I present a timedependent density functional study of the 20 lowlying excited states as well the ground states of the zinc dimer
Zinc dimer
The paper presents allelectron calculations on the lowestlying 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 spinfree Hamiltonian (SFH), without spinorbit coupling, with a comparison to a relativistic 4component DiracCoulomb Hamiltonian (DCH), spinorbit coupling included, in the framework of timedependent density functional theory (TDDFT) and its linearresponse approximation (LRA). The calculations are performed using DiracPackage (program for atomic and molecular direct iterative relativistic allelectron 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  DiracCoulomb Hamiltonian 
MP2  MøllerPlesst 2ndorder perturbation theory 
CCSD(T)  Coupled cluster singlesdoubles (triples) 
SFH  Relativistic spinfree Hamiltonian 
(TD)DFT  (Timedepended) density functional theory 
xc  Exchangecorrelation 
LR(A)  Linearresponse (approximation) 
ALR  Adiabatic LR 
srLDAMP2  Shortrange LDA, longrange MP2 
Timedependent 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 DiracPackage [
The ground state of the group 12 dimer has a (closedshell) valence orbitals configuration:
Lowest excited states and the corresponding asymptotes.











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 DiracCoulomb 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 densityresponse 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 groundstate 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 timedependencies in the past and takes only the instantaneous density
The reported results in this paper have been performed using a development version of the Dirac10Package [
The values of the spectroscopic constants
We employed the augccpVTZ (likewise augccpVQZ) 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 DiracPackage. 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 groundstate bond of
Groundstate


 

exp^{1}  25.7  0.034  
exp^{2}  4.19  25.9  0.035 
HFMP2^{Q}  3.611  29  0.049 
srLDAMP2^{Q}  3.445  31  0.0459 
PBE^{Q}  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 
GRACPBE0  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 augccpVTZ basis set and SFH. ^{Q}augccpVQZ basis set, for PBE, HFMP2 and srLDAMP2 (NR with parameter
The excited states shown in the pw are given in Table
At first we compare for PBE functional a 4component and spinfree result for the four lowest states calculated in augccpVTZ 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 4component DCH of the spectroscopic constant. Above






SFH  2.347  2.534  4.795  4.79 
4c. 
2.345  —  4.874  — 
4c. 
2.345  —  4.480  — 
4c. 
—  2.534  —  4.553 
4c. 
2.347  2.534  4.625  4.574 
4c. 
2.349  —  4.945  — 
 
SFH  219  172  7  27 
4c. 
220  —  6  — 
4c. 
220  —  13  — 
4c. 
—  172  —  33 
4c. 
219  172  13  34 
4c. 
219  —  8  — 
 
SFH  13097  10870  52  405 
4c. 
12934  —  52  — 
4c. 
13130  —  417  — 
4c. 
—  10486  —  533 
4c. 
12906  10680  235  550 
4c. 
13068  —  53  — 
Bond lengths
Method 









P^{Q}  2.345  2.532  4.254  4.765  2.350  2.596  4.735  2.573 
P^{T}  2.347  2.534  4.795  4.79  2.351  2.602  4.744  2.592 
W91^{T}  2.343  2.517  diss  4.546  2.347  2.621  dis  5.158 
P0^{T}  2.358  2.517  5.046  4.517  2.351  2.631  2.715  2.594 
GP0^{T}  2.356  2.522  diss  5.806  2.345  2.780  2.929  4.755 
CB3L^{T}  2.343  2.489  diss  diss  2.327  2.613  2.637  2.572 
B3L^{T}  2.371  2.566  diss  5.525  2.366  2.655  2.807  2.624 
BL^{T}  2.371  2.587  diss  4.882  2.376  2.648  diss  2.639 
B86^{T}  2.337  2.534  diss  4.583  2.341  2.611  4.647  5.370 
LDA^{T}  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 
[ 
2.35  2.50  4.11  diss  2.33  2.69  2.42  2.92 
[ 
2.41  2.70  diss  diss  2.33  3.22  2.40  3.05 
[ 
2.38  2.59  4.36  diss  2.38  2.64  2.65^{f}  2.65^{f} 
[ 
2.53  2.74  diss  —  2.51  2.97  2.64  3.07 
[ 
2.56  2.70  diss  diss  2.48  2.92  2.64  — 
[ 
2.372  2.53  
exp  —  —  4.49^{g}  —  —  3.0^{g}  —  — 
^{
T}Present work calculated with augccpVTZ and ^{Q}with augccpVQZ basis set. P, W91, P0, GP0, B86, BL, B3L, and CB3L denote PBE, BPW91, PBE0, GRACPBE0, BP86, BLYP, B3LYP, and CAMB3LYP, respectively. ^{a}With DKCASPT2. ^{b}With DKMRACPF. ^{c}With CI. ^{d}With MRCI. ^{e}With CCSD(T). ^{f}Value are ca. ^{g}From [
Vibrational frequencies
Method 









P^{Q}  220  173  9  28  219  136  12  142 
P^{T}  219  172  7  27  219  135  13  135 
W91^{T}  218  177  diss  33  220  129  diss  21 
P0^{T}  215  182  9  11  223  135  116  146 
GP0^{T}  216  181  diss  11  226  106  78  38 
CB3L^{T}  220  189  diss  diss  232  139  137  150 
B3L^{T}  211  167  diss  13  215  126  90  139 
BL^{T}  210  157  diss  27  207  122  diss  115 
B86^{T}  222  172  diss  37  222  131  14  29 
LDA^{T}  247  189  85  45  247  160  89  26 
[ 
231  200  23  diss  250  131  211  58 
[ 
220  208  32  diss  244  121  205  104 
[ 
211  169  diss  diss  212  77  175  112 
[ 
192  175  —  diss  210  134  178  — 
[ 
175  150  diss  diss  202  107  166  104 
exp 



—  — 


— 
For the acronyms, see Table
Dissociation energies (
Method 









P^{Q}  1.626  1.347  0.0065  0.050  1.703  0.579  0.0180  0.112^{*} 
P^{T}  1.624  1.348  0.0065  0.050  1.698  0.572  0.0175  0.08^{*} 
W91^{T}  1.423  1.23  diss  0.031  1.654  0.541  diss  0.027 
P0^{T}  1.481  1.332  0.0034  0.0031  2.387  1.247  0.413  0.10 
GP0^{T}  1.43  1.316  diss  0.0148  2.385  1.111  0.270  0.279 
CB3L^{T}  1.436  1.281  diss  diss  2.298  1.126  0.099  0.292^{*} 
B3L^{T}  1.45  1.189  diss  0.0033  2.226  1.125  0.393  0.148^{*} 
BL^{T}  1.514  1.181  diss  0.468  1.426  0.361  diss  0.542 
B86^{T}  1.593  1.312  diss  0.0673  1.688  0.546  0.008  0.058 
LDA^{T}  2.119  1.704  1.456  1.902  2.089  0.798  0.145  0.788 
[ 
1.502  1.225  0.026  diss  2.713  1.189  0.734  0.60 
[ 
1.457  1.204  0.110  diss  2.694  1.292  0.718  0.204 
[ 
0.91  0.90  diss  diss  2.35  0.71  —  — 
[ 
1.21  0.95  0.016  diss  2.26  1.12  0.63  0.32 
[ 
1.10  0.98  —  diss  2.43  1.13  0.66  — 
[ 
1.05  0.87  diss  diss  2.42  1.06  0.83  0.44 
[ 
1.41  1.21  —  —  —  —  —  — 
exp  —  — 

—  — 

—  — 
For the acronyms, see Table
Higher states corresponding to higher asymptotes see Table
State 





CB3L  P0  GP0  B3L  W91  B86  CB3L  P0  GP0  B3L  W91  B86  CB3L  P0  GP0  B3L  W91  B86  

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^{*} 

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 

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 

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 

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^{*} 

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 

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 

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 

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 

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 

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 

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 augccpVTZ basis set. For the acronyms, see Table
(a) Zn_{2} 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 longrange 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 augccpVTZ basis set and augccpVQZ 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 abovementioned reasons. Available approximations do not describe the longrange 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 longrange part of the twoelectron 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 GRACPBE0 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 wellknown and newly developed functional approximations. We performed the calculations with DiracPackage using relativistic 4component DCH and SFH. First, we showed that SFH is capable to achieve the same accuracy as 4components 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.