# Highly Excited States from a Time Independent Density Functional Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Outline of TI-DFT and Its COEP Implementation

_{0k}of the ground state KS determinant Φ

_{0}can be presented by imposition of some orthogonality constraints on the ES orbitals.

_{0eff}(r)

_{i}} of the excited state determinant Φ, generated by excitation of an electron from orbital ${\varphi}_{0k}^{\alpha}$ are determined by

- The constraint vector $|{\varphi}_{0k}^{\alpha}\rangle $ tends to an eigenvector of the modified operator$${h}_{\mathrm{mod}}=\left[-\frac{1}{2}{\nabla}^{2}+{{\displaystyle V}}_{eff}^{\alpha}(\overrightarrow{r})+\lambda {P}_{0k}^{\alpha}\right],\text{if and only if \lambda \u21d2 \xb1\u221e}$$

## 3. Matrix Kohn–Sham-Like Equations for Highly Excited States

^{double}produced by excitation of electrons from ${\varphi}_{0k}^{\alpha}$ and ${\varphi}_{0l}^{\beta}$ orbitals can be obtained by using the following orthogonality constraints imposed on orbitals {ϕ

_{j}} of the doubly excited state Slater determinant Φ

^{double}:

^{3}–10

^{4}hartrees. This value provides a target accuracy for $\langle {\varphi}_{0k}^{\alpha}|{\varphi}_{j}^{\alpha}\rangle $ and $\langle {\varphi}_{0l}^{\beta}|{\varphi}_{j}^{\beta}\rangle $ ~ 10

^{−5}[35] and leads to orthogonality of the KS determinants which describe excited states.

## 4. Results of Calculations and Their Discussion

^{2}ns (n = 3, …, 7) doublet states of the Li atom.

_{eff}proposed in [36] and further developed in [37,38] where an effective potential is a direct mapping of the external potential V

_{ext}and for N-electron atoms takes the form:

_{p}, were defined by the geometric series:

- (i)
- Common basis set adjusted to the ground state and the potential parameters optimized for a given excited state were employed (x-COEP-bgs column in Table 1). We restricted parameters to 3 excited states to show some tendencies.
- (ii)
- (iii)
- Both orbital basis sets and the potential parameters were optimized for a given excited state (x-COEP column in these tables).

^{exact}) based on a configuration interaction approach with the explicitly correlated Hylleraas basis set functions [40]. A comparison of excited state energies presented in x-COEP-bgs and x-COEP-Vgs columns of Table 1 with the fully optimized results (x-COEP column) shows that basis set optimization plays a crucial role for a correct description of excited states with respect to optimization of potential parameters. We observed that potential parameters for the ground state differ from those for excited states. However, we did not notice any trends in their behavior. In the case of helium the dependence of the results on the values of the parameters is relatively weak and the same effective potential correctly describes both the ground state and the excited states. The columns x-COEP-Vgs in Table 1 and Table 2 demonstrate that the ground state potential with parameters C = 3.982687 and d = 0.248872 and the optimization of the orbital basis for a given excited state can support a reasonable accuracy of excited energies.

^{COEP}− Е

^{exact}for different excited states are similar. As a result, excitation energies based on the x-COEP method are in good agreement with those computed with highly correlated methods (see Table 5).

^{2}ns (n = 3,…, 7) energies and excitation energies of the Li atom to the HF [38] and “exact” energies obtained with the most accurate configuration interaction wave function using the Hylleraas basis set [41]. It is known that high accuracy of calculation of the transition frequencies for Rydberg states of alkali metal atoms, in particular lithium, is a topical problem of theoretical methods for studying the electronic structure [44]. Such calculations for highly excited states have been carried out only recently [41]. Comparison of the calculated energies of excitation from the 1s

^{2}3s state with the high precision results [39] shows that the proposed implementation for ESs yields excellent agreement with the precision excitation energies (compare columns 5 and 6 of Table 6). Our calculation of the x-COEP ground state (1s

^{2}2s) energy yields E = −7.431724 hartrees. Comparison with the precision result E

^{exact}= −7.478060 hartrees gives E(xCOEP) – E

^{exact}= 0.046336 hartrees, which is inagreement with the accuracy of ES energy determination. For example, for the 1s

^{2}6s state E(xCOEP) – E

^{exact}= 0.046655 hartrees. For the Li atom we collected also the optimum values of parameters “C” and “d” defining the effective potential (see Table 7). We can see that, unlike He, there aresome trends in their behavior: the “C” parameter increases for higher excited states whereas the “d” parameter decreases.

## 5. Concluding Remarks

## Author Contributions

## Conflicts of Interest

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**Table 1.**x-COEP energies (hartrees) of triplet 1sns (n = 2, 3,…, 7) states of He and their comparison with MOM method (basis set consists of 42s functions).

State | x-COEP-bgs * | x-COEP-Vgs ** | x-COEP | MOM [43] | ∆ *** |
---|---|---|---|---|---|

1s2s ^{3}S | −2.171687 | −2.171687 | −2.171687 | −2.174251 | 2.564 |

1s3s ^{3}S | −2.000406 | −2.067214 | −2.067464 | −2.068485 | 1.021 |

1s4s ^{3}S | −0.525465 | −2.034747 | −2.035195 | −2.036436 | 1.241 |

1s5s ^{3}S | 3.256538 | −2.020897 | −2.021524 | −2.022583 | 1.059 |

1s6s ^{3}S | - | −2.013670 | −2.01424 | −2.015357 | 1.114 |

1s7s ^{3}S | - | −2.006859 | −2.009609 | −2.011118 | 1.509 |

_{eff}for each individual state are used; ** ground state V

_{eff}and optimized basis set adjusted to a given excited state are used; *** ∆ = E(xCOEP) − E(MOM) (mhartrees).

**Table 2.**Doubly excited energies (hartrees) computed at the xCOEP level and their comparison with the Hartree-Fock and “exact” values for the 2sns (n = 3, 4, …, 8) states of He; (basis set consists of 42s functions).

State | x-COEP-Vgs * | x-COEP | HF [39] | E^{exact} [40] | ∆ ** |
---|---|---|---|---|---|

2s3s ^{3}S | −0.583918 | −0.584750 | −0.584843 | −0.602578 | 17.828 |

2s4s ^{3}S | −0.540931 | −0.541915 | −0.541994 | −0.548841 | 6.926 |

2s5s ^{3}S | −0.524047 | −0.525104 | −0.525151 | −0.528414 | 3.310 |

2s6s ^{3}S | −0.515899 | −0.516738 | −0.516757 | −0.518546 | 1.808 |

2s7s ^{3}S | −0.503937 | −0.511622 | −0.511964 | −0.513046 | 1.424 |

2s8s ^{3}S | −0.485457 | −0.508428 | −0.508969 | −0.509673 | 1.245 |

_{eff}and optimized basis set adjusted to a given excited state are used; ** ∆ = E(x-COEP) − E

^{exact}(mhartrees).

**Table 3.**Doubly excited triplet energies (hartrees) computed at the xCOEP level and their comparison with the Hartree-Fock and “exact” values for the 3sns (n = 4, …, 8) states of He; (basis set consists of 42s functions).

State | x-COEP | HF | E^{exact} [40] | ∆ * (mhartrees) |
---|---|---|---|---|

3s4s | −0.272284 | −0.272245 | −0.287277 | 14.993 |

3s5s | −0.250622 | −0.250554 | −0.258134 | 7.512 |

3s6s | −0.240454 | −0.240598 | −0.244807 | 4.353 |

3s7s | −0.234329 | −0.235129 | −0.237672 | 3.343 |

3s8s | −0.231937 | −0.231791 | −0.233433 | 1.496 |

^{exact}.

**Table 4.**Doubly excited triplet energies (hartrees) computed at the xCOEP level and their comparison with the Hartree-Fock and “exact” values for the 4sns (n = 5, 6, 7, 8) states of He; (basis set consists of 42s functions).

State | x-COEP | HF | E^{exact} [40] | ∆ * (mhartrees) |
---|---|---|---|---|

4s5s | −0.157968 | −0.157982 | −0.169307 | 11.339 |

4s6s | −0.145387 | −0.145402 | −0.152122 | 6.735 |

4s7s | −0.138015 | −0.138020 | −0.143176 | 5.161 |

4s8s | −0.134051 | −0.134134 | −0.137961 | 3.91 |

^{exact}.

**Table 5.**Doubly excitation energies (eV) from triplet state of He computed at the x-COEP level of approximation.

Excitation | x-COEP | “Exact” [40] |
---|---|---|

2s3s → 3s4s | 8.5 | 8.58 |

2s4s → 3s5s | 7.93 | 7.91 |

2s5s → 3s6s | 7.75 | 7.72 |

2s6s → 3s7s | 7.68 | 7.64 |

2s7s → 3s8s | 7.61 | 7.6 |

3s4s → 4s5s | 3.11 | 3.21 |

3s5s → 4s6s | 2.86 | 2.88 |

3s6s → 4s7s | 2.79 | 2.77 |

3s7s → 4s8s | 2.73 | 2.71 |

**Table 6.**Excited doublet 1s

^{2}ns (n = 3, 4,…, 7) energies (hartrees) and excitation energies ∆Е (eV) computed at the x-COEP level with respect to the 1s

^{2}3s state and their comparison to “exact” values for the Li atom (basis set consists of 42s functions).

State | x-COEP | HF [39] | Exact * [41] | ∆Е (eV) | |
---|---|---|---|---|---|

x-COEP | «exact» | ||||

1s^{2}3s | −7.307322 | −7.310208 | −7.354098 | 0 | 0 |

1s^{2}4s | −7.273149 | −7.274884 | −7.318531 | 0.93 | 0.97 |

1s^{2}5s | −7.257268 | −7.259979 | −7.303552 | 1.36 | 1.38 |

1s^{2}6s | −7.249194 | −7.252317 | −7.295859 | 1.58 | 1.59 |

1s^{2}7s | −7.242843 | −7.247864 | −7.291392 | 1.75 | 1.71 |

**Table 7.**The optimum values of parameters of the effective potential for the different doublet states of Li.

State | C | d |
---|---|---|

1s^{2}2s | 1.094723 | 1.661917 |

1s^{2}3s | 1.691346 | 1.204497 |

1s^{2}4s | 4.151071 | 0.694042 |

1s^{2}5s | 9.623142 | 0.431399 |

1s^{2}6s | 18.66719 | 0.301418 |

1s^{2}7s | 65.39487 | 0.157559 |

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Glushkov, V.; Levy, M.
Highly Excited States from a Time Independent Density Functional Method. *Computation* **2016**, *4*, 28.
https://doi.org/10.3390/computation4030028

**AMA Style**

Glushkov V, Levy M.
Highly Excited States from a Time Independent Density Functional Method. *Computation*. 2016; 4(3):28.
https://doi.org/10.3390/computation4030028

**Chicago/Turabian Style**

Glushkov, Vitaly, and Mel Levy.
2016. "Highly Excited States from a Time Independent Density Functional Method" *Computation* 4, no. 3: 28.
https://doi.org/10.3390/computation4030028