.043 84 114.028 60 114.052 75 a)Ref. 102. b)Ref. 93. c)Ref. 141. Table 311: A comparison of the geometries of the n47r* states in formaldehyde, acetaldehyde, and acetone. Geometries are in Angstroms and degrees. Formaldehyde rcn 1.340 rCHa rCHb rcHc rCHd 0(oCC) 0(OCHa) 0(CCHb) 0(CCHc) O(CCHd) T(HaCOH) T(HaCOC) r(CCOC) T(HbCCO) r(HcCCO) T(HdCCO) 1.093 117.0 Acetaldehyde 1.359 1.494 1.095 1.091 1.099 1.094 116.0 114.3 110.2 110.6 110.2 153.0 149.9 177.5 61.6 57.1 Acetone 1.376 1.494 1.092 1.101 1.094 113.4 110.3 110.2 110.6 146.9 174.5 65.0 53.7 68 Table 312: A comparison of two possible assignments of the low frequency vibrations in the SI state of acetone to the calculated frequencies. All frequencies are in cm1. v12 torsion (antigearing) V24 torsion (gearing) V8 CCC bend v23 C=O outofplane wagging v19 C=O inplane wagging first possibility 155.5 172.5 373 177.5 a)Ref. 140. second possibility 155.5 172.5 373 333 465.4 PEOM MBPT(2) 206.8 190.1 353.8 329.7 368.4 CHAPTER 4 THE SIMILARITY TRANSFORMED EQUATION OFMOTION COUPLEDCLUSTER METHOD Introduction The equationofmotion coupledcluster (EOMCC) method,56 57 especially when based on a coupledcluster singles and doubles (CCSD)9 ground state, offers an attractive and unified formalism to extend single reference coupledcluster methods to excitation energies (EEEOMCCSD, sometimes just EOMCCSD),58 59 ionization potentials (IP EOMCCSD),67 and electron affinities (EAEOMCCSD).68 The method is conceptually single reference, in that the entire calculation is built upon one reference determinant. Therefore, the only choices necessary in the method are the choice of basis set, the choice of reference determinant, usually the HartreeFock solution, and the choice of excitation level to include in the ground and excited state operators. Several applications58, 142145, 123 of EEEOMCCSD and its twin, coupledcluster singles and doubles linear response,53, 54 have been made. The method has also been extended to include various effects of triple excitations.663 But EOMCCSD, and especially EEEOMCCSD, has some drawbacks. The first is the cost of the method. Every EEEOMCCSD excited state is approximately as expensive as a ground state CCSD calculation. Approximations can be made to reduce the cost,114 with varying degrees of success. The second drawback is the accuracy of the method. For ethylene, butadiene, and cyclopentadiene, the average error for the Rydberg states was 0.17 eV, but the average error for the valence states was 0.26 eV.123 70 The key to EOMCC theory is the similarity transformation of the Hamiltonian. After the ground state coupledcluster equations have been solved, the cluster amplitudes are used to transform the Hamiltonian. A similarity transformation changes the eigenvectors of an operator without changing its eigenvalues. Hence, the energies of the excited states, electron attached states, and ionized states, which are all eigenvalues of the Hamiltonian matrix in terms of the appropriate bases, have not changed. So far nothing has been accomplished. To calculate the energies of these states, it is still necessary to diagonalize a matrix that is, in principal, infinite. After a finite basis set has been introduced, the matrix is now finite, but its size grows as the size of the basis set factorial.22 Therefore, approximations must be introduced. The typical approximations involve truncating the Hamiltonian matrix to include only a subset of possible excitations. In EEEOMCCSD the matrix is limited to only single and double excitations, i.e. determinants where one or two virtual orbitals replace one or two occupied orbitals. For EAEOMCCSD the space is limited to determinants where an electron has been added to a single virtual orbital and determinants where along with the electron being added to a virtual orbital, an occupied orbital is replaced with a virtual orbital. These are the so called lp and 2plh states. For IPEOMCCSD the space consists of the lh and 2hlp determinants, which are those determinants with one electron removed from an occupied orbital and those determinants with one electron removed and one electron excited. After the truncation of the matrix is when the similarity transformation becomes important. The truncation will change the eigenvalue spectrum. Many eigenvalues will no longer appear, and those that remain will be shifted by an amount which will depend on how much of the true eigenvector lies in the omitted space. What gives the 71 similarity transformation its power is that by choosing the transformation so that the important eigenvectors are contained, to the greatest extent possible, in the space kept after the truncation, the effect of the truncation on the energies of the states of interest can be minimized. Note that an similarity transformation is needed, since a unitary transformation leaves the eigenvectors unchanged. An equivalent picture is to choose the transformation such that the parts of the matrix that couple the kept space with the excluded space are minimized, since these parts of the matrix determine the extent to which eigenvectors span both spaces. For singly excited states, the most important terms deleted in the truncation to singles and doubles are the triple excitations. In the untransformed Hamiltonian matrix, the terms that couple single excitations to triple excitations are the two electron integrals (abllij), where a, b, c, and d are unoccupied in the reference determinant and i, j, k, and I are occupied in the reference determinant. These are precisely the terms which are set to zero by the similarity transformation. The transformation does add new coupling terms, but they are of the form of a t2 amplitude times a different two electron integral. Since for a normal single reference problem, the largest t amplitudes are of the order of 0.1, the new coupling terms are at least an order of magnitude smaller than the former coupling terms. Therefore the "true" eigenvector for a singly excited state is contained more fully within the space of single and double excitations, and the eigenvalue for the state with the truncated similarity transformed Hamiltonian is much closer to the eigenvalue of the untruncated Hamiltonian than is the eigenvalue for that state with the truncated untransformed Hamiltonian. 72 In the similarity transformed equationofmotion coupledcluster (STEOMCC) theory6466 the blocking of the matrix by similarity transformations is carried one step farther. Starting with the EOMCC transformed Hamiltonian, a second similar ity transformation69 is performed in order to minimize the coupling between the singles space and the doubles space. The dominant terms that couple these two blocks in the EOMCC Hamiltonian are the (ablIci) and (ialljk) two electron integrals. The most important of these coupling terms are set to zero in the second similarity transformation, leaving as the main coupling term between the singles and doubles blocks again a t2 amplitude times a two electron integral. Once again, the coupling is reduced by at least an order of magnitude. Since the second similarity transformation is done in such a way as to preserve the reduction of the singlestriples coupling from the first similarity transformation, the eigenvectors for states dominated by single excitations are almost exclusively composed of only single excitations, meaning that the truncation needs to leave only the space of singles. Now the final diagonalization to get the eigenvalues and eigenvectors only scales as the fourth power of the basis set, as opposed to the sixth power scaling for EOM CCSD. The tradeoff is having to perform the second similarity transformation, but the second transformation only scales as the fifth power of the size of the basis set. If the final diagonalization is over the space where an electron has been removed from an occupied orbital and placed in an unoccupied orbital, then excited states are calculated, and the method is known as excitation energy STEOMCC (EESTEOM CC).64 66 If the final space is over states with two electrons removed, the method is the double ionization potential STEOMCC (DIPSTEOMCC).66 Finally, if the space is 73 over determinants with two electrons added, the method is the double electron attachment STEOMCC (DEASTEOMCC).66 Each of these methods provides an important tool for describing interesting chemistry, and although the rest of the dissertation will focus on the EESTEOMCC variant, everything said can be applied to the other two with simple modifications. The STEOMCC Method In STEOMCC theory, the orbital space is divided into occupied and virtual orbitals, sometimes referred to as holes and particles, based on the orbital's occupancy in the reference determinant. The orbital space is then subdivided into active and inactive occupied orbitals and active and inactive virtual orbitals. To distinguish the sets, the labels i, j, k, and I will refer to generic occupied orbitals, and a, b, c, and d will refer to generic virtual orbitals. Active occupied orbitals will be m and n while active virtual will be e andf. Explicitly inactive orbitals will be indicated with a prime, such as i' and a'. The labels p, q, r, and s can refer to any orbital. All such orbitals are spacial orbitals, as opposed to the previous chapters where spin orbitals have been used. Although a STEOMCC calculation could be based on any coupledcluster or many body perturbation theory treatment of the ground state, so far only CCSD and MBPT(2) have been used.64 This discussion will focus on STEOMCC with a CCSD9 ground state, giving the STEOMCCSD method.64 In coupledcluster theory, the ground state of the molecule is represented as IWo) = erl), (41) 74 where 10) is a closed shell single determinant, often the Restricted HartreeFock (RHF) determinant. For CCSD the operator T consists of pure excitation operators of the form T = Ti + T2 = {a + atibtj. (42) i,, The curly brackets mean that the operator is normal ordered with respect to 10). The untransformed Hamiltonian, in second quantization, is H = ho + E fpq pfq + Vqprs p,{rqs. (43) p,q Pq The constant term, ho, is the energy of the reference determinant. The fpq refer to the appropriate parts of the Fock operator. The Vqprs are nonantisymmetrized two electron integrals. After the first transformation the Hamiltonian becomes H = eTHe Sho + hfpq pfq} + pqrs.{prqs} p,q P'q (44) +36 hpqrstuPlsrqttrtu. + .... With a CCSD reference state, H will contain up to sixbody terms, but the four and higher body terms do not appear in the EOMCCSD or STEOMCCSD equations. Explicit equations for these H elements, as used here, can be found elsewhere.146 In terms of H, the CCSD equations can be written as (I1H >0) = lh, = 0., (45) (jiflH10) = /bi = 0. The CCSD energy is ECCSD = (0oHl0) = ho. (46) 75 Thus, solving the CCSD equations is equivalent to setting the one and twobody pure excitation parts of H to zero. In STEOMCC, a second manybody similarity transformation69 is performed in order to also set to zero selected remaining terms which increase the net excitation level by one. But since, in diagrammatic language, the necessary operators have a line on the bottom, they do not commute and can connect to each other. Thus instead of using a transformation of the form e as originally done by Stolarczyk and Monkhorst,37 a normal ordered exponential operator {es} is used. The normal ordered exponential, introduced by Lindgren,33 excludes terms inside the normal ordering from connecting to each other, and this simplifies the equations. The S operator used in the transformation consists of two parts S = S+ + S, where S+ = S+ + S+ i< {a'te} + + sa {at j}, (47) a',c a.b Je, and S = S + S2 : Zs{m?} + 1 bs.i{mbtj}. (48) hi',mj It is the presence of the active qannihilation operators which cause the components of S to, in general, not commute.66 The double similarity transformed Hamiltonian takes the form G = es H es} = go + gpq{pq} + pqrs {ptrqs} + .... (49) p,q PIq rs 76 The similarity transformation preserves the zeros for the one and twobody pure exci tation parts, so that66 9ai = hai = 0, (410) gabij habij = 0. Also, 90 = ho = ECCSD. (411) In principle, like the CCSD equations for T above, the S equations could be derived by setting the appropriate parts of G to zero. These are66 9mi' = (i, jIGIm) = 0, gae = (a' IGIe) = 0, j (412) gabej = (a IG) = 0. Note that the second two equations set to zero the primary coupling terms between singly and doubly excited determinants. In practice, the S coefficients are not calculated this way. Doing so would lead to systems of nonlinear equations which in some situations can be numerically unstable.64 Instead, the equivalence of IPEOMCC and EAEOMCC to Fock space coupledcluster theory'47 is exploited to rewrite S in terms of IPEOMCCSD and EAEOMCCSD eigenvectors. These equations are66 m b (413) ba= ba 1(A)r A 77 The A's stand for the set of IP and EAEOMCCSD eigenvectors. There is one eigenvector per active orbital. The terms r1 stand for the coefficients in the inverse of matrix constructed from the active components of the eigenvectors A.66 After the S coefficients are determined, the problem remains of how to determine the G coefficients. Since the inverse of the normal ordered exponential is not known,69 Eq. (49) cannot be used directly. Instead the equation {es}G = {es} (414) is used. It has been shown69 that this equation is equivalent to (IeS G) = (ft eS}) (415) or G= (l{eS}) (es 1 G), (416) where the c means that the equations must be connected. In practice, all terms involving S1 are dropped.66 Since S1 does not change the net excitation level, it is not involved in the decoupling of excitation blocks. Ignoring all of the S1 terms is equivalent to a simple rotation of the eigenvector, and since the G matrix is diagonalized over all singly excited determinants this "rotation" is automatically compensated for in the diagonalization. Therefore, including the Si terms does not change the final answer, and for convenience they are left out. Leaving out S1 also simplifies Eq. (416). Since only the singlessingles block of G is needed, the second term does not contribute,66 leaving straightforward, linear equations for G in terms of H and S. The G equations in orbital form can be found in Ref. 66. 78 Finally, the G matrix is diagonalized over the space of single excitations to get the excited states. This is actually the fastest step in the process, since the final matrix to be diagonalized is so small. Afterwards, excited state and transition properties can be calculated, subject to certain approximations.66 Discussion About STEOMCC Formally, the introduction of active and inactive orbitals is not needed in STEOM CC. All orbitals could be considered active. It would simply mean that S would have no onebody component and that an IPEOMCCSD eigenvector would need to be found for each occupied orbital and an EAEOMCCSD eigenvector would be needed for each virtual orbital. In practice, this would never work, except for trivial cases. For example, there is normally a onetoone relation between IPEOMCCSD states and high lying occupied orbitals, but as the spectrum moves into the inner valence region, shakeup states appear.13 At that point, many states appear per orbital, and even the concept of principal ionization potential becomes unclear. The same will happen for EAEOM CCSD. Eventually, the iterative diagonalization procedure99 will have trouble converging. Even if it were possible to find an IPEOMCCSD or EAEOMCCSD eigenvector for every orbital, it would not be desirable. The cost of each IPEOMCCSD state is occ virt times the number of iterations needed for the diagonalization. The cost of each EAEOMCCSD state is n, cNv i per iteration. If the number of states equals the number of orbitals, then the cost of calculating S becomes as large as the cost of calculating T. It is not necessary to include all of the orbitals in the active space, anyway. The second similarity transformation serves to describe the differential dynamic correlation between the ground and excited states. But relatively few orbitals actually are involved in the 79 excitation, and it is only those orbitals involved in the excitation that need to be in the active space. The relationship of STEOMCCSD to both Fock space coupledcluster (FSCCSD) theory3245 and EOMCCSD58 59 have been discussed at some length.66 Only the highlights will be reviewed here. If the active space includes all orbitals, then STEOM CCSD and FSCCSD are numerically identical for singly excited states.66 For doubly excited or higher states, the two remain quite different. If the active space does not include all orbitals, then the two are different for singly excited states. The first difference is that STEOMCCSD can be viewed as a series of successive similarity transformations, whereas FSCCSD has just one transformation. The second difference is that in STEOM CCSD the final diagonalization is over all singly excited determinants, while the final diagonalization in FSCC is over those determinants where only the active orbitals change occupation. To consider the difference between STEOMCCSD and EOMCCSD, it is useful to fully expand out the STEOMCCSD wavefunction. It is IIr)=e {es}RI0) eo 0)ro =eT(l+S+ S2)R1 0) +eT ) 'ro (417) where RI is the single excitation part of the STEOMCCSD eigenvector and r0 is the contribution of the reference determinant. This should be compared to the EOMCCSD wavefunction, which is I) = eT(R + R2)10) + eO)ro. (418) 80 Since S1 does not change the excitation level, and is not included in the calculation anyway, it can be ignored in this discussion. It is possible to directly equate terms in Eqs. (417) and (418). They match up as follows: eTI0)ro eT0)ro eTRIO0) eTjlZ 10) eTS2R110) eT210) (419) eTS2R10) < 2 The STEOMCCSD term is on the left and the EOMCCSD term is on the right. The primary differences between the two methods are therefore that the doubles term in EOMCCSD has the flexibility to determine its optimum weight, while the doubles term in STEOMCCSD is a product of a fixed R1 and a fixed S2. On the other hand, the only triple excitations included in the EOMCCSD wavefunction are of the form eTR, while the STEOMCCSD also includes a S22RI term. Thus STEOMCCSD has a more complete description of triple excitations but gives up some flexibility in describing the doubles.66 The cost of a STEOMCC calculation in almost dominated by the cost of calculating the ground state CCSD solution and the ground state left hand state, lambda.107 Lambda is needed for transition properties.66 Both of these are iterative n2 4Ni steps. The next most expensive step is the formation of H, which also scales as ncc Nirt but is done only once. The costs of the IPEOMCCSD and EAEOMCCSD calculations were discussed above. Calculating the S coefficients will scale as occNirNirt,aN c where Nvirt,act is the number of active virtual orbitals. Forming G will scale as noccN irtNvirt,act. The final diagonalization is only an iterative N4irt step. 81 What all of these scalings imply is that the cost of the calculation is effectively independent of the number of excited states desired. In fact, the only practical effect that the number of states has on the cost is that as higher lying excited states are calculated, more orbitals will need to be included in the active space. This independence of the cost of the calculation to the number of excited states is in contrast to almost every other excited state method in use today, and it makes STEOMCC well suited to calculating large numbers of excited states. However, STEOMCC does have some drawbacks. The primary one is that the method is only suitable for states dominated by single excitations. It would be possible to derive a STEOMCC variant whereby the final diagonalization takes place over the space of both singles and doubles. This would allow the description of both doubly excited states and states which are mixed singles and doubles. The problem is that the equations become much more complicated. For example, the second term in Eq. (416) would contribute. It would also be at least as expensive as an EOMCCSD calculation. STEOMCC also has strong and poorly understood dependencies on the size and the nature of the active space. It is known that the component of the excited state vector which lies within the active space should be at least 9599%. But it is not clear how the error in the energy and properties depends on that percentage. This is an area that will require much more practical experience before it becomes clear how best to choose the active space. Finally, the calculation of excited state and transition properties in STEOMCC is unsatisfactory. Several approximations66 are currently made in the properties calculations. There is some hope, at least for the excited state properties, in that the advent of 82 gradients for STEOMCC will give a way to calculate one electron excited state properties rigorously. CHAPTER 5 THE SPECTRUM OF FREE BASE PORPHIN Introduction Because of their importance in such biological processes as photosynthesis, electron transfer, and oxygen absorption and transport, the porphyrins have been extensively studied.148 As the base molecule for the porphyrins, the electronic spectrum of free base porphin has received much attention, both with semiemperical (see, for example, Ref. 149) and ab initio methods.150, 75, 151153, 65 The interesting part of the spectrum consists of, in order, two visible peaks known as the Q bands, a very intense peak known as the B (or Soret) band, a shoulder on the B band, called the N band, and two other small peaks, the L and M bands. The spectrum can be found in Ref. 154 and Ref. 153. The traditional interpretation of the spectrum is that, with the molecule in the xy plane and with the two internal hydrogens along the x axis, the lowest energy band, the Qx band, comes from exciting to the 1'B3u state. The Qy band then comes from the I B2u, with the B band assigned to the 2'B3u and 2'B2u states.150 Nakatsuji et al.,153 based on their SACCI (symmetry adapted clusterconfiguration interaction) calculations,'55 reassigned the spectrum. They agreed with the assignments of the Q bands, but they claimed that the B band should be assigned to only the 21B3u state, with the N band being the 2'B2u state. However, their results have three significant weaknesses. The first is the method used. In principle SACCI energies (but not oscillator strengths) could be equivalent to EOM CCSD (equationofmotion coupledcluster singles and doubles) 58 and coupledcluster singles and doubles linear response53 energies. In practice, though, some nonlinear terms 84 in the underlying ground state coupledcluster result are always omitted. Also, in these calculations many double excitations were omitted based on a perturbation selection.153 For complex organic molecules even the untruncated EOMCCSD may not be sufficient. In a study of benzene and the azabenzenes, EOMCCSD had an average error of 0.32 eV for the 7r+7r* states.156 SACCI should do no better than this, unless it has some fortuitous error cancellations. The second criticism is the basis set used. Nakatsuji et al.153 discuss the importance of a rearrangement to the excitation energies, yet their basis set only had 2s type and 2p type functions on the carbons and nitrogens, giving it no flexibility to describe the 2s orbitals. They also dropped some of the occupied and virtual orbitals corresponding to the 2s orbitals, along with all of the Is orbitals on the carbons and nitrogens. The result is that the basis set had limited flexibility, no polarization functions, and no diffuse functions. The lack of diffuse and polarization functions corresponds to the conventional viewpoint that, for at least the lowest states of free base porphin, polarization and diffuse functions are not needed.152 Finally, there is a problem with the oscillator strengths. The N band appears as a shoulder to the B band, but Nakatsuji et al.153 calculate the excitation to the 21B2u state to have an oscillator strength sixtyeight percent larger than the excitation to the 2'B3u state. They argue that the N band is actually quite broad, with the B band being a narrow peak on top of it. But if that is true, then the splitting between the vertical excitation energies for the B and N bands would be less than the 0.32 eV reported.154 To answer some of these questions, a series of STEOMCCSD (similarity transformed equationofmotion coupledcluster singles and doubles)64 excited state calculations have been performed. 85 Computational Details STEOMCC In a STEOMCCSD64 66 calculation (i.e. a STEOMCC calculation64 based on a CCSD9 ground state), two similarity transformations are applied to the second quantized Hamiltonian, such that the one and twobody terms in the Hamiltonian that increase the excitation level are set to zero. This effectively blocks the Hamiltonian matrix by excitation level, so that the single excitations can be accurately calculated with a diago nalization over just the singlessingles block. The higher excitation parts of the excited state wavefunction are then implicitly included through the similarity transformations. In this way, STEOMCC has the same conceptual appeal as CI singles, but now in a fully correlated structure. The STEOMCCSD method has been implemented within ACES II. The first similarity transformation, the same as in EOMCCSD,58 involves the T amplitudes from a ground state CCSD9 calculation. The new Hamiltonian is then H = eTHeT. (51) The next step is to solve for a set of states with one electron removed and one electron added by means of the IPEOMCCSD (ionization potential EOMCCSD)67 and the EA EOMCCSD (electron attachment EOMCCSD)68 methods. These involve diagonalizing H over the space of lh2hlp and lp2plh determinants, respectively. These eigenvectors are used to determine the S coefficients in the second similarity transformation G = eS1 }. (52) The final double similarity transformed Hamiltonian G is then diagonalized over the space of single excitations. For a detailed description of the method, see Ref. 66. 86 In the current implementation the final wavefunction in terms of G has only single excitations, limiting the method to only singly excited states, but in terms of the normal Hamiltonian, the wavefunction actually includes all possible excited determinants (with only the singles having optimized coefficients). Specifically, the wavefunction contains double excitations in the form of S2R1 and T1R1, and triple excitations in the form of S2R1, T2R1, and T1S2R1, where R1 is the STEOMCC single excitation eigenvector. Since the higher excitations and the differential correlation between the ground and excited states are described via the S operator, it is critical that the excitation be described within the set of active orbitals chosen for the IPEOMCC and EAEOMCC calculations and included in S. In the calculations presented here, the active components of the excitations are almost always above 99% and in all singlets are above 98%. In test calculations the energy seems to be converged to within 0.05 eV when the active component is 98%.64 Basis Set and Geometry The basis set used for the calculations presented here come from the large ANO basis set of Widmark, Malmqvist, and Roos.157 This is a very large, generally contracted basis set. It has 14s9p4d3f primitives for carbon and nitrogen and 8s4p3d primitives for hydrogen. From this set, the first 3s and 2p contractions were used for C and N and the first s contraction was selected for H. This basis set, consisting of 230 functions, was the same as used by Merchin et al.152 It is the same number of contracted functions as that used by Nooijen and Bartlett,65 but their basis set had a much smaller set of primitives. This set is also significantly larger than the one used by Nakatsuji et al.153 87 This basis set was then extended in two different ways. First, to gauge the effect of polarization functions on the excitation energies, the first d contraction from the ANO set was added on the carbon and nitrogen atoms, while the second s function was added to the hydrogens. This gave a [3s2pld] set on C and N and a [2s] set on H, with a total of 364 contracted functions. In other calculations, to gauge the effect of diffuse functions on the excited states, a set of 2s and 2p uncontracted functions were added to the center of the molecule and to the geometrical center of each ring. The diffuse exponents were taken from Ref. 158, and have been used for naphthalene158 and biphenyl,159 where they were placed at the center of the molecule. Since two functions of each type were used, the exact exponent chosen should not matter significantly. In all calculations the first 24 occupied orbitals, corresponding to the ls orbitals on carbon and nitrogen, were left uncorrelated. Two different geometries were used in this study. The first is an idealized xray structure,160 where the molecule, without the two internal hydrogens, would be of D4h symmetry. The hydrogens make the structure D2h. It was used in several of the previous studies153, 65 and will be referred to as "D4h". With the [3s2p/ls] basis set, this geometry had a SCF energy of 982.955034 Hartrees and a CCSD energy of 985.154443 Hartrees. The [3s2pld/2s] basis set gave a SCF energy of 983.416854 Hartrees and a CCSD energy of 986.676253 Hartrees. The basis with the diffuse functions gave a SCF energy of 982.967070 Hartrees and a CCSD energy of 985.182042 Hartrees. The other geometry is a B3PW91/631G* optimized geometry,'61 which will be re ferred to as "Opt". It also has D2h symmetry. This geometry gave a SCF energy of 982.964858 Hartrees and a CCSD energy of 985.163441 Hartrees with the [3s2p/ls] 88 basis set. The diffuse basis gave a SCF energy of 982.977085 Hartrees and a CCSD en ergy of 985.191231 Hartrees. The [3s2p d/2s] basis gave a SCF energy of 983.430681 Hartrees and a CCSD energy of 986.688179 Hartrees, putting this geometry 7.5 kcal/mol below the D4h geometry. Results Ionized and Electron Attached States In all of the current calculations, the occupied part of the active space consisted of eleven orbitals. Therefore, eleven IPEOMCCSD states were calculated. In Table 51 the current results, with the three basis sets, are compared with the SACCI results,153 with the previous STEOMCC results,65 and with experiment.162 Because of the low resolution of the experimental spectrum, it is very difficult to relate the measured peaks to the calculated states. Therefore, the assignments given should be considered tentative. While adding the diffuse functions does little to change the ionization potentials, every other time that the basis set was enlarged, from the previous STEOMCC results,65 to the DZ (the [3s2p/ls] basis) results, to the polarized (the [3s2pld/2s] basis) results, the electron became more bound. Adding the polarization functions had the most dramatic effect. It caused the first two states to switch and the sixth and seventh states to switch. However, the differences are close to an order of magnitude smaller than the errors in the calculations, so nothing can be said definitively. Going from the "D4h" geometry to the "Opt" geometry had effects of less than 0.2 eV. The "Opt" geometry typically had larger IP's. Twentythree virtual orbitals (4 ag, 2 big, 3 b2g, 3 b3g, 3 au, 2 blu, 3 b2u, and 3 b3u) were included in the active space for the DZ and the polarized basis sets. For the diffuse 89 basis set several more orbitals had to be included. The diffuse basis added forty Rydberg orbitals, and most of them had orbital eigenvalues very close to zero. Therefore, to include all of the equivalent orbitals that were included for the DZ basis, the active space needed to consist of fiftynine virtual orbitals. But that many orbitals caused convergence problems. The final set included in the active space consisted of fiftyone virtual orbitals (10 ag, 4 big, 5 b2g, 5 b3g, 3 au, 8 blu, 5 b2u, and 5 b3u). If an IPEOMCC or an EAEOMCC eigenvector is less than 70% singles, the code automatically excludes it from the second similarity transformation. For the diffuse basis, one of the B3g states was excluded, leaving fifty states for the similarity transformation. In each of the calculations two positive electron affinities were predicted. They are listed in Table 52. The other electron attached states, even though they do not correspond to stable states, are still essential for the calculation of the excited states. They go into the second similarity transformation and help describe the differential correlation between the ground and excited states. Excited States The singlet excited states of free base porphin are listed in Tables 53 to 57, and the triplet excited states are in Tables 58 to 512. The important states will be discussed later, but some general comments can be made. The first is that the Rydberg states start about 4.5 eV, which is right in the region of most interest. The second is that the addition of the diffuse functions never changes the energy of the valence states by more than 0.02 eV. This has several consequences. It implies that the DZ basis set has diffuse enough tails that the diffuse functions are not needed to describe the valence region. It also means that there is essentially no mixing between Rydberg and valence excited 90 states. Finally, it suggests that adding diffuse functions to the polarized basis would not significantly change the energetic. Adding diffuse functions might, however, have an effect on the oscillator strengths. Those tend to change a little with the addition of the diffuse functions. In general, going from the "D4h" geometry to the "Opt" geometry has little effect on the spectrum. The excitation energy typically increases by about 0.1 eV. The important exceptions are the IB2u state which drops from 5.22 eV to 5.07 eV and the 1B3u state which drops from 5.33 eV to 5.23 eV. A note should be made about the oscillator strengths. The oscillator strengths are calculated as f = l,< /],)( [, (53) where w is the excitation energy. The righthand ground state is the coupledcluster wavefunction, the lefthand ground state is the lambda solution from coupledcluster theory,107 and the righthand excited state is the STEOMCC state. Currently, several approximations are introduced when calculating the lefthand excited state.64 These sometimes can cause numerical problems with the properties calculated. When the oscillator strength is listed as (), it means that the approximations are too severe, and the calculated oscillator strength is unreliable. Since the energies are calculated with the right hand excited state wavefunction, they are still correct. For the polarized basis results, another approximation is made. Because the cost of calculating the ground state lambda vector is prohibitive for that large a basis set, the lefthand ground state is estimated with the same approximations as the excited state lefthand wavefunction. In 91 a test calculation with the DZ basis, this makes a difference in the oscillator strengths of less than 20% in every case. The "D4h" geometry, polarized basis results for the singlet valence states from Tables 53 to 57 are listed in Table 513, along with CASPT2,152 SACCI, 153 previous STEOMCC,65 and EOMCCSD results. The EOMCCSD calculation is at the "D4h" geometry with the DZ basis. The valence states are numbered for convenience, but these numbers are only accurate through 3'Big. After that the Rydberg states should enter into the numbering. The energy of the first triplet state has been measured in a solvent/ethyl iodide mixture at 77K.163 The phosphorescence peak is at 1.58 eV, well above the STEOMCC polarized basis result of 1.26 eV. The CASPT2 result is between the two at 1.37 eV.152 Discussion The recent controversy over the assignment of the spectrum centers around how to assign the N band. The argument of Nakatsuji et al.,153 essentially, is that they calculated no other optically allowed states in the 34 eV range, and therefore, by default, the N band must be 21B2u. Table 513 shows the poor quality of their results. Had they used a decent basis set and had they not made the approximations that are always used in their SACCI calculations,153 their calculated energies would have approached the EOMCCSD excitation energies, meaning that they all would have increased. At that point, it becomes impossible to draw any meaningful conclusions about the assignment of the spectrum. The CASPT2152 results for the B band are too low. This could have been caused by the size of their active space. It has been shown several times164, 151, 153, 65 that for the 21B3u and 2'B2u states, Gouterman's four orbital model165167 is not sufficient; the 4blu 92 orbital must also be included. Merchan et al.152 did not include it in their active space. More CASPT2 calculations, with a larger active space and with d functions, would be very informative. The Qx and Qy bands belong to the 11B3u and 1 'B2u states. The current results show fortuitously good agreement with the Qy band. For the Qx band all of the methods predict too low an excitation energy. A fluorescence spectrum of free base porphin taken in a supersonic jet expansion168 placed the 00 transition for the Qx band at 2.0234 eV and the 00 transition for the Qy band at 2.4653 eV, slightly higher than the vertical excitation energies reported by Edwards et al.154 The addition of polarization functions increased the Qx excitation energy by 0.05 eV, so the use of much larger basis sets may further increase the calculated excitation energy, moving it towards the experimental number. The current results strongly support the original assignment for the B band being both the 21B3u and 21B2u states. The N band is then assigned to the 3'B3u state. If the assignments of Nakatsuji et al. were correct, it would mean that the polarized basis STEOMCC energies would have to be 0.19 eV too low for the first L peak and 0.32 eV too low for the second L peak. It is highly unlikely that the STEOMCC would consistently underestimate the excitation energy like that. Instead, this assignment puts 21B3u 0.14 eV above the B peak and 2'B2u 0.29 eV above the B peak. The 3'B3u state is 0.41 eV above the N peak. This error is uncomfortably large. However, the energies of all of these states dropped substantially when the polarizations functions were added. Larger basis set calculations should further decrease these gaps. The intensities still present a problem. The experimental oscillator strength of the N band is less than 0.1, and the calculated oscillator strength is 0.93. Edwards et al.154 93 mention that the shape of the B and N bands would be consistent with the two states of the B bands being split by 1500 cm1 and an intense N band donating intensity into the B band. These calculations give a splitting of 1200 cm1 between the 21B3u and 21B2u states. Thus these calculations agree with the possible interpretation given by Edwards et al.154 A B band splitting of 240 cm1 was measured in a low temperature crystal spectrum and assigned to the energy difference between the two electronic states,169 but this assignment has been disputed.170 These assignments then leave the two peaks of the L band assigned to the 3'B2u and 4'B2u states. The diffuse basis energy for the 3'B2u state agrees well with the first L peak, but the 41B2u state is 0.33 eV higher than the second L peak. This difference should also be reduced with larger basis sets. The experimental oscillator strength for the two states combined is about 0.1. The calculated oscillator strengths are well above that. The 1'Blu state sits under the L peak, but its intensity is so small that it is not visible. The one remaining problem with this assignment is the M peak. The calculated excitation energy for the 4'B3u state is 5.17 eV, 0.33 eV lower than the M band maximum. The intensities match well, though, and there is no other assignment that makes sense. To say that 4'B3u is part of the L band would require that the excitation energy to drop by 0.5 eV, and it would cause problems assigning the other states. Still, it is not clear why the calculated excitation energy would be so low. Finally, a point should be made about the cost of these calculations. For the polarized basis calculation there are 57 occupied and 283 virtual functions. This gave 2479 T, and 34 170 895 T2 amplitudes. The time taken for each step on a Cray C90 is listed in Table 514. It took only 36 seconds per excited state to calculate the energies and properties. 94 Most of that time was used calculating the properties. The energies took 1.4 seconds each. This is a very important advantage of the STEOMCC method. The total time for the calculation is effectively independent of the number of excited states calculated. That makes it possible to study entire spectra instead of just a few selected states. Conclusions STEOMCCSD64 calculation on free base porphin using a [3s2pld/2s] basis set are presented. These are the first reported calculations on excited states above the lowest that use polarization functions. For the important optically allowed excited states, the polarization functions have a significant effect. These calculations strongly support the traditional interpretation of the spectrum, in that the intense B band is assigned to both the 2'B3u and 21B2u states. This study is possible only because the second similarity transformation in STEOM CCSD means that the excited states can be represented in terms of single excitations with respect to a highly modified Hamiltonian. Thus it is possible to efficiently calculate many excited states at once. 
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