Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 901–904 c International Academic Publishers
Vol. 47, No. 5, May 15, 2007
Time-Dependent Density-Functional Calculations for Optical Spectra of Na2 and Na4 Clusters∗ ZHANG Yan-Ping,1,2,4,† ZHANG Feng-Shou,1,3 MENG Ke-Lai,1,2,4 and XIAO Guo-Qing1,2 1
Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China 2 Institute of Modern Physics, the Chinese Academy of Sciences, Lanzhou 730000, China 3
Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China
4
Graduate School of the Chinese Academy of Sciences, Beijing 100049, China
(Received May 26, 2006; Revised July 17, 2006)
Abstract With the frame of the time-dependent local density approximation, an efficient description of the optical response of clusters has been used to study the photo-absorption cross section of Na2 and Na4 clusters. It is shown that our calculated results are in good agreement with the experiment. In addition, our calculated spectrum for the Na4 cluster is in better agreement with experiment than the GW absorption spectrum. PACS numbers: 36.40.Mr, 61.20.Lc
Key words: time-dependent local density approximation, optical response of cluster
1 Introduction Electronic and structure calculations for clusters can help explain such phenomena as quantum confinement, surface reconstruction, and crystal growth, describe the formation of surface and bulk defects, and predict the properties of porous and disordered materials. However, due to a large number of atoms and the lack of general symmetry, computer simulations for clusters present enormous challenges for traditional theoretical methods. Simple classical and cost-efficient methods based on empirical force fields or interatomic potentials do not work well for clusters. Interatomic interactions derived from the crystalline state are difficult to describe clusters, because of the construction of surfaces and the strong delocalization.[1] A direct quantum mechanical approach is required for accurate calculations of clusters. Among such methods, ab initio pseudopotential techniques based on density-functional theory (DFT) within the local density approximation (LDA) attract special interest.[2] Unfortunately, conventional DFT cannot deal with electronic excitations that are related to many important physical properties such as optical absorption and emission, response to time-dependent fields, and the dynamical dielectric function. Some complex computational techniques, such as the configuration interaction (CI) method,[3] quantum Monte Carlo (QMC) simulations,[4] and the Greens’s function method based on the GW approximation[5] can properly describe electronic excitations and calculate excitation energies and absorption spectra. But these methods are very computationally demanding and limited to very small systems. Recently, a time-dependent density func∗ The
tional approach was developed by Deb and Ghosh[6] and by Bartolotti,[7] and subsequently generalized by Runge and Gross,[8] as well as by Dhara and Ghosh.[9] The Kohn– Sham DFT was extended to individually excited state,[10] quantal density functional theory (Q-DFT) of singly or multiply excited bound nondegenerate states.[11] As a simplest level of approximation to the time-dependent density functional theory (TDDFT), the time-dependent local density approximation (TDLDA) includes the proper representation of excited states and allows one to compute the true excitation energies of a many-electron system from the conventional time-independent Kohn–Sham transition energies and wave functions. Compared to other theoretical methods for excited states, the TDLDA technique has considerably low computational demands. For these reasons, TDLDA has been widely applied to describe the photo-absorption of atoms, molecules, clusters, and solids, starting with jellium model by Ekardt.[12] With the developments of experiment techniques, spectral profiles and absolute values for the absorption cross section can be provided. The photo-absorption spectra of sodium clusters have been investigated experimentally and theoretically.[13−17] In this paper, we apply the TDLDA technique combined with the pseudopotential approach to calculate the complex response of small free sodium clusters to an external field.
2 Theory In this section, we briefly describe TDDFT, which is the basis of our electronic quantum mechanical calculations. The central theorem of DFT is that the external
project supported by National Natural Science Foundation of China under Grant Nos. 10405025, 10575012, 10435020, and 10535010
[email protected]
† E-mail:
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ZHANG Yan-Ping, ZHANG Feng-Shou, MENG Ke-Lai, and XIAO Guo-Qing
potential and the ground-state energy of an interacting N electron system are uniquely determined by the groundstate density.[2] Within the TDDFT, the main theorem of the density-functional formalism is extended to timedependent systems and the physics of a system of interacting N -electrons is mapped by a model system of noninteracting electrons with equivalent density and energy. The time-dependent density is obtained through a set of single-electron wavefunctions φj (r, t) which satisfy the time-dependent Kohn–Sham equation (TDKS),[18] ∂ ˆ Ks φj (r, t) i φj (r, t) = H ∂t ∇2 = − + Veff (r, t))φj (r, t , 2 j = 1, 2, 3, . . . , N . (1) The TDKS effective potential Veff is decomposed into an P ionic background potential Vion = I Vps (r − RI ) from the ionic core at positions {RI }, external force potential Vext , a time-dependent Hartree part VH , and a so-called exchange-correlation (xc) potential Vxc , Veff [n](r, t) = Vion (r, t) + Vext (r, t) + VH [n](r, t) + Vxc [n](r, t) ,
(2)
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To calculate the linear optical absorption cross section of the system, an external field, Vext (r, t) = κq δ(t)Dˆq is applied instantaneously at t = 0. Here, Dˆq shows the dipole operator in the three orientations along the principal axes of the system ~eq (q = 1, 2, 3). An initial state is achieved by transforming the ground-state wave functions according to φj (r, t0 ) = e iκq rq φstat (r) , j
(7)
and then propagating these wave-functions for some time. (r) are wave functions for the ground state of the φstat j system. The optical absorption cross section σ(ω) can be obtained from the following expression: 4πω Im α(ω) . (8) σ(ω) = c The dynamical polarizability α(ω) is essentially the Fourier transform of the dipole moment of the system dq (t), Z 1 X 1 T dt exp(iωt−λt)[dq (t)−dq (0)] , (9) α(ω) = − 3 q=1,2,3 κq 0 where λ is a smoothing parameter. To remove the signal noise induced by the fitness of the time interval, an exponential damping function exp(−λt) is used.
where the electronic density n is written by n(r, t) =
N X
3 Results and Discussions |φj (r, t)|2 ,
(3)
j=1
and the Hartree potential VH [n](r, t) is defined as Z n(r 0 , t) . VH [n](r, t) = d 3 r0 |r − r 0 |
(4)
The xc potential Vxc [n](r, t) is a functional of the timedependent density and has many approximation choices. In this work, we use the simplest approximation TDLDA, which is defined as dhom (n) TDLDA Vxc [n](r, t) = xc , (5) dn n=n(r,t) where hom xc (n) is the xc energy density of homogeneous electron gas, for which the parametrization of Perdew and Zunger is used.[19] The norm-conserving pseudopotentials Vps in the fully separable forms are constructed following the method of Troullier and Martins.[20] The TDKS equations are integrated with approximated enforced time-reversal symmetry (AETRS) method,[21] n ∆t o ˆ + ∆t) φj (t + ∆t) = exp −i H(t 2 n ∆t o ˆ × exp −i H(t) φj (t) , (6) 2 where a time step ∆t of 0.002 eV−1 is chosen. To calculate the exponential of the Hamiltonian, an approximation to the fourth-order Taylor expansion is used. This approximation has been shown to conserve energy to high accuracy.[22]
In this section, we apply the TDLDA to calculate absorption spectra of small Na2 and Na4 clusters. In calculations, a boundary domain with a radius of 25 a.u. and a grid spacing of 0.8 a.u. are used. In order to make systems stay in a linear regime, we use a small excitation κq ≈ 5.8 × 10−5 /r0 along the diagonal (1, 1, 1). Here r0 is Bohr radius.
Fig. 1 (a) The average dipole response δd of Na2 ; (b) The same as (a) but for Na4 .
Firstly, the average dipole responses δd of Na2 and Na4 clusters are calculated and shown in Figs. 1(a) and 1(b),
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Time-Dependent Density-Functional Calculations for Optical Spectra of Na2 and Na4 Clusters
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respectively. Then we obtain the absorption spectra of Na2 and Na4 from their average dipole responses by using Eqs. (8) and (9). The lower inset in Fig. 2 shows the experimental photoabsorption cross section of the Na2 cluster, which is adapted from Ref. [15]; while the upper inset shows the photoabsorption cross section of our TDLDA calculation which is plotted on the same scale. The experimental photoabsorption spectrum shows three excitation peaks around 1.91, 2.52, and 3.71 eV, which belong to the 1 + 1 + 1 1 + 1 X 1 Σ+ g → A Σu , X Σg → B Πu , and X Σg → C Πu
transitions in the Na2 cluster. In our TDLDA calculation spectrum, the three peaks are 2.04, 2.65, and 3.68 eV, respectively. The 2.04 eV peak comes from the vibration mode parallel to symmetry axis which is the line connecting two atoms of Na2 . The 2.65 eV peak is generated from vibration mode orthogonal to the symmetry axis. There is a peak at the energy of 4.73 eV which is consistent with the experimental ionization energy at 4.87 eV for Na2 → Na+ 2 process. All the peak positions agree with experiment to within 0.03 ∼ 0.15 eV.
Fig. 2 The calculated and experimental optical spectra of Na2 cluster. The experimental spectra are adapted from Ref. [15]. A Lorentzian convolution of 0.06 eV is used to simulate finite broadening of the calculated TDLDA spectra.
Fig. 3 The calculated and experimental optical spectra of Na4 cluster. The experimental spectra are adapted from Ref. [15]. A Lorentzian convolution of 0.06 eV is used to simulate finite broadening of the calculated TDLDA spectra.
A comparison between the measured absorption photoabsorption cross section and our TDLDA calculated energies for allowed transitions of Na4 with the structure of D2h is presented in Fig. 3. The lower inset shows the measured photo-absorption cross section of the Na4 cluster, which is adapted from Ref. [15]; the upper inset shows our TDLDA calculated photoabsorption cross section, which is plotted on the same scale. Compared with Na2 , the optical response spectrum of Na4 exhibits very rich patterns. The experimental photoabsorption cross section of Na4 cluster shows six peaks at the energies of 1.80, 1.98, 2.18, 2.51, 2.81, 3.15, 3.33 eV, which belong to 1 Ag → 11 B3u , 1 Ag → 21 B3u , 1 Ag → 11 B1u , 1 Ag → 21 B2u /31 B2u , 1 Ag → 21 B1u , 1 Ag → 31 B1u /31 B3u , and 1 Ag → 41 B2u transitions, respectively. The six peaks are also produced in our calculated spectrum of Na4 cluster. Their peak positions are 1.87, 2.12, 2.28, 2.73, 3.08, 3.39, and 3.58 eV, respectively. All the calculated peak positions are in good agreement with experiment to within 0.04 ∼ 0.27 eV. Table 1 shows the energies of selected transitions in the measured and our TDLDA calculated absorption spectra of Na2 and Na4 cluster. As a reference, the self-consistent ionization potentials, the ionization energies of the highestoccupied Kohn–Sham LDA orbits and assignments are included in Table 1. The self-consistent vertical ionization potential ΩSCF ion are calculated as the differences between the total energies of the neutral and the positively charged clusters, calculated at the equilibrium atomic coordinates for the neutral clusters. It is well known that the Kohn– Sham LDA ionization energies −LDA HOMO are obtained by the negative energies for the highest occupied orbitals. It has been shown that TDLDA calculations tend to underestimate electronic excitation energies above −LDA HOMO threshold. Although many energies listed in Table 1 are close to or even higher than −LDA HOMO , we can see that our TDLDA calculations for Na2 and Na4 clusters follow perfectly the transition rules in the case of dipole radiation and agree with experiment well. Furthermore, the calculated result for Na4 cluster is in better agreement with the experiment than the GW absorption spectrum.[17] In our TDLDA calculated spectra, the information about the transitions between singlet and multiplet, as well as the transitions from bound states to continuum states are not provided because the spin-orbit coupling effects are not included and the continuum states are not well reproduced.
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ZHANG Yan-Ping, ZHANG Feng-Shou, MENG Ke-Lai, and XIAO Guo-Qing
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Table 1 Energies and assignments of selected transitions in the measured and our TDLDA calculated absorption spectra of Na2 and Na4 clusters. Experimental transition energies are adapted from Ref. [15]. Kohn–Sham LDA “ionization” energies −LDA HOMO and self-consistent LDA ionization potential ΩSCF ion are given as a reference. All energies are in eV. Cluster
Na2 (Dcoh )
Na4 (D2h )
Experiment
TDLDA
−LDA HOMO
ΩSCF ion
Assignment
1.91
2.04
3.21
5.17
1 + X 1 Σ+ g → A Σu
2.52 3.71
2.65 3.68
1.80 1.98 2.18 2.51 2.81 3.15 3.33
1.87 2.12 2.28 2.73 3.08 3.39 3.58
1 X 1 Σ+ g → B Πu 1Π X 1 Σ+ → C u g
2.77
4.38
1A g 1A g 1A g
→ 11 B3u → 21 B3u → 11 B1u 1 A → 21 B /31 B g 2u 2u 1 A → 21 B g 1u 1 A → 31 B /31 B g 1u 3u 1 A → 41 B g 2u
4 Conclusion In this paper, we use the real-time TDLDA to calculate the dynamic polarizabilities for Na2 with the structure of Dcoh symmetry and Na4 with the structure of D2h symmetry. By the Fourier transform of the dynamic polarizabilities, the photoabsorption cross sections of the system are obtained. It is shown that our results for Na2 and Na4 clusters follow perfectly the transition rules in the case of dipole radiation. The calculated energies of allowed transitions in the system are in good agreement with the experiments. For the spectrum of Na4 , our calculated result is in better agreement with the experiment than the GW spectrum. With the developments of experimental techniques, the research on cluster physics is beyond the regime of linear response. In the future, we will apply TDLDA to high-order harmonic generation, electron emission spectra, the kinetic spectra of emitted fragments, etc.
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