Getting excited states of \(Ir^{16+}\)

In the following we are interested in getting the ground (\(^{2}S_{1/2}\)) and two closest excited states (\(^{2}P_{1/2}\), \(^{2}P_{3/2}\)) of the \(Ir^{16+}\) cation and their correlation energies. The electronic configurations of these states are

(^2S)     : [Xe] 4f(14) 5s(1) 5p1/2(0)  5p3/2(0)
(^2P_1/2) : [Xe] 4f(14) 5s(0) 5p1/2(1)  5p3/2(0)
(^2P_3/2) : [Xe] 4f(14) 5s(0) 5p1/2(0)  5p3/2(1)

For simplicity, we will work with the wo-component Hamiltonian (.X2C) and employ the smallest v2z decontracted basis set by K.Dyall. Due to the convergence problem of the standalone AMFI atomic SCF code we keep the +2 charge (.AMFICH) for mean-field orbitals.

There are two ways to obtain excited states - (i) from the converged SCF state, and, (ii) only at the correlated level from the 2P_aver SCF state.

For subsequent Coupled Cluster (CC) correlated calculations please soften the DHOLU variable in the subroutine DENOM (file src/relccsd/cceqns.F) to the value of 5.0D-4 and recompile DIRAC. Note that this is not recommended approach as the p32 states are in general not well described at the CC level. Nevertheless, with this little trick we can compare desired excited states calculated with both CC and Fock-space CC methods.

The \(^{2}S_{1/2}\) ground state

Thanks to the linear symmetry, having two irreps, one can place the unpaired electron the 1st irrep to get the \(^{2}S_{1/2}\) ground state. SCF calculations are followed by two CC calculations, where in the second one uses orbital energies for denominators.

pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2S12.scf_cc33e.2fs.inp
pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2S12.scf_cc33e_oe.2fs.inp

Input files to download are Ir.dyall_v2z.lsym.mol, Z61.x2c.2S12.scf_cc33e.2fs.inp, Z61.x2c.2S12.scf_cc33e_oe.2fs.inp. Corresponding output files are Z61.x2c.2S12.scf_cc33e.2fs_Ir.dyall_v2z.lsym.out, Z61.x2c.2S12.scf_cc33e_oe.2fs_Ir.dyall_v2z.lsym.out.

The SCF \(^{2}P_{1/2}\), \(^{2}P_{3/2}\) excited states

By placing the unpaired electron into 2nd irrep one gets the \(^{2}P_{1/2}\) first excited state:

pam  --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2P12.scf_cc33e.2fs.inp --get "DFCOEF=DFCOEF.v2z.2P12"
pam  --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2P12.scf_cc33e_oe.2fs.inp

Input files to download Z61.x2c.2P12.scf_cc33e.2fs.inp, Z61.x2c.2P12.scf_cc33e_oe.2fs.inp. Corresponding output files are Z61.x2c.2P12.scf_cc33e.2fs_Ir.dyall_v2z.lsym.out, Z61.x2c.2P12.scf_cc33e_oe.2fs_Ir.dyall_v2z.lsym.out.

How to obtain the \(^{2}P_{3/2}\) second excited state at the SCF level, and, consequently, at the CC level ? For that, we utilize the .REORDER MO keyword with reading of the DFCOEF.v2z.2P12 file from the previous run. Likewise one uses overlap selection in the SCF step:

pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2P32.scf_cc33e.2fs.inp --put "DFCOEF.v2z.2P12=DFCOEF"
pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2P32.scf_cc33e_oe.2fs.inp --put "DFCOEF.v2z.2P12=DFCOEF"

Corresponding input files to download are Z61.x2c.2P32.scf_cc33e.2fs.inp, Z61.x2c.2P32.scf_cc33e_oe.2fs.inp. Output files are Z61.x2c.2P12.scf_cc33e.2fs_Ir.dyall_v2z.lsym.out, Z61.x2c.2P12.scf_cc33e_oe.2fs_Ir.dyall_v2z.lsym.out.

The CCSD(T) \(^{2}P_{1/2}\), \(^{2}P_{3/2}\) excited states

The other option is to start from the \(^{2}P_{aver}\) averaged single determinant state and distinguish between individual \(^{2}P_{1/2}\) and \(^{2}P_{3/2}\) states at the Coupled Cluster correlated level thanks to the linear symmetry.

First we test the averaged, \(^{2}P_{aver}\), state:

pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2Paver.scf_cc33e.2fs.inp
pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2Paver.scf_cc33e_oe.2fs.inp

Files to download are Z61.x2c.2Paver.scf_cc33e.2fs.inp, Z61.x2c.2Paver.scf_cc33e_oe.2fs.inp.

Afterwards we can proceed to the individual spin-orbit distinguished states, based on \(M_{J}\) splitted occupation of each fermion irrep at the Coupled Cluster level.

First the first excited state, \(^{2}P_{1/2}\):

pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2Paver.scf_cc33e_2P12.2fs.inp
pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2Paver.scf_cc33e_oe_2P12.2fs.inp

Input files to download are Z61.x2c.2Paver.scf_cc33e_2P12.2fs.inp, Z61.x2c.2Paver.scf_cc33e_oe_2P12.2fs.inp. Corresponding output files are Z61.x2c.2Paver.scf_cc33e_2P12.2fs_Ir.dyall_v2z.lsym.out, Z61.x2c.2Paver.scf_cc33e_oe_2P12.2fs_Ir.dyall_v2z.lsym.out.

Then we proceed to the second excited state, \(^{2}P_{3/2}\):

pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2Paver.scf_cc33e_2P32.2fs.inp
pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.2Paver.scf_cc33e_oe_2P32.2fs.inp

Files to download are Z61.x2c.2Paver.scf_cc33e_2P32.2fs.inp, Z61.x2c.2Paver.scf_cc33e_oe_2P32.2fs.inp. Corresponding output files are Z61.x2c.2Paver.scf_cc33e_2P32.2fs_Ir.dyall_v2z.lsym.out, Z61.x2c.2Paver.scf_cc33e_oe_2P32.2fs_Ir.dyall_v2z.lsym.out.

The \(^{2}S_{1/2}\), \(^{2}P_{1/2}\) and \(^{2}P_{3/2}\) FSCCSD states

Simple and very stable approach to obtain ground and multiple excited states in one step is through the Fock-space Coupled Cluster method. Starting from the closed-shell system, \(Ir^{17+}\), one gets - by solving (01) sector all three correlated states of interest, \(^{2}S_{1/2}\), \(^{2}P_{1/2}\) and \(^{2}P_{3/2}\) :

pam --noarch --mw=120  --mol=Ir.dyall_v2z.lsym.mol --inp=Z61.x2c.scf_fscc01_33ce_5s5p.2fs.inp

The input file to download is Z61.x2c.scf_fscc01_33ce_5s5p.2fs.inp. Corresponding output file is Z61.x2c.scf_fscc01_33ce_5s5p.2fs_Ir.dyall_v2z.lsym.out.

Overview of excitation energies

In the following table we summarize excitation energies. All values are in a.u. Energies in the Table are not rounded, the are cut to 8 decimal places (“oe” means orbital energies used in CC denominators, otherwise recalculated diagonal Fock matrix elements).

Method	^2S_{1/2}	^2P_{1/2}	^P_{3/2}	2S12-2P12	2S12-2P32
(SCF ref)
SCF	-17751.10181462	-17749.67796221	-17748.96107014	1.42385	2.14074
CCSD	-17751.90589433	-17750.48885480	-17749.77167089	1.41704	2.13422
CCSDoe	-17751.90589433	-17750.48885479	-17749.77167088	1.41704	2.13422
CCSD(T)	-17751.90998637	-17750.49777405	-17749.78015403	1.41221	2.12983
CCSD(T)oe	-17751.90999205	-17750.49779342	-17749.78020119	1.41220	2.12979
(CC ref)
CCSD	-17751.90589433	-17750.48895732	-17749.77154045	1.41694	2.13435
CCSDoe	-17751.90589433	-17750.48895728	-17749.77154043	1.41694	2.13435
CCSD(T)	-17751.90998637	-17750.49779338	-17749.78012458	1.41219	2.12986
CCSD(T)oe	-17751.90999205	-17750.49784867	-17749.78024342	1.41214	2.12974
FSCCSD	-17751.90324662	-17750.48431436	-17749.76770431	1.41893	2.13554

It seems that quality of computed excitation energies increases in the line SCF-FSCCSD-CCSD-CCSD(T). Triple excitations (CCSD(T) results) are significant.