I present a time-dependent density functional study of the 20 low-lying excited states as well the ground states of the zinc dimer
Zinc dimer
The paper presents all-electron calculations on the lowest-lying excited states as well as the ground state. The first 8 lowest exited states are discussed with a comparison to experimental and literature values, and several other higher excited states are presented and discussed. Earlier works investigated the lowest 8 excited states using different wave function methods. Ellingsen et al. [
In this work, we use a relativistic spin-free Hamiltonian (SFH), without spin-orbit coupling, with a comparison to a relativistic 4-component Dirac-Coulomb Hamiltonian (DCH), spin-orbit coupling included, in the framework of time-dependent density functional theory (TDDFT) and its linear-response approximation (LRA). The calculations are performed using Dirac-Package (program for atomic and molecular direct iterative relativistic all-electron calculations) [
The paper is organized as follows. Section
Some of the acronyms used in this work.
HF | Hartree Fock method |
NR | Nonrelativistic |
DHF | Dirac or relativistic HF |
DCH | Dirac-Coulomb Hamiltonian |
MP2 | Møller-Plesst 2nd-order perturbation theory |
CCSD(T) | Coupled cluster singles-doubles (triples) |
SFH | Relativistic spin-free Hamiltonian |
(TD)DFT | (Time-depended) density functional theory |
xc | Exchange-correlation |
LR(A) | Linearresponse (approximation) |
ALR | Adiabatic LR |
srLDAMP2 | Short-range LDA, long-range MP2 |
Time-dependent density functional theory (TDDFT) currently has a growing impact and intensive use in physics and chemistry of atoms, small and large molecules, biomolecules, finite systems, and solidstate. For excited states resulting from a single excitation that present a single jump from the ground state to an excited state, I used in this work the LRA as implemented in Dirac-Package [
The ground state of the group 12 dimer has a (closed-shell) valence orbitals configuration:
Lowest excited states and the corresponding asymptotes.
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We will discuss the lowest 20 excited states dissociating to the atomic asymptotes (NR notation) given in Table
Density functional theory [
In the relativistic Dirac theory in absence of electromagnetic field, the DCH has the same generic form as the NR Hamiltonian (for molecules) [
The Dirac equation with the Dirac-Coulomb Hamiltonian (DCH) describes the important relativistic effects for chemical calculation, which become large for systems with large
In this section, we briefly introduce TDDFT formulation with a special emphasis on the linear density-response function and its connection to the electronic excitation spectrum, a more extensive derivations and wide discussions can be found in refs [
In the special case of the response of the ground-state density to a weak external field, that is, the case in the most optical applications, the slightly perturbed system, which can be written in a series expansion
In the adiabatic approximation which is the most common in TDDFT, one ignores all time-dependencies in the past and takes only the instantaneous density
The reported results in this paper have been performed using a development version of the Dirac10-Package [
The values of the spectroscopic constants
We employed the aug-cc-pVTZ (likewise aug-cc-pVQZ) Gaussian basis sets of Dunning and coworkers [
In this section, we discuss our computational result based on our calculations with the linear response adiabatic TDDFT module in Dirac-Package. Our main concern will be (beside the correctness of our computational result) to compare the behavior of different density functional approximations (and in comparison to other methods) to draw conclusions on the performance, the quality, and the validity of the different functional approximations, also in regard to applications to similar systems and possibly enlighten improvements of the DFT approximations in future works. The comparison with the literature values is accompanying our discussion, where works with different computational methods are available and with experimental values as far as available to judge the quality of our result.
As already mentioned, the ground-state bond of
Ground-state
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exp1 | 25.7 | 0.034 | |
exp2 | 4.19 | 25.9 | 0.035 |
HF-MP2Q | 3.611 | 29 | 0.049 |
srLDAMP2Q | 3.445 | 31 | 0.0459 |
PBEQ | 3.157 | 48 | 0.678 |
PBE | 3.156 | 49 | 0.683 |
PBE0 | diss | diss | diss |
BPW91 | 3.225 | 41 | 0.0154 |
BP86 | 3.181 | 46 | 0.036 |
BLYP | diss | diss | diss |
B3LYP | diss | diss | diss |
GRAC-PBE0 | 3.338 | 40.0 | 0.045 |
CAMB3LYP | 4.219 | 11 | 0.001 |
LDA | 2.846 | 85 | 0.225 |
a | 3.959 | 22 | 0.024 |
b | 3.96 | 22.5 | 0.030 |
c1 | 4.03 | 20.4 | 0.0205 |
c2 | 4.03 | 20.4 | 0.0205 |
pw using aug-cc-pVTZ basis set and SFH. Qaug-cc-pVQZ basis set, for PBE, HF-MP2 and srLDAMP2 (NR with parameter
The excited states shown in the pw are given in Table
At first we compare for PBE functional a 4-component and spin-free result for the four lowest states calculated in aug-cc-pVTZ basis set and demonstrate that SFH describes accurately the main relevant contributions of the relativistic effects. As seen in Table
Comparison between SFH (NR state assignment) and 4-component DCH of the spectroscopic constant. Above
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SFH | 2.347 | 2.534 | 4.795 | 4.79 |
4-c. |
2.345 | — | 4.874 | — |
4-c. |
2.345 | — | 4.480 | — |
4-c. |
— | 2.534 | — | 4.553 |
4-c. |
2.347 | 2.534 | 4.625 | 4.574 |
4-c. |
2.349 | — | 4.945 | — |
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SFH | 219 | 172 | 7 | 27 |
4-c. |
220 | — | 6 | — |
4-c. |
220 | — | 13 | — |
4-c. |
— | 172 | — | 33 |
4-c. |
219 | 172 | 13 | 34 |
4-c. |
219 | — | 8 | — |
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SFH | 13097 | 10870 | 52 | 405 |
4-c. |
12934 | — | 52 | — |
4-c. |
13130 | — | 417 | — |
4-c. |
— | 10486 | — | 533 |
4-c. |
12906 | 10680 | 235 | 550 |
4-c. |
13068 | — | 53 | — |
Bond lengths
Method |
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PQ | 2.345 | 2.532 | 4.254 | 4.765 | 2.350 | 2.596 | 4.735 | 2.573 |
PT | 2.347 | 2.534 | 4.795 | 4.79 | 2.351 | 2.602 | 4.744 | 2.592 |
W91T | 2.343 | 2.517 | diss | 4.546 | 2.347 | 2.621 | dis | 5.158 |
P0T | 2.358 | 2.517 | 5.046 | 4.517 | 2.351 | 2.631 | 2.715 | 2.594 |
GP0T | 2.356 | 2.522 | diss | 5.806 | 2.345 | 2.780 | 2.929 | 4.755 |
CB3LT | 2.343 | 2.489 | diss | diss | 2.327 | 2.613 | 2.637 | 2.572 |
B3LT | 2.371 | 2.566 | diss | 5.525 | 2.366 | 2.655 | 2.807 | 2.624 |
BLT | 2.371 | 2.587 | diss | 4.882 | 2.376 | 2.648 | diss | 2.639 |
B86T | 2.337 | 2.534 | diss | 4.583 | 2.341 | 2.611 | 4.647 | 5.370 |
LDAT | 2.265 | 2.454 | 2.764 | 4.364 | 2.267 | 2.485 | 2.702 | 5.414 |
[ |
2.33 | 2.48 | 3.99 | diss | 2.30 | 2.64 | 2.40 | 2.74 |
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2.35 | 2.50 | 4.11 | diss | 2.33 | 2.69 | 2.42 | 2.92 |
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2.41 | 2.70 | diss | diss | 2.33 | 3.22 | 2.40 | 3.05 |
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2.38 | 2.59 | 4.36 | diss | 2.38 | 2.64 | 2.65f | 2.65f |
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2.53 | 2.74 | diss | — | 2.51 | 2.97 | 2.64 | 3.07 |
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2.56 | 2.70 | diss | diss | 2.48 | 2.92 | 2.64 | — |
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2.372 | 2.53 | ||||||
exp | — | — | 4.49g | — | — | 3.0g | — | — |
TPresent work calculated with aug-cc-pVTZ and Qwith aug-cc-pVQZ basis set. P, W91, P0, GP0, B86, BL, B3L, and CB3L denote PBE, BPW91, PBE0, GRAC-PBE0, BP86, BLYP, B3LYP, and CAMB3LYP, respectively. aWith DK-CASPT2. bWith DK-MRACPF. cWith CI. dWith MRCI. eWith CCSD(T). fValue are ca. gFrom [
Vibrational frequencies
Method |
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PQ | 220 | 173 | 9 | 28 | 219 | 136 | 12 | 142 |
PT | 219 | 172 | 7 | 27 | 219 | 135 | 13 | 135 |
W91T | 218 | 177 | diss | 33 | 220 | 129 | diss | 21 |
P0T | 215 | 182 | 9 | 11 | 223 | 135 | 116 | 146 |
GP0T | 216 | 181 | diss | 11 | 226 | 106 | 78 | 38 |
CB3LT | 220 | 189 | diss | diss | 232 | 139 | 137 | 150 |
B3LT | 211 | 167 | diss | 13 | 215 | 126 | 90 | 139 |
BLT | 210 | 157 | diss | 27 | 207 | 122 | diss | 115 |
B86T | 222 | 172 | diss | 37 | 222 | 131 | 14 | 29 |
LDAT | 247 | 189 | 85 | 45 | 247 | 160 | 89 | 26 |
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231 | 200 | 23 | diss | 250 | 131 | 211 | 58 |
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220 | 208 | 32 | diss | 244 | 121 | 205 | 104 |
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211 | 169 | diss | diss | 212 | 77 | 175 | 112 |
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192 | 175 | — | diss | 210 | 134 | 178 | — |
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175 | 150 | diss | diss | 202 | 107 | 166 | 104 |
exp |
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For the acronyms, see Table
Dissociation energies (
Method |
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PQ | 1.626 | 1.347 | 0.0065 | 0.050 | 1.703 | 0.579 | 0.0180 | 0.112* |
PT | 1.624 | 1.348 | 0.0065 | 0.050 | 1.698 | 0.572 | 0.0175 | 0.08* |
W91T | 1.423 | 1.23 | diss | 0.031 | 1.654 | 0.541 | diss | 0.027 |
P0T | 1.481 | 1.332 | 0.0034 | 0.0031 | 2.387 | 1.247 | 0.413 | 0.10 |
GP0T | 1.43 | 1.316 | diss | 0.0148 | 2.385 | 1.111 | 0.270 | 0.279 |
CB3LT | 1.436 | 1.281 | diss | diss | 2.298 | 1.126 | 0.099 | 0.292* |
B3LT | 1.45 | 1.189 | diss | 0.0033 | 2.226 | 1.125 | 0.393 | 0.148* |
BLT | 1.514 | 1.181 | diss | 0.468 | 1.426 | 0.361 | diss | 0.542 |
B86T | 1.593 | 1.312 | diss | 0.0673 | 1.688 | 0.546 | 0.008 | 0.058 |
LDAT | 2.119 | 1.704 | 1.456 | 1.902 | 2.089 | 0.798 | 0.145 | 0.788 |
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1.502 | 1.225 | 0.026 | diss | 2.713 | 1.189 | 0.734 | 0.60 |
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1.457 | 1.204 | 0.110 | diss | 2.694 | 1.292 | 0.718 | 0.204 |
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0.91 | 0.90 | diss | diss | 2.35 | 0.71 | — | — |
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1.21 | 0.95 | 0.016 | diss | 2.26 | 1.12 | 0.63 | 0.32 |
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1.10 | 0.98 | — | diss | 2.43 | 1.13 | 0.66 | — |
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1.05 | 0.87 | diss | diss | 2.42 | 1.06 | 0.83 | 0.44 |
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1.41 | 1.21 | — | — | — | — | — | — |
exp | — | — |
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— | — |
For the acronyms, see Table
Higher states corresponding to higher asymptotes see Table
State |
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CB3L | P0 | GP0 | B3L | W91 | B86 | CB3L | P0 | GP0 | B3L | W91 | B86 | CB3L | P0 | GP0 | B3L | W91 | B86 | |
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2.527 | 2.546 | 2.711 | 2.578 | 2.531 | 2.532 | 168 | 164 | 115 | 150 | 163 | 160 | 0.914 | 0.938 | 0.174 | 0.636* | 0.555 | 0.644* |
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2.737 | 2.769 | 5.772 | 2.802 | 2.71 | 2.714 | 185 | 196 | 23 | 168 | 193 | 186 | 0.533 | 0.728 | 0.596 | 0.421 | 0.118 | 0.094 |
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2.60 | 2.630 | 2.787 | 2.679 | 2.622 | 2.605 | 149 | 142 | 92 | 120 | 134 | 140 | 0.839 | 0.677 | 0.231 | 0.583 | 0.513 | 0.539 |
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3.444 | 3.388 | 8.434 | 3.449 | 3.256 | 3.21 | 174 | 146 | 19 | 118 | 131 | 139 | 0.339 | 0.333 | 0.383 | 0.097 | 0.152 | 0.153 |
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2.919 | 3.080 | 3.162 | 3.352 | 3.323 | 3.451 | 99 | 82 | 72 | 59 | 51 | 45 | 1.416 | 0.95 | 0.90 | 0.646* | 0.039 | 0.040* |
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2.487 | 2.504 | 4.748 | 2.524 | 2.491 | 2.485 | 178 | 174 | 41 | 163 | 171 | 172 | 1.140 | 0.434 | 0.635 | 0.213* | 0.143 | 0.20 |
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2.519 | 2.532 | diss | 2.551 | 2.546 | 2.506 | 172 | 171 | diss | 158 | 166 | 164 | 0.905 | 0.270 | diss | 0.515 | 0.482 | 0.480 |
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2.569 | 2.583 | diss | 2.603 | 2.513 | 2.563 | 153 | 150 | diss | 145 | 140 | 150 | 0.247 | 0.158* | diss | 0.163* | 0.150 | 0.157 |
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3.650 | 5.750 | 6.209 | 9.026 | diss | diss | 123 | 14 | 22 | 12 | diss | diss | 1.50 | 0.483 | 0.486 | 0.274 | diss | diss |
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2.459 | 2.482 | 6.317 | 2.495 | 2.472 | 2.465 | 190 | 184 | 26 | 174 | 180 | 182 | 1.417 | 0.344 | 0.482 | 0.46* | 0.43 | 0.393 |
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2.534 | 2.555 | diss | 2.585 | 2.537 | 2.533 | 169 | 167 | diss | 155 | 162 | 159 | 1.125 | 0.302* | diss | 0.50* | 0.561 | 0.560 |
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2.704 | 2.682 | diss | 4.237 | 2.616 | 2.583 | 281 | 288 | diss | 296 | 244 | 210 | 0.517* | 0.298 | diss | 0.218* | 0.146 | 0.165 |
All values with SFH and aug-cc-pVTZ basis set. For the acronyms, see Table
(a) Zn2 PBE functional, with SFH (left) ground state (lowest curve) and 8 lowest excited state (corresponding to the two asymptotes (
In Figure
In Figure
Obviously, a crucial point in calculating the excited states in TDDFT is that the most of the DFT approximations are semilocal, the long-range interaction is incorrectly described, consequently a disturbed potential curves is obtained, especially near the avoiding crossing point where the disturbed curves show enhanced effects. This can be clearly seen for the
In Tables
First, we look at the PBE values using aug-cc-pVTZ basis set and aug-cc-pVQZ basis set. As we see from Tables
Looking at the Tables
From Tables
To deal with more higher excited states is difficult because of the above-mentioned reasons. Available approximations do not describe the long-range behavior correctly and/or fail to offer the correct asymptotic limit or predict it accurately [
The general conclusion of this section is that CAMB3LYP gives the best result due to its better treatment of the long-range part of the two-electron interaction and its asymptotically better behavior (tail of the potential curve) apparently due to including a suitable amount of exact exchange, PBE0 gives a comparable result, the main problem here is the tail of the potential curve. BPW91, BP86, and B3LYP are less satisfactory but still show acceptable result, whereas (most likely) the result of GRAC-PBE0 is not useful.
In the present work, we have studied the ground as well the 20 lowest exited states of the zinc dimer in the framework of DFT and TDDFT using well-known and newly developed functional approximations. We performed the calculations with Dirac-Package using relativistic 4-component DCH and SFH. First, we showed that SFH is capable to achieve the same accuracy as 4-components DCH and can describe quantitatively the main relevant contributions of the relativistic effects. In analyzing the results obtained from different functional approximations, comparing them with each other, with literature and experimental values as far as available, we drew some conclusions. The results show that the linear response in the adiabatic approximation with the known DFT approximations give good performance for the 8 lowest excited states of
The author gratefully acknowledges fruitful discussions with Dr. Trond Saue, Laboratoire de Chimie et Physique Quantique, Université de Toulouse (France), and the kindly support from him. Dr. Radovan Bast, Tromsø University (Norway), is acknowledged for his kindly support and the kindly support from the Laboratoire de Chimie Quantique, CNRS et Université de Strasbourg.