ACES2 Labs. Part B: Calculating electronically excited, ionized, and electro-attached states using equation of motion coupled cluster calculations. I. Electronically excited states. Purpose: - Calculation and characterization of electronic excitation spectra - Valence vs. Rydberg transitions. - CIS calculations. - EOMCC calculations of excited states - Difficulty of doubly excited states. I.a. CI singles calculations using the ACES2 program. Let us return to ethylene, the molecule we investigated using Gaussian calculations. We will now perform calculations using the ACES2 program. Use the 6-31++G** basis, as in the Gaussian CIS calculation (6-31G++(d,p). Let us first redo the CIS calculation in ACES2. You can take the geometry of the ground state from Gaussian, and adjust to ACES2 format. Then you can use cartesian coordinates by specifying COORDINATES =CARTESIAN. That would allow a direct comparison with Gaussian. Alternately the following input file uses the same geometry specified using a zmat. We took the bonddistances and angles from the Gaussian output file after geometry optimization. C2H4 CIS calculation C C 1 CC H 2 CH 1 A H 2 CH 1 A 3 D180 H 1 CH 2 A 3 D0 H 1 CH 2 A 5 D180 CC = 1.3241 CH = 1.0827 A = 121.7397 D180=180 D0=0.0 *ACES2(BASIS=6-31++G*,CALC=MBPT(2),PROGRAM=MN_A3 EXCITE=TDA,EE_SYM=3-3-3-3-3-3-3-3/3-3-3-3-3-3-3-3) I specified a large number of roots, as CIS calculations are cheap. The PROGRAM=MN_A3 keyword indicates a special branch through the ACES2 program. It is software developed in my research group, much of it written by myself. Specify EXCITE=TDA, (Tamm- Dancoff approximation is the physicists name for CI singles) and for example EE_SYM=3-3-3-3-3-3-3-3/3-3-3-3-3-3-3-3, to request 3 states per symmetry-block of both singlet and triplet spin (which are separated by the dashed line). The relevant part of the output occurs near the string:
Summary of excitation energy TDA calculation Multiplet root irrep energy diff (ev) osc. strength % singles total energy Singlet 1 [5] 7.22320144 0.9407E-01 100.00-77.77749914 Singlet 1 [7] 7.84828482 0.0000E+00 100.00-77.75452776 Singlet 1 [2] 7.87846174 0.5514E+00 100.00-77.75341878 Singlet 1 [6] 8.03389727 0.0000E+00 100.00-77.74770663 Singlet 1 [1] 9.05311758 0.0000E+00 100.00-77.71025099 Singlet 1 [8] 9.17131785 0.0000E+00 100.00-77.70590721 This little table summarizes the excited state calculation. It lists the spin multiplicity, root # and symmetry, indicated by the irrep number of the computational point group, e.g. [5]. Then it lists the excitation energy (in ev), the oscillator strength (in a.u.) and in the final column the total electronic energy (in a.u.). If you scroll down this Table in the output file, you will also see the triplet states listed: Triplet 1 [2] 3.68290135 0.0000E+00 100.00-77.90760273 Triplet 1 [5] 7.00961314 0.0000E+00 100.00-77.78534836 Triplet 1 [7] 7.74454580 0.0000E+00 100.00-77.75834010 Triplet 1 [6] 7.89157083 0.0000E+00 100.00-77.75293703 Triplet 1 [1] 8.28098859 0.0000E+00 100.00-77.73862619 Triplet 2 [7] 8.67664441 0.0000E+00 100.00-77.72408611 The meaning of the symmetry labels is most easily extracted if you look at the SCF orbital energies: 6 3-0.5907724960-16.0757441583 Ag Ag (1) 7 38-0.5058141270-13.7639083603 B1g B1g (4) 8 47-0.3777695387-10.2796364009 B1u B1u (5) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 9 4 0.0497778150 1.3545238215 Ag Ag (1) 10 17 0.0623478622 1.6965723502 B3u B3u (2) 11 30 0.0642354476 1.7479361817 B2u B2u (3) The numbers in the final column here are the same as the numbers of the irreps, in the socalled computational symmetry group, which is always Abelian. Hence [5] is the B 1u state in D 2h symmetry group. Besides the energies we can also look at properties, in particular their dipole and quadrupole moments.
Properties from EE-TDA calculation State Energy Dipole Moments Electronic Second Moments 2S+1 # irp E(eV) <x> <y> <z> <xx> <yy> <zz> <r^2> ---- parent state 0.000 0.000 0.000 49.6 21.4 12.0 82.9 1 1 [1] 9.05 0.000 0.000 0.000 54.9 27.1 28.8 110.8 1 2 [1] 12.66 0.000 0.000 0.000 77.3 48.8 20.0 146.1 1 3 [1] 13.07 0.000 0.000 0.000 57.7 38.7 21.8 118.1 1 1 [2] 7.88 0.000 0.000 0.000 55.9 23.2 16.8 96.0. 3 1 [1] 8.28 0.000 0.000 0.000 54.6 26.8 28.5 109.9 3 2 [1] 11.63 0.000 0.000 0.000 61.4 35.0 16.8 113.2 3 3 [1] 12.74 0.000 0.000 0.000 65.8 41.8 20.0 127.6 3 1 [2] 3.68 0.000 0.000 0.000 51.5 21.4 12.2 85.1 This information is of help in characterizing each excited state as valence or Rydberg by examining the second moments of the charge distributions, in particular the <r^2> value. Rydberg states have larger <r^2> value as they describe excitation of a compact molecular orbital into a more diffuse virtual orbital. These Rydberg orbitals are different from the antibonding orbitals, which often have more or less the same spatial extent as the bonding orbitals, but a different nodal pattern (and an antibonding nature). For the ground state of ethylene <r^2> is 82.9. For other states it is usually significantly larger. The first state of symmetry [2] has comparable second moment 96.0, but it actually also has substantial Rydberg character. The corresponding triplet state has a much clearer triplet character, with <r^2>=85.1, very close to that of the ground state. In solution phase Rydberg states are suppressed, and photochemical processes typically involve valence excited states. This is why we like to distinguish Rydberg from valence excited states. Another way to analyse the Rydberg vs. valence character is to match up corresponding pairs of singlet and triplet states by looking at the character of the eigenvectors and/or their symmetry. For matching pairs of Rydberg states the energy splitting is usually small (a few tenth s of a electronvolt to even much smaller differences, as states get more diffuse), while for valence states it is huge, here 4.20 ev, the difference between 7.88 and 3.68 ev. Let me copy a piece of the output that describes the character of the states:
***************************************** * * * TDA Excitation Energies * * Spin State : Singlet * * * ***************************************** @delfile, file 5, DERGAM will be deleted root # 1 [1], Active component 100.00 eigenvalue = -1 [5] -> 1 [5] 0.695843 48.4 % root # 2 [1], Active component 100.00 eigenvalue = -1 [4] -> 1 [4] 0.683709 46.7 % root # 1 [2], Active component 100.00 eigenvalue = -1 [5] -> 1 [6] 0.660833 43.7 % -1 [5] -> 2 [6] -0.201176 4.0 % root # 1 [5], Active component 100.00 eigenvalue = -1 [5] -> 1 [1] 0.670950 45.0 % -1 [5] -> 3 [1] -0.173779 3.0 % 9.053 ev 12.656 ev 7.878 ev 7.223 ev and the corresponding section(s) for triplet states: ***************************************** * * * TDA Excitation Energies * * Spin State : Triplet * * * ***************************************** @delfile, file 5, DERGAM will be deleted root # 1 [1], Active component 100.00 eigenvalue = -1 [5] -> 1 [5] 0.701499 49.2 % root # 1 [2], Active component 100.00 eigenvalue = -1 [5] -> 1 [6] 0.534897 28.6 % -1 [5] -> 2 [6] -0.453317 20.5 % root # 1 [5], Active component 100.00 eigenvalue = -1 [5] -> 1 [1] 0.649717 42.2 % -1 [5] -> 3 [1] -0.230719 5.3 % 8.281 ev 3.683 ev 7.010 ev You may notice that the character of the states only adds up to 50%. This is because there is both an α β and a β α part of the excitation (see notes on lab: excitations in Gaussian). I don t want you to forget that! Root #1 of [2] is the π π * valence excitation in ethylene. It has a huge singlet-triplet splitting (due to exchange effects). You
can also see that the character of the state is somewhat different for singlet and triplet. We say that the singlet state has acquired some Rydberg character. I. b: Excited states using Equation of Motion Coupled Cluster Theory: EOMCC. Let us now perform a EOM-CCSD calculation on ethylene using the same 6-31++G** basis set and geometry. We will also calculate properties and use the following input. C2H4 EOMCC calculation C C 1 CC H 2 CH 1 A H 2 CH 1 A 3 D180 H 1 CH 2 A 3 D0 H 1 CH 2 A 5 D180 CC = 1.3241 CH = 1.0827 A = 121.7397 D180=180 D0=0.0 *ACES2(BASIS=6-31++G**,CALC=CCSD,PROGRAM=MN_A3,DROPMO=1-2 EXCITE=EOMEE,EE_SYM=2-2-2-2-2-2-2-2 ESTATE_PROP=EXPECTATION) We drop the two 1s orbitals from the correlated calculation (DROPMO=1-2, or DROPMO=1>2, to specify a range). I selected the first two roots (following the prior CIS calculation) by assigning the EE_SYM keyword. The program first calculates the CCSD ground state, and then it calculates a left hand eigenstate, the so-called Lambda state corresponding to the ground state in order to define transition properties. In the EOMCC step both left and right hand vectors of the transformed Hamiltonian T H e He T = are calculated, and this allows the calculation of properties (as specified by the estate_prop keyword). The output follows the same logic as the output from a TDA calculation. Summary of excitation energy eom-cc calculation Multiplet root irrep energy diff (ev) osc. strength % singles total energy Singlet 1 [5] 7.32803275 0.7951E-01 95.22-78.07930845 Singlet 1 [7] 8.01624975 0.0000E+00 95.41-78.05401695 Singlet 1 [6] 8.05818721 0.0000E+00 94.87-78.05247578 Singlet 1 [2] 8.30224164 0.3847E+00 96.49-78.04350695 Singlet 2 [7] 8.84635147 0.0000E+00 93.54-78.02351128 Now the entry %singles also attains a significant meaning. It indicates the fraction of the excited state that can be characterized as singly excited. If this fraction drops below 85-90% then the results are increasingly suspicious. More elaborate tools would be needed to obtain accurate results. We also get properties to characterize the states:
Properties from EE-EOM calculation State Energy Dipole Moments Electronic Second Moments 2S+1 # irp E(eV) <x> <y> <z> <xx> <yy> <zz> <r^2> ---- parent state 0.000 0.000 0.000 49.8 21.4 11.7 82.9 1 1 [1] 9.16 0.000 0.000 0.000 52.9 25.5 27.8 106.2 1 2 [1] 11.79 0.000 0.000 0.000 72.9 46.1 19.2 138.2 1 1 [2] 8.30 0.000 0.000 0.000 55.3 22.6 17.3 95.2 and we can investigate the excitation patterns that define these states, e.g. find this section, which describes the valence state. Look for the section: Solving for right hand root # 1 [2] Convergence reached after 10 cycles. Eigenvalue: 0.305102 a.u. 8.302242 ev 66962.07 cm-1 149.3383 nm Maximumum residual :.423E-05 Contribution from single excitations 96.49 % Single excitation coefficients a [irrep] i [irrep] 5 [2], -1 [1] ; 0.057536 6 [2], -1 [1] ; 0.035407 1 [1], -1 [2] ; -0.034197 4 [1], -1 [2] ; -0.032228 2 [4], -1 [3] ; 0.032937 1 [3], -1 [4] ; 0.043653 2 [3], -1 [4] ; -0.052835 3 [3], -1 [4] ; -0.036439 1 [6], -1 [5] ; 0.659001 2 [6], -1 [5] ; -0.170289 Or this section, which describes the lowest Rydberg state Solving for right hand root # 1 [5] Convergence reached after 10 cycles. Eigenvalue: 0.269300 a.u. 7.328033 ev 59104.55 cm-1 169.1917 nm Maximumum residual :.590E-05 Contribution from single excitations 95.22 % Single excitation coefficients a [irrep] i [irrep] 1 [5], -2 [1] ; -0.009021 1 [6], -1 [2] ; -0.004831 3 [6], -1 [2] ; 0.001896 1 [1], -1 [5] ; 0.655457 2 [1], -1 [5] ; -0.055268 3 [1], -1 [5] ; -0.169868
4 [1], -1 [5] ; 0.119054 5 [1], -1 [5] ; 0.008529 6 [1], -1 [5] ; 0.008425 7 [1], -1 [5] ; -0.012344 Double excitation coefficients a [irrep] a [irrep] i [irrep] i [irrep] 1 [5], 1 [1] ; -1 [5] -1 [5] -0.042168 1 [1], 1 [5] ; -1 [5] -1 [5] -0.042168 Here also some double excitation coefficients are printed. This printing is suppressed if the coefficients are too small to be of interest. You can compare the results from EOMCC with results from the CIS calculation. Which states correspond? Characterize the Rydberg character of the states. In general valence excited states are calculated to be a bit high in EOMCC calculations. Here the error may be close to 0.5 ev. For Rydberg states the results from EOMCC calculations are expected to be very accurate, as long diffuse basis sets are used. Here the 6-31++G** basis, where the ++ indicates diffuse functions. Note: Starting from an RHF ground state you will only find singlet states in an EOM- CCSD calculation. Do not specify triplet states through the EE_SYM keyword. We can also do geometry optimizations using TDA or EOMCC. This will be discussed in a later lab when we discuss the similarity transformed EOMCC approach. Assignments excitation energy EOMCC calculations. 1. Perform a EOM-CCSD calculation on BH (R=1.2266 Å), and use the 6-31+G* basis set (This is a reasonable basis set for excitation energies). Analyse the excitation character of the excited states. Which states have appreciable double excitation character? The eigenvalues of these latter states are suspect at the EOM-CCSD level. What are proper symmetry labels for the states? The experimental values for the low lying vertical excitations are 2.87, 5.72, 6.49, 6.86, 7.58 and 7.76 ev. Can you make an assignment of the spectrum based on your calculations? Make clear which assignments are uncertain. 2. Calculate the EOM-CCSD spectrum of formaldehyde, using the ground state geometry optimized at the CC-PVTZ/CCSD level. Use the 6-31-++G** basis to calculate excitation energies. Compare CIS and EOM-CCSD. You can run a similar calculation in Gaussian, and also compare to CAM-B3LYP TD-DFT excitation energies.
Part B II: Ionized states using the IP-EOMCC approach. Purpose. - Calculating photo-electron spectra. - Optimizing geometries for ionized states, and correlating structures with orbitals. - Calculating vibrational frequencies for radicals and ionized states. Consider N 2 at R=1.097 Å. Let us calculate the vertical ionization spectrum using the Dunning cc-pvtz basis set, using the following input. N2 photo-elelectron spectrum N N 1 R R=1.097 *ACES2(CALC=CCSD,BASIS=CC-PVTZ IP_CALC=IP_EOMCC,IP_SYM=3-1-1-0-2-0-0-0) The relevant output is given in the summary table: Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [1] 15.57981922 93.15-108.80847705 Doublet -1 [3] 17.19921155 95.99-108.74896550 Doublet -1 [2] 17.19921155 95.99-108.74896550 Doublet -1 [5] 18.81929532 89.37-108.68942853 Doublet -2 [1] 38.60795827 53.75-107.96220883 Doublet -2 [5] 410.54290685 82.20-94.29385610 Doublet -3 [1] 410.65130104 82.16-94.28987269 The ionization potentials can be compared with the orbital energies from a Hartree-Fock calculation. The negative of the orbital energy (positive) is a approximation of the ionization potential. Here they are in the cc-pvtz basis set: ORBITAL EIGENVALUES (ALPHA) (1H = 27.2113957 ev) MO # E(hartree) E(eV) FULLSYM COMPSYM ---- -------------------- -------------------- ------- --------- 1 1-15.6813946958-426.7126361945 SGg+ Ag (1) 2 31-15.6778414692-426.6159479393 SGu- B1u (5) 3 2-1.4701240009-40.0041259168 SGg+ Ag (1) 4 32-0.7772128143-21.1490454343 SGu- B1u (5) 5 3-0.6324092631-17.2087387037 SGg+ Ag (1) 6 14-0.6123161192-16.6619762144 PIu B3u (2) 7 21-0.6123161192-16.6619762140 PIu B2u (3) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
You can see that the Koopmans orbital energies give the wrong ordering of the σ and π orbitals in N 2. This is a famous observation. If you read current quantum mechanic text books, like McQuarry, they discuss the ordering of σ and π orbitals based on molecular orbital theory. It is only when electron correlation is included that the σ orbital (really σ state!) has a lower ionization potential than the π orbital. We can learn something else from the IP-EOMCC output. Many of the deeper lying ionized states have a very low %singles character, e.g. 53.75, 89.37, 82.20%. This means that these results cannot be trusted at this level of theory. Experimentally one would observe several other states that are very close in energy. They are called shake-up states by spectroscopists. Finally: How did I know which symmetries to specify in the IP_SYM keyword? I did a prior HF calculation, and took the symmetries from the occupied orbitals, in the orbital table, or alternatively from the output section: @VSCF: Final occupancies: Alpha population by irrep: 3 1 1 0 2 0 0 0 Beta population by irrep: 3 1 1 0 2 0 0 0 Note that sometimes only initial populations are specified, if the populations do not change in the SCF calculation. Let us do a similar calculation for ethylene. If you use the same geometry as before, and the cc-pvtz basis set, you should get the following results: Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [5] 10.70673818 95.69-78.06211254 Doublet -1 [4] 13.05844860 94.25-77.97568880 Doublet -1 [1] 14.95790295 93.44-77.90588517 Doublet -1 [3] 16.29278157 90.95-77.85682930 Doublet -1 [2] 19.58511142 87.96-77.73583845 Doublet -2 [1] 24.48888570 77.92-77.55562813 Doublet -2 [2] 291.62601126 80.53-67.73852309 Doublet -3 [1] 291.67996953 80.49-67.73654016 Again you can compare the ordering of levels and orbital eigenvalues at Koopmans level and the correlated level. Are all states calculated reliably? Why? Geometry optimization and vibrational frequencies for IP-EOMCC - Consider the NO 2 anion from a previous lab. Calculate the ionization spectrum (i.e. radical states of the neutral) using an IP-EOMCC calculation (see above). You can use DROPMO=1>3, to exclude the three core-orbitals from the calculation. Remember that the reference state is the anion, so CHARGE=-1. Also perform a Gaussian calculation to analyse the orbitals. By analysing the orbitals you can predict geometry changes upon ionization from a particular orbital. For the three lowest states perform an IP-EOMCC
optimization. For this you need to set the proper keywords. Here is an input file, using the TZ2P basis set. NO2 anion optimization O N 1 NO* O 2 NO* 1 A* NO = 1.2612565945 A = 115.9110636345 *ACES2(BASIS=TZ2P,CHARGE=-1,CALC=CCSD,DROPMO=1>3 IP_CALC=IP_EOMCC,IP_SYM=1-1-1-1) *mrcc_gen gradient_type=ip igrad_sym=3 igrad_state=-1 igrad_mult=2 *end The gradient_type keyword indicates the method that is used to define the gradient (Here an IP calculation). The other other keywords specify the state you want to optimize. The igrad_mult keyword specifies the spin-multiplicity and the igrad_sym keyword indicates the symmetry of the state of interest. The igrad_state keyword indicates the orbital you ionize from, counting down from the Fermi level. The counting is done per symmetry block i.e. you have state 1 [1], -2 [1], -1 [2], etc. The labeling follows the summary output in the program (see below). At each new geometry the calculation calculates all of the states you specified by IP_SYM keyword. Following the Summary the program tells you which state it is optimizing, i.e. for which state it is minimizing the energy by changing geometry: Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [1] 2.42839340 93.03-204.68612299 Doublet -1 [3] 3.40507957 92.09-204.65023045 Doublet -1 [4] 3.89101949 92.53-204.63237249 Doublet -1 [2] 8.64905509 85.38-204.45751796 Track state to follow in optimization @track_state: Initialize TRACEVEC Selected state for gradient Total energy of Doublet state 1 [3] -204.6502304459 The state tracking algorithm is sometimes useful if two state of the same symmetry are close in energy, and you want to follow a state of particular character. This is what the program prints at subsequent geometries:
Track state to follow in optimization @track_state: kroot, length, nao 1 3 72 3 2 Current overlaps in MO basis: (new, old) column 1 row 1 0.98058411817 @track_state, robust = T Update tracking vectors on Jobarc Selected state for gradient Total energy of Doublet state 1 [3] -204.6670717618 To verify that you are following the correct state in the optimization you can use the grep command in UNIX, e.g. on my output file ip_3.out0, I use [nooijen@head no2]$ grep "Total energy of Doublet state" ip_3.out0 Total energy of Doublet state 1 [3] -204.6502304459 Total energy of Doublet state 1 [3] -204.6670717618 Total energy of Doublet state 1 [3] -204.6675156140 Total energy of Doublet state 1 [3] -204.6675745992 Total energy of Doublet state 1 [3] -204.6675754151 Total energy of Doublet state 1 [3] -204.6675754743 You can see how the energy converges to a minimum in subsequent geometrical updates. Here we restricted the molecular symmetry to be C 2 V, using the zmatrix input. After optimization you can compare results of the optimizations with your predictions. The final geometry is summarized near the end of the output file (search backwards from the end for Summary): Summary of optimized internal coordinates (Angstroms and degrees) 1 NO = 1.2548193498 A = 101.0533880089 You can also calculate vibrational frequencies. To this end get rid of the * s in your ZMAT and start from the optimized geometry (as indicated above). Use VIB=FINDIF, indicating that the force field is calculated from finite differences of energy gradients. During finite displacement calculations of vibrations the symmetry is distorted from C 2 V symmetry. Therefore you cannot specify the state of interest by symmetry. However the displacements are so small that you can specify the state by total energy. To do this use the following input file
NO2 anion optimization O N 1 NO O 2 NO 1 A NO = 1.2548193498 A = 101.0533880089 *ACES2(BASIS=TZ2P,CHARGE=-1,CALC=CCSD,DROPMO=1>3 IP_CALC=IP_EOMCC,IP_SYM=4-2-1-1,VIB=FINDIF) *mrcc_gen gradient_type=ip igrad_mult=2 grad_energ=-204.6675754743 *end I took the energy from the last occurrence in the geometry optimization: Total energy of Doublet state 1 [3] -204.6675754743 Another potentially confusing aspect of the input is the IP_SYM keyword. Because the symmetry of the molecule changes when calculating the force constant matrix I need to make sure that the states are calculated at all stages of geometrical displacment. That is why I ask for 4 states in symmetry block 1. This way I will get the 4 states of interest even if the molecule has no symmetry at all! Here it will always have Cs symmetry, but asking for 4 states does not hurt. To clarify this further let us do a grep on the selected energies in the output file: [nooijen@head no2]$ grep "Total energy of Doublet state" vib_3.out0 Total energy of Doublet state 1 [3] -204.6675750112 Total energy of Doublet state 1 [3] -204.6675750117 Total energy of Doublet state 1 [3] -204.6675745703 Total energy of Doublet state 1 [3] -204.6675746879 Total energy of Doublet state 4 [1] -204.6675749825 Here you see that in the last line the symmetry of the state and its ordinal number (4 instead of 1) has changed, but the program picks up the state that is closest in energy to the energy specified in the input (grad_energ keyword). It is clear that the same state has been calculated at all displaced geometries. The final vibrational frequencies are Vibrational frequencies after rotational projection of Cartesian force constants: 7 730.1106 8 777.0081 9 1470.1709 Zero-point vibrational energy = 4.2562 kcal/mole. You can now do the calculations on all the other states of the NO 2 radical. You can verify that all frequencies are real, except for those of the state of symmetry [4], i.e. the state of
A 2 symmetry. You may verify that we reached a similar conclusion in the previous lab. This state has an imaginary frequency! What does this mean? This means that we have found a first order saddle point for this state. In this state the symmetry is broken, implying (in this simple case) that the NO bond distances are not both equal. It is another case where we have (accidentally) located a transition state by imposing symmetry. Let us see if we can find the true minimum. First we need to find the symmetry of the state if we distort the molecule. I will distort the molecule slightly from the transition state. Notice the change in ZMAT below. I use two different symbols for the two NO bond distances. NO2 anion Cs single point O N 1 NO1 O 2 NO2 1 A NO1 = 1.265 NO2 = 1.275 A = 109.8 *ACES2(BASIS=TZ2P,CHARGE=-1,CALC=CCSD,DROPMO=1>3 IP_CALC=IP_EOMCC,IP_SYM=2-2) From the output we see: Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [1] 2.79028427 92.86-204.66928422 Doublet -2 [1] 3.03823964 92.19-204.66017204 Doublet -1 [2] 3.70609065 92.48-204.63562897 Doublet -2 [2] 8.56112067 84.99-204.45720997 You can see that the state is now the first state of symmetry [2]. We can commence the optimization. Using the optimized geometry, we run a vibrational frequency calculation. At the moment of writing these notes, the ACES2 program has a problem. If you remove the DROPMO=1>3 keyword it works. We are looking into this bug Assignment on IP-EOMCC Jahn-Teller distortions in the NO 3 radical. Optimize the geometry of the closed shell NO 3 - anion using a DZP basis set and the CCSD approach. You can use DROPMO=1>4. The basis set is rather small, but the calculations in this exercise are multitude, so let us speed it up a bit. At the optimized geometry, which has D 3 h symmetry, run an IP-EOMCC calculation. I used the following input after optimization of the geometry of the anion.
NO3- photo-detachment spectrum X N 1 NX O 2 NO 1 A90 O 2 NO 1 A90 3 D120 O 2 NO 1 A90 4 D120 NO = 1.2610337703 NX=1.0 A90=90.0 D120=120.0 *ACES2(BASIS=DZP,CHARGE=-1,CALC=CCSD,DROPMO=1>4 IP_CALC=IP_EOMCC,IP_SYM=1-2-1-1) You should get the following results: Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [2] 3.37757233 92.30-279.57096544 Doublet -1 [3] 4.58424867 91.75-279.52662092 Doublet -1 [4] 4.58424868 91.75-279.52662092 Doublet -2 [2] 5.41334455 92.24-279.49615222 Doublet -1 [1] 5.41334455 92.24-279.49615222 There are two sets of degenerate states that belong to different symmetries. Going back to the SCF orbital energies we can classify them: 12 31-0.2847028677-7.7471623895 E' B2 (2) 13 8-0.2847028666-7.7471623610 E' A1 (1) 14 46-0.2317834974-6.3071524646 E'' B1 (3) 15 55-0.2317834957-6.3071524169 E'' A2 (4) 16 32-0.2201773124-5.9913319711 A2' B2 (2) +++++++++++++++++++++++++++++++++++++++++++++++++++++ Now we wish to optimize the geometries and calculate vibrational frequencies for all five states. The non-degenerate state (relating to the homo) will turn out to have D 3 h symmetry, and we can easily optimize. If we optimize the generate states the molecule will distort from D 3 h symmetry. This is the famous Jahn-teller distortion, which we already investigated for trimethylene methane. The primary distortion is in the bond angles. In order to figure out how to set up the calculations I run two other calculations, in which I distort the geometry to C 2 V. Please note how I adjusted the ZMAT.
In the first calculation use a pair of dihedral angles D and md and I made the angle smaller than 120 degrees. I call this the acute geometry. NO3- photo-detachment spectrum. Acute geometry X N 1 NX O 2 NO1 1 A90 O 2 NO2 1 A90 3 D O 2 NO2 1 A90 3 md NO1=1.26 NO2=1.27 D=115.0 md=-115.0 NX=1.0 A90=90.0 *ACES2(BASIS=DZP,CHARGE=-1,CALC=CCSD,DROPMO=1>4 IP_CALC=IP_EOMCC,IP_SYM=2-2-2-2) Here are the results I got for the IP s after these displacements at the acute geometry. Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [3] 3.26497378 92.27-279.56633505 Doublet -1 [2] 4.34089277 91.75-279.52679577 Doublet -1 [4] 4.77211913 91.63-279.51094850 Doublet -1 [1] 5.18014928 92.29-279.49595368 Doublet -2 [3] 5.64206214 92.06-279.47897870 Doublet -2 [1] 10.94023948 90.81-279.28427434 Doublet -2 [2] 11.43078945 87.00-279.26624696 You can see that the degeneracy is lifted. Using this starting geometry we can optimize states -1 [2] and -1 [1], as their energies are lowered upon distortion. You can calculate vibrational frequencies afterwards. Be careful about the number of states you ask for using IP_SYM, as the geometry will not remain C 2V in the vibrational frequency calculation. The other geometry is obtained by making the dihedrals larger than 120 degrees, in what I refer to as the obtuse geometry. Here is the input
NO3- photo-detachment spectrum X N 1 NX O 2 NO1 1 A90 O 2 NO2 1 A90 3 D O 2 NO2 1 A90 3 md NO1=1.26 NO2=1.27 D=122.0 md=-122.0 NX=1.0 A90=90.0 *ACES2(BASIS=DZP,CHARGE=-1,CALC=CCSD,DROPMO=1>4 IP_CALC=IP_EOMCC,IP_SYM=2-2-2-2) And I get the following results at this obtuse displacement: Summary of ionization-potential eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet -1 [2] 3.37070373 92.26-279.56969584 Doublet -1 [4] 4.44743221 91.75-279.53012682 Doublet -1 [3] 4.70717905 91.66-279.52058130 Doublet -2 [2] 5.35152963 92.28-279.49690186 Doublet -1 [1] 5.46695509 92.12-279.49266005 Doublet -2 [1] 11.10702179 90.81-279.28539149 Doublet -2 [3] 11.44204413 87.14-279.27307965 This is a good starting geometry to optimize states -1 [4] and -2 [2], as they lower in energy upon the distortion. If you return to the previous acute results, you can see that somehow irreps 2 and 3 got interchanged. This happens because of a renaming of x, y and z axis in the ACES2 program. It is a bit confusing, and we will just have to deal with it: The acute and obtuse geometries label the states differently. This can potentially cause trouble in geometry optimizations if the program decides to reorient in the middle of an optimization, and we specify the state to optimize by symmetry. The keyword NOREORI=ON (no reorientation) solves the problem. It tells the program to keep the axis fixed. So to optimize any of the states use an input file like
NO3- photo-detachment spectrum. Optimize E_a state (accute) X N 1 NX O 2 NO1* 1 A90 O 2 NO2* 1 A90 3 D* O 2 NO2* 1 A90 3 md* NO1=1.26 NO2=1.27 D=117.0 md=-117.0 NX=1.0 A90=90.0 *ACES2(BASIS=DZP,CHARGE=-1,CALC=CCSD,DROPMO=1>4 NOREORI=ON IP_CALC=IP_EOMCC,IP_SYM=1-1-2-1) *mrcc_gen gradient_type=ip igrad_mult=2 igrad_state=-1 igrad_sym=2 *end Once you have found the optimized geometry you can calculate vibrational frequencies, now specifying the energy of the state you are interested in. The program will make distortions from C 2 V symmetry. This is similar to what happens for the NO 2 radical. Part B III: Electron affinities using EA-EOMCC calculations. - Electron affinities. - Excited states of radicals. - Limitations of single reference Coupled Cluster methods. The most straightforward use of EA-EOMCC calculations is to use the method to calculate electron affinities. Start from a closed shell molecule like F 2, and optimize the geometry of the neutral at the CCSD level using for example the 6-31+G* basis set. We see that the lowest virtual orbital energy has symmetry [5]. It is the σ* = σ u orbital of the F 2 molecule. Now we calculate the vertical electron affinity using EA-EOMCC, using the following input. F2 anion optimization F F 1 R R = 1.4361180194 *ACES2(BASIS=6-31+G*,CALC=CCSD,DROPMO=1-2 DAMP_TYPE=DAVIDSON EA_CALC=EA_EOMCC,EA_SYM=0-0-0-0-1-0-0-0) The relevant output is as before in the Summary section
Summary of electron attachment eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet 1 [5] -0.21941366 92.79-199.06136621 The negative value indicates that F 2 can bind an electron at its neutral geometry with -0.219 ev. Let us attempt a geometry optimization, using analogous input as for IP- EOMCC: F2 anion optimization F F 1 R* R = 1.4361180194 *ACES2(BASIS=6-31+G*,CALC=CCSD,DROPMO=1-2 DAMP_TYPE=DAVIDSON EA_CALC=EA_EOMCC,EA_SYM=0-0-0-0-1-0-0-0) *mrcc_gen gradient_type=ea igrad_sym=5 igrad_mult=2 igrad_state=1 *end Something funny happens. The program terminates in an error. @GETLST: Assertion failed. List ( 1, 160 ) does not exist. @ACES_EXIT: An ACES error has occurred. return status = 1 This is pretty non-descriptive, and we should fix this in the program! If you browse back in the output, you can find the following output. ccsd_iter xcisd F T 1.000000000000000 1.000000000000000 ccsd_iter xcisd F T 1.000000000000000 1.000000000000000 ccsd_iter xcisd F T 1.000000000000000 1.000000000000000 ccsd_iter xcisd F T 1.000000000000000 1.000000000000000 *** Warning : Very large t2 amplitudes, maximum 0.3039467015947211 We suggest stopping the calculation and analyzing the results! To run this calculation you have to switch on the flag -- get_nonsense -- in the *mrcc_gen namelist We will print out t-amplitudes obtained and stop the calculation Double excitation coefficients a [irrep] a [irrep] i [irrep] i [irrep] 1 [5], 1 [5] ; -1 [1] -1 [1] -0.303947 1 [5], 2 [5] ; -1 [1] -1 [1] 0.070465
The program detects that the methodology is no longer valid. The t-amplitudes (- 0.303947) are too large and single reference methodology is not applicable. You can get results, but you have to switch on a delicate and very descriptive flag: get_nonsense=on. Let us do it. Here is the input: F2 anion optimization F F 1 R* R = 1.4361180194 *ACES2(BASIS=6-31+G*,CALC=CCSD,DROPMO=1-2 DAMP_TYPE=DAVIDSON EA_CALC=EA_EOMCC,EA_SYM=0-0-0-0-1-0-0-0) *mrcc_gen get_nonsense=on gradient_type=ea igrad_sym=5 igrad_mult=2 igrad_state=1 *end Now the program runs to completion. But the author of the program (that is me) does not think these results should be published or trusted. This is the final geometry. Summary of optimized internal coordinates (Angstroms and degrees) 1 R = 1.8728626618 If you go back in the output, the program prints out this info when solving for the t- amplitudes.. *** Warning : Very large t2 amplitudes, maximum 0.4017605983488065 We suggest stopping the calculation and analyzing the results! ccsd_iter xcisd F T 1.000000000000000 1.000000000000000 *** Warning : Very large t2 amplitudes, maximum 0.4017605943794813 We suggest stopping the calculation and analyzing the results! Convergence reached after 19 cycles Correlation energies (S, D, S+D) 0.00000000-0.43823960-0.43823960 Vacuum state energy -198.5737725885942 Total correlated energy -199.0120121868627 Maximum residuals 0.430E-07 0.679E-07 Double excitation coefficients a [irrep] a [irrep] i [irrep] i [irrep] 1 [5], 1 [5] ; -1 [1] -1 [1] -0.401761 1 [5], 2 [5] ; -1 [1] -1 [1] 0.071884 1 [5], 4 [5] ; -1 [1] -1 [1] 0.029586 The program tries hard to tell you something is not right. Better not to switch on the flag get_nonsense=on, for initial users! The EA-EOMCC can be used to calculate excitation spectra of radicals in a convenient way. Consider the MgF (R=1.752) molecule, using a PBS basis set. Start from the cation
(CHARGE=1) and use the EA-EOM-CCSD method to obtain excited states. CALC=CCSD, EA_CALC=EA_EOMCC, EA_SYM=2-2-2-2, e.g. To speed up the calculation also specify DROPMO=1-2. Other keywords to be included are familiar. Repeat the calculation at the EA-EOM-MBPT[2] level, using CALC=MBPT(2). As you go through the output you can identify familiar features you can recognize from other EOM calculations. Input: MgF excited states MG F 1 R R=1.752 *ACES2(BASIS=PBS,DROPMO=1-2,CALC=CCSD,CHARGE=1 EA_CALC=EA_EOMCC,EA_SYM=2-2-2-2,ESTATE_PROP=EXPECTATION) Relevant parts of output: Summary of electron attachment eom-cc calculation Multiplet orb. irrep energy diff (ev) % singles total energy Doublet 1 [1] -7.82075213 99.07-299.42699572 Doublet 1 [3] -4.34031236 99.23-299.29909196 Doublet 1 [2] -4.34031236 99.23-299.29909196 Doublet 2 [1] -3.05322662 99.13-299.25179245 Doublet 2 [3] -1.63665403 99.84-299.19973439 Doublet 2 [2] -1.63665403 99.84-299.19973439 Doublet 1 [4] 0.49463002 99.03-299.12141117 Doublet 2 [4] 13.27356590 28.93-298.65179409 You can see we shouldn t have asked for two states in symmetry block 4! You can check the output. It goes haywire, but it does no harm. Some of these states are just not meaningful, and the program is not able to converge. You can fnd the statement that at some points it stops and fakes convergence, in order not to mess up the rest of the calculation, which is fine. A nice feature of the EA-EOMCC is that it calculates excitation spectra of the radical automatically:
Symmetry allowed transitions from lowest state ------------------------------------------------------------------------ Transition energy (cm-1) Transition Dipoles (a.u.) Osc. Str. initial final <x> <y> <z> f (a.u.) ------------------------------------------------------------------------ Transitions between Doublet spin states 1 [1] -> 1 [3] 28071.63 0.0000E+00-0.1770E+01 0.0000E+00 0.2673E+00 1 [2] 28071.63-0.1770E+01 0.0000E+00 0.0000E+00 0.2673E+00 2 [1] 38452.67 0.0000E+00 0.0000E+00-0.1540E+01 0.2769E+00 2 [3] 49878.10 0.0000E+00 0.7717E-01 0.0000E+00 0.9023E-03 2 [2] 49878.10 0.7717E-01 0.0000E+00 0.0000E+00 0.9023E-03 ------------------------------------------------------------------------ We can also optimize geometries, using similar tools as before. For example to optimize the ground state of the radical use MgF excited states MG F 1 R* R=1.752 *ACES2(BASIS=PBS,DROPMO=1-2,CALC=CCSD,CHARGE=1 EA_CALC=EA_EOMCC,EA_SYM=1-0-0-0) *mrcc_gen gradient_type=ea igrad_sym=1 igrad_mult=2 igrad_state=1 *end The resulting optimized geometry is Summary of optimized internal coordinates (Angstroms and degrees) 1 R = 1.7672718068 EA-EOMCC calculations are a nice tool to keep a proper symmetry of the states. This can be illustrated by a very simple calculation on the B atom. Use a PBS basis set and do a UHF calculation on the neutral. Analyse the orbital energies. What happened to the degeneracies of the virtual 2p, 3p, 3d etc. levels of the atom? Now perform an RHF calculation on the cation B +. If you look at the orbital energies you find back the symmetry of the atom. Proceed to calculate correlated excited states using EA-EOMCC. Symmetry restored!
Assignment on EA-EOMCC 1. Consider the Li...H 2 complex at two geometries, where the H 2 distance is fixed at 0.74 Å. The first geometry is a linear conformation. Take the Li...H 2 distance as 3.5 Å. Perform EA-EOM-CCSD on the cation to obtain excited states of the neutral. Explain the splitting of the levels you observe. Also calculate the Li atom in the same basis set (PBS). Compare your results. Next consider the T-shaped geometry where the Li atom is at a distance of 3.5 Å from the midpoint of H 2 (perpendicular). 2. Consider the NO and OH radicals. Set up proper IP/and or EA calculations to optimize the geometry of these radicals and calculate their vibrational frequencies. You will want to start from a closed shell parent state, and obtain a final wavefunction of proper symmetry. You will want the reference state to be nicely closed shell, and not break symmetry. This should guide you to use either and IP or EA calculation. Compare with corresponding UHF based MBPT(2) and CCSD calculations on the radicals directly. What is the electron affinity of the OH radical in the gas phase? Consider vertical and adiabatic electron affinities and also photo-detachment energies at the IP/EA-EOM- CCSD level. Use the 6-31+G* basis set.