Dyson orbitals can be used to compute total photodetachment/photoionization cross-sections, as well as angular distribution of photoelectrons. A Dyson orbital is the overlap between the N-electron molecular wave function and the / electron wave function of the corresponding cation/anion:
For the Hartree-Fock wave functions and within Koopmans’ approximation, these are just the canonical HF orbitals. For correlated wave functions, Dyson orbitals are linear combinations of the reference molecular orbitals:
The calculation of Dyson orbitals is straightforward within the EOM-IP/EA-CCSD methods, where cation/anion and initial molecule states are defined with respect to the same MO basis. Since the left and right CC vectors are not the same, one can define correspondingly two Dyson orbitals (left and right):
The norm of these orbitals is proportional to the one-electron character of the transition.
Dyson orbitals also offer qualitative insight visualizing the difference between molecular and ionized/attached states. In ionization/photodetachment processes, these orbitals can be also interpreted as the wave function of the leaving electron. For additional details, see Refs. 824 and 825. Dyson orbitals can be used for computing total and differential photoelectron cross-sections using a stand-alone ezDyson code.Gozem:2021 Dyson orbitals can be computed both for valence states and core-level states;1162 see Section 7.10.8 for calculations of Dyson orbitals within the FC-CVS-EOM framework.
The calculation of Dyson orbitals is implemented for the ground (reference) and excited states ionization/electron attachment. To obtain the ground state Dyson orbitals one needs to run an EOM-IP/EA-CCSD calculation, request transition properties calculation by setting CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE. The Dyson orbitals decomposition in the MO basis is printed in the output, for all transitions between the reference and all IP/EA states. At the end of the file, also the coefficients of the Dyson orbitals in the AO basis are available.
Two implementations of Dyson orbitals are currently available: (i) the original implementation in CCMAN; and (ii) new implementation in CCMAN2. The CCMAN implementation is using a diffuse orbital trick (i.e., EOM_FAKE_IPEA will be automatically set to TRUE in these calculations). Note: this implementation has a bug affecting the values of norms of Dyson orbitals (the shapes are correct); thus, using this code is strongly discouraged. The CCMAN2 implementation has all types of initial states available: Dyson orbitals from ground CC, excited EOM-EE, and spin-flip EOM-SF states; it is fully compatible with all helper features for EOM calculations, like FNO, RI, Cholesky decomposition. The CCMAN2 implementation can use a user-specified EOM guess (using EOM_USER_GUESS keyword and $eom_user_guess section), which is recommended for highly excited states (such as core-ionized states). In addition, CCMAN2 can calculate Dyson orbitals involving meta-stable states (see Section 7.10.9) and core-level states (see Section 7.10.8).
For calculating Dyson orbitals between excited or spin-flip states from the reference configuration and IP/EA states, same CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE keywords have to be added to the combination of usual EOM-IP/EA-CCSD and EOM-EE-CCSD or EOM-SF-CCSD calculations. (However, note the separate keyword CC_DO_DYSON_EE = TRUE for CCMAN.) The IP_STATES keyword is used to specify the target ionized states. The attached states are specified by EA_STATES. The EA-SF states are specified by EOM_EA_BETA. The excited (or spin-flipped) states are specified by EE_STATES and SF_STATES. The Dyson orbital decomposition in MO and AO bases is printed for each EE/SF-IP/EA pair of states first for reference, then for all excited states in the order: CC-IP/EA1, CC-IP/EA2,, EE/SF1 - IP/EA1, EE/SF1 - IP/EA2,, EE/SF2 - IP/EA1, EE/SF2 - IP/EA2,, and so on. CCMAN implementation keeps reference transitions separate, in accordance with separating keywords.
Dyson orbitals are most easily visualized by setting IQMOL_FCHK = TRUE (equivalently, GUI = 2) and reading the resulting checkpoint file into IQmol. In addition to the canonical orbitals, the Dyson orbitals will appear under the Surfaces item in the Model View. For step-by-step instructions, see the ezDyson manual.389 Alternatively Dyson orbitals can be plotted using IANLTY = 200 and the $plots utility. Only the sizes of the box need to be specified, followed by a line of zeros:
$plots comment 10 -2 2 10 -2 2 10 -2 2 0 0 0 0 $plots
All Dyson orbitals on the Cartesian grid will be written in the resulting plot.mo file (only CCMAN). For RHF(UHF) reference, the columns order in plot.mo is:
In addition, setting the MAKE_CUBE_FILES keyword to TRUE will create cube files for Dyson orbitals which can be viewed with VMD or other programs (see Section 10.5.5 for details). This option is available for CCMAN and CCMAN2. The Dyson orbitals will be written to files mo.1.cube, mo.2.cube, in the order . For meta-stable states, the real and imaginary parts of the Dyson orbitals are written to separate files in the order
Note: Visualization via the MolDen format is currently not available.
$molecule 0 3 O 0.000 0.000 0.000 O 1.222 0.000 0.000 $end $rem BASIS 6-31G* METHOD eom-ccsd IP_STATES [1,0,0,0,0,0,1,0] Target EOM-IP states CC_TRANS_PROP true request transition OPDMs to be calculated CC_DO_DYSON true calculate Dyson orbitals IANLTY 200 $end $plots plots excited states densities and trans densities 10 -2 2 10 -2 2 10 -2 2 0 0 0 0 $plots
$molecule -1 1 H 0.000 0.000 0.000 O 1.000 0.000 0.000 $end $rem METHOD eom-ccsd BASIS 6-31G* IP_STATES [1,0,0,0] states of HO radical EE_STATES [1,1,0,0] excited states of HO- CC_TRANS_PROP 2 calculate transition properties CC_DO_DYSON true calculate Dyson orbitals for ionization from ex. states IANLTY 200 $end $plots plot excited states densities and trans densities 10 -2 2 10 -2 2 10 -2 2 0 0 0 0 $plots
$molecule 0 1 O C O 1.131 $end $rem CORRELATION CCSD BASIS cc-pVDZ PURECART 111 5d, will be required for ezDyson IP_STATES [1,0,1,0] (A1,A2,B1,B2) CCMAN2 true CC_DO_DYSON true CC_TRANS_PROP true necessary for Dyson orbitals job PRINT_GENERAL_BASIS true will be required for ezDyson $end
$molecule 0 1 O H1 O 0.955 H2 O 0.955 H1 104.5 $end $rem CORRELATION CCSD BASIS cc-pVTZ PURECART 111 5d, will be required for ezDyson IP_STATES [1,0,0,0] (A1,A2,B1,B2) EOM_USER_GUESS 1 on, further defined in $eom_user_guess CCMAN2 true CC_DO_DYSON true CC_TRANS_PROP true necessary for Dyson orbitals job PRINT_GENERAL_BASIS true will be required for ezDyson N_FROZEN_CORE false $end $eom_user_guess 1 $end
$molecule +1 1 N 0.00000 0.00000 0.00000 O 0.00000 0.00000 1.02286 $end $rem CORRELATION CCSD BASIS aug-cc-pVTZ PURECART 111 5d, will be required for ezDyson EA_STATES [1,0,0,1] (A1,A2,B1,B2) CCMAN2 true CC_DO_DYSON true CC_TRANS_PROP true necessary for Dyson orbitals job PRINT_GENERAL_BASIS true will be required for ezDyson $end
$molecule 0 1 N 0.0 0.0 0.55 N 0.0 0.0 -0.55 GH 0.0 0.0 0.0 $end $rem METHOD EOM-CCSD EA_STATES [0,0,2,0,0,0,0,0] CC_MEMORY 5000 MEM_STATIC 1000 BASIS GEN COMPLEX_CCMAN TRUE CC_TRANS_PROP TRUE CC_DO_DYSON TRUE MAKE_CUBE_FILES TRUE IANLTY 200 $end $complex_ccman CS_HF 1 CAP_TYPE 1 CAP_X 2760 CAP_Y 2760 CAP_Z 4880 CAP_ETA 400 $end $plots plot Dyson orbitals 50 -10.0 10.0 50 -10.0 10.0 50 -10.0 10.0 0 0 0 0 $end $basis N 0 S 8 1.000000 1.14200000E+04 5.23000000E-04 1.71200000E+03 4.04500000E-03 3.89300000E+02 2.07750000E-02 1.10000000E+02 8.07270000E-02 3.55700000E+01 2.33074000E-01 1.25400000E+01 4.33501000E-01 4.64400000E+00 3.47472000E-01 5.11800000E-01 -8.50800000E-03 S 8 1.000000 1.14200000E+04 -1.15000000E-04 1.71200000E+03 -8.95000000E-04 3.89300000E+02 -4.62400000E-03 1.10000000E+02 -1.85280000E-02 3.55700000E+01 -5.73390000E-02 1.25400000E+01 -1.32076000E-01 4.64400000E+00 -1.72510000E-01 5.11800000E-01 5.99944000E-01 S 1 1.000000 1.29300000E+00 1.00000000E+00 S 1 1.000000 1.78700000E-01 1.00000000E+00 P 3 1.000000 2.66300000E+01 1.46700000E-02 5.94800000E+00 9.17640000E-02 1.74200000E+00 2.98683000E-01 P 1 1.000000 5.55000000E-01 1.00000000E+00 P 1 1.000000 1.72500000E-01 1.00000000E+00 D 1 1.000000 1.65400000E+00 1.00000000E+00 D 1 1.000000 4.69000000E-01 1.00000000E+00 F 1 1.000000 1.09300000E+00 1.00000000E+00 S 1 1.000000 5.76000000E-02 1.00000000E+00 P 1 1.000000 4.91000000E-02 1.00000000E+00 D 1 1.000000 1.51000000E-01 1.00000000E+00 F 1 1.000000 3.64000000E-01 1.00000000E+00 **** GH 0 S 1 1.000000 2.88000000E-02 1.00000000E+00 S 1 1.000000 1.44000000E-02 1.00000000E+00 S 1 1.000000 0.72000000E-02 1.00000000E+00 S 1 1.000000 0.36000000E-02 1.00000000E+00 S 1 1.000000 0.18000000E-02 1.00000000E+00 S 1 1.000000 0.09000000E-02 1.00000000E+00 P 1 1.000000 2.45000000E-02 1.00000000E+00 P 1 1.000000 1.22000000E-02 1.00000000E+00 P 1 1.000000 0.61000000E-02 1.00000000E+00 P 1 1.000000 0.305000000E-02 1.00000000E+00 P 1 1.000000 0.152500000E-02 1.00000000E+00 P 1 1.000000 0.076250000E-02 1.00000000E+00 D 1 1.000000 0.755000000E-01 1.00000000E+00 D 1 1.000000 0.377500000E-01 1.00000000E+00 D 1 1.000000 0.188750000E-01 1.00000000E+00 D 1 1.000000 0.094375000E-01 1.00000000E+00 D 1 1.000000 0.047187500E-01 1.00000000E+00 D 1 1.000000 0.023593750E-01 1.00000000E+00 **** $end
$comment EOM-IP-CCSD/6-311+G* and EOM-EA-CCSD/6-311+G* levels of theory, UHF reference. Start from O2: 1) detach electron - ionization of neutral (alpha IP). 2) attach electron, use EOM-EA w.f. as initial state - ionization of anion (beta EA). $end $molecule 0 3 O 0.00000 0.00000 0.00000 O 0.00000 0.00000 1.30000 $end $rem CORRELATION CCSD BASIS 6-311(3+)G* PURECART 2222 6d, will be required for ezDyson EOM_IP_ALPHA [0,0,0,1,0,0,0,0] (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u) EOM_EA_BETA [0,0,0,1,0,0,0,0] (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u) CCMAN2 true CC_DO_DYSON true CC_TRANS_PROP true necessary for Dyson orbitals job PRINT_GENERAL_BASIS true will be required for ezDyson $end
$molecule 0 1 O 1.535338855 0.000000000 -0.438858006 C 1.535331598 -0.000007025 0.767790994 H 1.535342484 0.937663512 1.362651452 H 1.535342484 -0.937656488 1.362672535 $end $rem CORRELATION CCSD BASIS 6-31G* PURECART 2222 6d, will be required for ezDyson CCMAN2 true new Dyson code EE_STATES  EOM_IP_ALPHA  EOM_IP_BETA  CC_TRANS_PROP true necessary for Dyson orbitals job CC_DO_DYSON true PRINT_GENERAL_BASIS true will be required for ezDyson $end
$molecule +1 1 Li 0.00000 0.00000 0.00000 $end $rem CORRELATION CCSD BASIS 6-311+G* PURECART 2222 6d, will be required for ezDyson CCMAN2 true new Dyson code EE_STATES [1,0,0,0,0,0,0,0] EA_STATES [1,0,0,0,0,0,0,0] EOM_NGUESS_SINGLES 5 to converge to the lowest EA state CC_TRANS_PROP true necessary for Dyson orbitals job CC_DO_DYSON true PRINT_GENERAL_BASIS true will be required for ezDyson $end
$molecule 0 3 C H 1 rCH H 1 rCH 2 aHCH rCH = 1.1167 aHCH = 102.07 $end $rem CORRELATION CCSD BASIS 6-31G* SCF_GUESS core CCMAN2 true new Dyson code CC_SYMMETRY false SF_STATES  EOM_IP_ALPHA  one should be careful to request EOM_EA_BETA  meaningful spin for EA/IP state(s) CC_TRANS_PROP true necessary for Dyson orbitals job CC_DO_DYSON true IQMOL_FCHK true generate formatted checkpoint file for IQMol $end
$comment SO-, calculating Dyson orbitals using EOM-EA to describe the anion states and EOM-SF to describe the neutral; both sets of EOM states are generated using triplet reference. $end $molecule 0 3 S 0.0000000 0.0000000 -0.5241891 O 0.0000000 0.0000000 1.0676951 $end $rem JOBTYPE¯¯¯SP METHOD¯ ¯EOM-CCSD BASIS ¯6-31G* PURECART¯¯111 needed for ezDyson EA_BETA¯¯ [0,0,0,1] anion state SF_STATES [2,2,0,0] neutral states CC_DO_DYSON¯¯true CC_TRANS_PROP¯¯true PRINT_GENERAL_BASIS¯true needed for ezDyson $end $trans_prop state_list ea_beta 4 1 !state 1 sf_states 1 1 !state 2 sf_states 1 2 !state 3 sf_states 2 1 sf_states 2 2 end_list state_pair_list 1 2 ! transition 1 <-> 2 1 3 1 4 1 5 end_pairs calc dyson $end