# Development:Excited-State Calculations

### From NWChem

# CIS, TDHF, TDDFT

## Overview

NWChem supports a spectrum of single excitation theories for vertical excitation energy calculations, namely, configuration interaction singles (CIS), time-dependent Hartree-Fock (TDHF or also known as random-phase approximation RPA), time-dependent density functional theory (TDDFT), and Tamm-Dancoff approximation to TDDFT. These methods are implemented in a single framework that invokes Davidson's trial vector algorithm (or its modification for a non-Hermitian eigenvalue problem). The capabilities of the module are summarized as follows:

- Vertical excitation energies,
- Spin-restricted singlet and triplet excited states for closed-shell systems,
- Spin-unrestricted doublet, etc., excited states for open-shell systems,
- Tamm-Dancoff and full time-dependent linear response theories,
- Davidson's trial vector algorithm,
- Symmetry (irreducible representation) characterization and specification,
- Spin multiplicity characterization and specification,
- Transition moments and oscillator strengths,
- Geometrical first and second derivatives of vertical excitation energies by numerical differentiation,
- Disk-based and fully incore algorithms,
- Multiple and single trial-vector processing algorithms,
- Frozen core and virtual approximation,
- Asymptotically correct exchange-correlation potential by van Leeuwen and Baerends (R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421 (1994)),
- Asymptotic correction by Casida and Salahub (M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub, J. Chem. Phys. 108, 4439 (1998)),
- Asymptotic correction by Hirata, Zhan, Aprà, Windus, and Dixon (S. Hirata, C.-G. Zhan, E. Aprà, T. L. Windus, and D. A. Dixon, J. Phys. Chem. A 107, 10154 (2003)).

These are very effective way to rectify the shortcomings of TDDFT when applied to Rydberg excited states (see below).

## Performance of CIS, TDHF, and TDDFT methods

The accuracy of CIS and TDHF for excitation energies of closed-shell systems are comparable to each other, and are normally considered a zeroth-order description of the excitation process. These methods are particularly well balanced in describing Rydberg excited states, in contrast to TDDFT. However, for open-shell systems, the errors in the CIS and TDHF excitation energies are often excessive, primarily due to the multi-determinantal character of the ground and excited state wave functions of open-shell systems in a HF reference.[ref] The scaling of the computational cost of a CIS or TDHF calculation per state with respect to the system size is the same as that for a HF calculation for the ground state, since the critical step of the both methods are the Fock build, namely, the contraction of two-electron integrals with density matrices. It is usually necessary to include two sets of diffuse exponents in the basis set to properly account for the diffuse Rydberg excited states of neutral species.

The accuracy of TDDFT may vary depending on the exchange-correlation functional. In general, the exchange-correlation functionals that are widely used today and are implemented in NWChem work well for low-lying valence excited states. However, for high-lying diffuse excited states and Rydberg excited states in particular, TDDFT employing these conventional functionals breaks down and the excitation energies are substantially underestimated. This is because of the fact that the exchange-correlation potentials generated from these functionals decay too rapidly (exponentially) as opposed to the slow − 1 / *r* asymptotic decay of the true potential. A rough but useful index is the negative of the highest occupied KS orbital energy; when the calculated excitation energies become close to this threshold, these numbers are most likely underestimated relative to experimental results. It appears that TDDFT provides a better-balanced description of radical excited states. This may be traced to the fact that, in DFT, the ground state wave function is represented well as a single KS determinant, with less multi-determinantal character and less spin contamination, and hence the excitation thereof is described well as a simple one electron transition. The computational cost per state of TDDFT calculations scales as the same as the ground state DFT calculations, although the prefactor of the scaling may be much greater in the former.

A very simple and effecive way to rectify the TDDFT's failure for Rydberg excited states has been proposed by Tozer and Handy (D. J. Tozer and N. C. Handy, J. Chem. Phys. 109, 10180 (1998)) and by Casida and Salahub (see previous reference). They proposed to splice a − 1 / *r* asymptotic tail to an exchange-correlation potential that does not have the correct asymptotic behavior. Because the approximate exchange-correlation potentials are too shallow everywhere, a negative constant must be added to them before they can be spliced to the − 1 / *r* tail seamlessly in a region that is not sensitive to chemical effects or to the long-range behavior. The negative constant or the shift is usually taken to be the difference of the HOMO energy from the true ionization potential, which can be obtained either from experiment or from a ΔSCF calculation. Recently, we proposed a new, expedient, and self-contained asymptotic correction that does not require an ionization potential (or shift) as an external parameter from a separate calculation. In this scheme, the shift is computed by a semi-empirical formula proposed by Zhan, Nichols, and Dixon (C.-G. Zhan, J. A. Nichols, and D. A. Dixon, J. Phys. Chem. A 107, 4184 (2003)). Both Casida-Salahub scheme and this new asymptotic correction scheme give considerably improved (Koopmans type) ionization potentials and Rydberg excitation energies. The latter, however, supply the shift by itself unlike to former.

## Input syntax

The module is called TDDFT as TDDFT employing a hybrid HF-DFT functional encompasses all of the above-mentioned methods implemented. To use this module, one needs to specify TDDFT on the task directive, e.g.,

TASK TDDFT ENERGY

for a single-point excitation energy calculation, and

TASK TDDFT OPTIMIZE

for an excited-state geometry optimization (and perhaps an adiabatic excitation energy calculation), and

TASK TDDFT FREQUENCIES

for an excited-state vibrational frequency calculation. The TDDFT module first invokes DFT module for a ground-state calculation (regardless of whether the calculations uses a HF reference as in CIS or TDHF or a DFT functional), and hence there is no need to perform a separate ground-state DFT calculation prior to calling a TDDFT task. When no second argument of the task directive is given, a single-point excitation energy calculation will be assumed. For geometry optimizations, it is usually necessary to specify the target excited state and its irreducible representation it belongs to. See the subsections TARGET and TARGETSYM for more detail.

Individual parameters and keywords may be supplied in the TDDFT input block. The syntax is:

TDDFT [(CIS||RPA) default RPA] [NROOTS <integer nroots default 1>] [MAXVECS <integer maxvecs default 1000>] [(SINGLET||NOSINGLET) default SINGLET] [(TRIPLET||NOTRIPLET) default TRIPLET] [THRESH <double thresh default 1e-4>] [MAXITER <integer maxiter default 100>] [TARGET <integer target default 1>] [TARGETSYM <character targetsym default 'none'>] [SYMMETRY] [ALGORITHM <integer algorithm default 0>] [FREEZE [[core] (atomic || <integer nfzc default 0>)] \ [virtual <integer nfzv default 0>]] [PRINT (none||low||medium||high||debug) <string list_of_names ...>] END

The user can also specify the reference wave function in the DFT input block (even when CIS and TDHF calculations are requested). See the section of Sample input and output for more details.

Since each keyword has a default value, a minimal input file will be

GEOMETRY Be 0.0 0.0 0.0 END BASIS Be library 6-31G** END TASK TDDFT ENERGY

Note that the keyword for the asymptotic correction must be given in the DFT input block, since all the effects of the correction (and also changes in the computer program) occur in the SCF calculation stage. See DFT (keyword CS00 and LB94) for details.

## Keywords of TDDFT input block

### CIS and RPA -- the Tamm-Dancoff approximation

These keywords toggle the Tamm-Dancoff approximation. CIS means that the Tamm-Dancoff approximation is used and the CIS or Tamm-Dancoff TDDFT calculation is requested. RPA, which is the default, requests TDHF (RPA) or TDDFT calculation.

The performance of CIS (Tamm-Dancoff TDDFT) and RPA (TDDFT) are comparable in accuracy. However, the computational cost is slightly greater in the latter due to the fact that the latter involves a non-Hermitian eigenvalue problem and requires left and right eigenvectors while the former needs just one set of eigenvectors of a Hermitian eigenvalue problem. The latter has much greater chance of aborting the calculation due to triplet near instability or other instability problems.

### NROOTS -- the number of excited states

One can specify the number of excited state roots to be determined. The default value is 1. It is advised that the users request several more roots than actually needed, since owing to the nature of the trial vector algorithm, some low-lying roots can be missed when they do not have sufficient overlap with the initial guess vectors.

### MAXVECS -- the subspace size

This keyword limits the subspace size of Davidson's algorithm; in other words, it is the maximum number of trial vectors that the calculation is allowed to hold. Typically, 10 to 20 trial vectors are needed for each excited state root to be converged. However, it need not exceed the product of the number of occupied orbitals and the number of virtual orbitals. The default value is 1000.

### SINGLET and NOSINGLET -- singlet excited states

SINGLET (NOSINGLET) requests (suppresses) the calculation of singlet excited states when the reference wave function is closed shell. The default is SINGLET.

### TRIPLET and NOTRIPLET -- triplet excited states

TRIPLET (NOTRIPLET) requests (suppresses) the calculation of triplet excited states when the reference wave function is closed shell. The default is TRIPLET.

### THRESH -- the convergence threshold of Davidson iteration

This keyword specifies the convergence threshold of Davidson's iterative algorithm to solve a matrix eigenvalue problem. The threshold refers to the norm of residual, namely, the difference between the left-hand side and right-hand side of the matrix eigenvalue equation with the current solution vector. With the default value of 1e-4, the excitation energies are usually converged to 1e-5 hartree.

### MAXITER -- the maximum number of Davidson iteration

It typically takes 10-30 iterations for the Davidson algorithm to get converged results. The default value is 100.

### TARGET and TARGETSYM-- the target root and its symmetry

At the moment, the first and second geometrical derivatives of excitation energies that are needed in force, geometry, and frequency calculations are obtained by numerical differentiation. These keywords may be used to specify which excited state root is being used for the geometrical derivative calculation. For instance, when TARGET 3 and TARGETSYM a1g are included in the input block, the total energy (ground state energy plus excitation energy) of the third lowest excited state root (excluding the ground state) transforming as the irreducible representation a1g will be passed to the module which performs the derivative calculations. The default values of these keywords are 1 and none, respectively.

The keyword TARGETSYM is essential in excited state geometry optimization, since it is very common that the order of excited states changes due to the geometry changes in the course of optimization. Without specifying the TARGETSYM, the optimizer could (and would likely) be optimizing the geometry of an excited state that is different from the one the user had intended to optimize at the starting geometry. On the other hand, in the frequency calculations, TARGETSYM must be none, since the finite displacements given in the course of frequency calculations will lift the spatial symmetry of the equilibrium geometry. When these finite displacements can alter the order of excited states including the target state, the frequency calculation is not be feasible.

### SYMMETRY -- restricting the excited state symmetry

By adding this keyword to the input block, the user can request the module to generate the initial guess vectors transforming as the same irreducible representation as TARGETSYM. This causes the final excited state roots be (exclusively) dominated by those with the specified irreducible representation. This may be useful, when the user is interested in just the optically allowed transitions, or in the geometry optimization of an excited state root with a particular irreducible representation. By default, this option is not set. TARGETSYM must be specified when SYMMETRY is invoked.

### ALGORITHM -- algorithms for tensor contractions

There are four distinct algorithms to choose from, and the default value of 0 (optimal) means that the program makes an optimal choice from the four algorithms on the basis of available memory. In the order of decreasing memory requirement, the four algorithms are:

- ALGORITHM 1 : Incore, multiple tensor contraction,
- ALGORITHM 2 : Incore, single tensor contraction,
- ALGORITHM 3 : Disk-based, multiple tensor contraction,
- ALGORITHM 4 : Disk-based, single tensor contraction.

The incore algorithm stores all the trial and product vectors in memory across different nodes with the GA, and often decreases the MAXITER value to accommodate them. The disk-based algorithm stores the vectors on disks across different nodes with the DRA, and retrieves each vector one at a time when it is needed. The multiple and single tensor contraction refers to whether just one or more than one trial vectors are contracted with integrals. The multiple tensor contraction algorithm is particularly effective (in terms of speed) for CIS and TDHF, since the number of the direct evaluations of two-electron integrals is diminished substantially.

### FREEZE -- the frozen core/virtual approximation

Some of the lowest-lying core orbitals and/or some of the highest-lying virtual orbitals may be excluded in the CIS, TDHF, and TDDFT calculations by this keyword (this does not affect the ground state HF or DFT calculation). No orbitals are frozen by default. To exclude the atom-like core regions altogether, one may request

FREEZE atomic

To specify the number of lowest-lying occupied orbitals be excluded, one may use

FREEZE 10

which causes 10 lowest-lying occupied orbitals excluded. This is equivalent to writing

FREEZE core 10

To freeze the highest virtual orbitals, use the virtual keyword. For instance, to freeze the top 5 virtuals

FREEZE virtual 5

### PRINT -- the verbosity

This keyword changes the level of output verbosity. One may also request some particular items in the table below.

Item | Print Level | Description |

"timings" | high | CPU and wall times spent in each step |

"trial vectors" | high | Trial CI vectors |

"initial guess" | debug | Initial guess CI vectors |

"general information" | default | General information |

"xc information" | default | HF/DFT information |

"memory information" | default | Memory information |

"convergence" | debug | Convergence |

"subspace" | debug | Subspace representation of CI matrices |

"transform" | debug | MO to AO and AO to MO transformation of CI vectors |

"diagonalization" | debug | Diagonalization of CI matrices |

"iteration" | default | Davidson iteration update |

"contract" | debug | Integral transition density contraction |

"ground state" | default | Final result for ground state |

"excited state" | low | Final result for target excited state |

## Sample input

The following is a sample input for a spin-restricted TDDFT calculation of singlet excitation energies for the water molecule at the B3LYP/6-31G*.

START h2o TITLE "B3LYP/6-31G* H2O" GEOMETRY O 0.00000000 0.00000000 0.12982363 H 0.75933475 0.00000000 -0.46621158 H -0.75933475 0.00000000 -0.46621158 END BASIS * library 6-31G* END DFT XC B3LYP END TDDFT RPA NROOTS 20 END TASK TDDFT ENERGY

To perform a spin-unrestricted TDHF/aug-cc-pVDZ calculation for the CO+ radical,

START co TITLE "TDHF/aug-cc-pVDZ CO+" CHARGE 1 GEOMETRY C 0.0 0.0 0.0 O 1.5 0.0 0.0 END BASIS * library aug-cc-pVDZ END DFT XC HFexch MULT 2 END TDDFT RPA NROOTS 5 END TASK TDDFT ENERGY

A geometry optimization followed by a frequency calculation for an excited state is carried out for BF at the CIS/6-31G* level in the following sample input.

START bf TITLE "CIS/6-31G* BF optimization frequencies" GEOMETRY B 0.0 0.0 0.0 F 0.0 0.0 1.2 END BASIS * library 6-31G* END DFT XC HFexch END TDDFT CIS NROOTS 3 NOTRIPLET TARGET 1 END TASK TDDFT OPTIMIZE TASK TDDFT FREQUENCIES

TDDFT with an asymptotically corrected SVWN exchange-correlation potential. Casida-Salahub scheme has been used with the shift value of 0.1837 a.u. supplied as an input parameter.

START tddft_ac_co GEOMETRY O 0.0 0.0 0.0000 C 0.0 0.0 1.1283 END BASIS SPHERICAL C library aug-cc-pVDZ O library aug-cc-pVDZ END DFT XC Slater VWN_5 CS00 0.1837 END TDDFT NROOTS 12 END TASK TDDFT ENERGY

TDDFT with an asymptotically corrected B3LYP exchange-correlation potential. Hirata-Zhan-Apra-Windus-Dixon scheme has been used (this is only meaningful with B3LYP functional).

START tddft_ac_co GEOMETRY O 0.0 0.0 0.0000 C 0.0 0.0 1.1283 END BASIS SPHERICAL C library aug-cc-pVDZ O library aug-cc-pVDZ END DFT XC B3LYP CS00 END TDDFT NROOTS 12 END TASK TDDFT ENERGY