Beryllium Triatomic Cluster - A Comparative Study of the LR and EOM Approaches

7.3 A Comparative Study of the LR and EOM Approaches

7.3.1 Beryllium Triatomic Cluster

The Beryllium trimer (Be3) is a prominent multireference system as the quasi-degeneracy

of the 2s-2p orbitals of Be increases significantly with the formation of the trimer. Due to the same reason, there are several excited states of this system, dominated mainly by double excitations, appearing at relatively low energies. A good description of the ground state of Be3 has been obtained using FCI [162], MRCI [163, 164] and also with

the single-reference CCSDt methods [165]. Calculations of the excited state energies have also been attempted using single-reference EOM methods like CR-EOM-CCSD(T) and EOM-CCSDt [165] and compared with respect to available FCI results [166]. Both the EOM and LR variants of the ic-MRCCSD theory are applied here to calculate these excited states and the results are compared among each other and with the single-reference methods mentioned earlier.

100 Chapter 7 Excitation Energies from ic-MRCC

Table 7.4: A comparison of vertical excitation energies (in eV) for several singlet states

of Be3 obtained using ic-MRCC-EOM, ic-MRCC-LRT, different variants of single-

reference based EOMCC as a difference with respect to FCI.

State type1 _{CR-EOMCCSD(T)}2 _EOMCCSDt2 _ic-MRCCSD

CAS(6e,9o) CAS(6e,12o)

(I) (II) EOM LR EOM LR

1E00 s -0.023 -0.012 0.008 0.001 -0.002 0.003 0.003 1A001 d 0.097 0.28 0.235 0.262 0.057 0.053 1E0 s -0.052 -0.009 0.023 -0.006 -0.008 -0.003 -0.005 2E00 d 0.073 0.234 0.219 0.265 0.050 0.045 2E0 d 0.352 0.028 0.200 0.034 0.034 0.027 0.026 1A0 2 s -0.100 -0.019 -0.019 0.004 0.005 0.005 0.004 2A0 1 s -0.014 0.027 0.065 0.010 0.011 0.006 0.002 3E00 _d _0.506 _0.075 _0.274 _0.113 _0.093 _0.028 _0.023 3E0 _d _0.258 _0.167 _0.367 _0.380 _0.385 _0.050 _0.045 1A00 2 s,d 0.198 0.142 0.277 0.018 -0.001 0.007 0.016 2A00 1 d 0.291 0.129 0.317 0.228 0.281 0.034 0.028 4E0 _d _0.153 _0.061 _0.263 _0.392 _0.395 _0.010 _0.010 3A0 1 d 0.393 0.206 0.348 0.325 0.316 0.037 0.012 4E00 _d _0.054 _0.118 _0.312 _0.184 _0.254 _0.041 _0.034 3A00 1 s,d -0.099 0.122 0.218 0.062 0.096 0.010 0.009 2A00 2 d 0.264 0.063 0.254 0.234 0.305 0.0042 0.060

1. Character of excitation with respect to the Hartree-Fock determinant: s = single-excitation dominated d = double-excitation dominated

2. Ref. [165]

The ic-MRCC calculations to get excitation energies of the Be3 molecule are done

using the same geometry and ANO basis which have been used in Refs. [165, 166]. The original symmetry of Be3, D3h, is used here to assign its excited states, although the

calculations are done using C2v symmetry. Three core 1s orbitals of Be are kept frozen in

order to compare the results with that of FCI. The results of the ic-MRCCSD excitation energies for both of the variants, along with the results obtained from the EOM-CCSD , CR-EOM-CCSD(T) and EOM-CCSDt calculations, are presented as differences with respect to the FCI results in Tab. 7.4. The ic-MRCCSD calculations are done using two different active spaces: (6e,9o) and (6e,12o). CAS(6e,9o) uses the lone 2s orbitals and the 2p orbitals residing in the molecular plane (2px and 2px in this case) as the active

orbitals. For CAS(6e,12o), on the other hand, the rest of the 2p orbitals (2pz) of the

system are added to the existing active orbitals of CAS(6e,9o). The orbitals involved in the ic-MRCCSD calculations are obtained from consecutive Hartree-Fock and CASSCF calculations with all the core 1s orbitals being frozen for the latter. This ensures that the

7.3 A Comparative Study of the LR and EOM Approaches 101 same orbital space is used while correlating electrons for all the calculations presented in Tab. 7.4. In the EOM-CCSDt method, the definition of the three body excitation operators needs the use of active orbitals. Among the two different variants of EOM- CCSDt used here, variant (I) has at least one active occupied and at least one active unoccupied spin-orbital indices in its definition of three body excitation operators. For the EOM-CCSDt(II), on the other hand, triple excitations contain at least two active occupied and two active unoccupied spin-orbitals thus spanning a relatively smaller space than the first variant. The EOM-CCSDt results, obtained from Ref. [165] and presented in Tab. 7.4, use the same active space of (6e,12o) as mentioned above to define the three body excitation operators.

The choice of the active space for ic-MRCCSD changes the accuracy of the excitation energies only for the doubly excited states. The results obtained using the active space of (6e,12o) are invariably more accurate for these states compared with those obtained for the CAS(6e,9o) as the former involves more configurations to describe the reference wave functions. The excitation energies do not change significantly between the use of the two variants of ic-MRCCSD. Though the differences between the results for EOM and LR are greater for the lower active active space, the highest among these differences is 0.05 eV obtained for the 2A00₁ which is a doubly excited state.

Between the results of the two variants of EOM-CCSDt, the variant (I) provides excitation energies with better accuracy as it includes more configurations obtained through three-body excitations than the other variant. EOM-CCSDt(I) also provides better results than CR-EOM-CCSD(T) with the exceptions of two higher-lying excited states, 4E00 and 3A00₁. The ic-MRCCSD results obtained using either of the active spaces are better than the single-reference results for all the excited states dominated by single excitation. For the doubly excited states, ic-MRCCSD provides accuracy comparable to the EOM-CCSDt(II) when the multireference calculations are done using the lower active space of (9e,6o). For these states, ic-MRCCSD results are better than that of EOM- CCSDt(II) for all the states with an exception being the state 3A0₁. However, by using the higher active space of (6e,12o), ic-MRCCSD produces excitation energies which are always more accurate even than those obtained for EOMCCSDt(I). The highest error in excitation energies with respect to FCI results is 0.057 for ic-MRCCSD/CAS(6e,12o) method, where EOM-CCSDt(I) produces an error as big as 0.2. Moreover, the errors in excitation energies obtained using ic-MRCCSD are always of similar magnitudes even for the higher lying states, where the magnitude of the errors increase for these states when the single-reference methods are used.

102 Chapter 7 Excitation Energies from ic-MRCC

In document Excitation energies and response properties of molecules using internally contracted multireference coupled cluster methods (Page 109-112)