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Instability of Dianions of Alkali-Metal Clusters

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Instability of Dianions of Alkali-Metal Clusters Materials Transactions, Vol. 46, No. 6 (2005) pp. 1103 to 1105 Special Issue on Computer Modeling of Materials and Processes #2005 The Japan Institute of Metals Instability of Dianions of Alkali-Metal Clusters Yoshifumi Noguchi, Soh Ishii and Kaoru Ohno Department of Physics, Graduate School of Engineering, Yokohama National University, Yokohama 240-8501, Japan In this paper, we discuss the instability of dianions of small alkali-metal clusters (Li2,Na2, and K2) starting from the state-of-the-art GW approximation and taking account of multiple scattering of two particles (hole–hole or electron–electron). The present approach is based on the first principles T-matrix theory in the many-body perturbation theory. (Received December 28, 2004; Accepted February 21, 2005; Published June 15, 2005) Keywords: first principles, ladder diagram, Bethe–Salpeter equation, T-matrix theory, all-electron mixed-basis approach, double electron affinity 1. Introduction starting point. The wave functions are expanded in a linear combination of both atomic orbitals (AO’s) and plane waves The Coulomb repulsive interaction between two electrons (PW’s). This all-electron mixed-basis approach4–7) can plays an important role for the double electron affinity, which describe all electronic states from a core state to a free is the energy gain in attaching two electrons to the neutral electron state. All AO’s are generated using Herman– system. Particularly, for a small size system, the interaction is Skillman’s atomic code8) within the non-overlapping atomic very strong because two electrons are confined to a small spheres and 2975 PW’s, corresponding to 3.1 Ry cuttoff region. energy, are used. We employ an fcc supercell with a cubic It is well known that the electron affinity as well as the first edge of 50 a.u. (¼ 2:7 nm). We introduce a spherical cutoff of ionization potential can be evaluated accurately using the the long tail of the Coulomb interaction in order to avoid the GW approximation (GWA) within a one-particle picture.1–6) effect of the periodic boundary condition. This approach is based on the many-body perturbation theory and the fact that the electron affinity is given by a pole of the 2.1 One-particle picture one-particle Green’s function. On the other hand, in contract The GWA represents the electron self-energy operator as a to the electron affinity, the double electron affinity cannot be product of the one-particle Green’s function G and the treated within a one-particle picture due to the existence of dynamically screened Coulomb interaction W. Here, using a the strong Coulomb repulsive interaction between the two short-hand notation (r1; t1Þ1, these functions are expressed electrons attached. In other words, the double electron as affinity is not simply given by a sum of the two one-particle Gð1; 10Þ¼iÂðt0 À t ÞhÉj y ð10Þ ð1ÞjÉi energies. Instead, a more reliable treatment based on a two- 1 1 H H particle picture is required. Using the electron–electron two- À iÂðt À t0 ÞhÉj ð1Þ y ð10ÞjÉi; ð2Þ 1 Z1 H H particle Green’s function, which describes the propagation of 0 0 0 two particles, this problem can be solved very elegantly since Wð1; 1 Þ¼Uð1; 1 Þþ d2d3Uð1; 2ÞPð2; 3ÞWð3; 1 Þ; ð3Þ the double electron affinity is given by a pole of the electron– GWA 0 0 þ 0 electron two-particle Green’s function. Æ ð1; 1 Þ¼iGð1; 1 ÞWð1 ; 1 Þ¼Æx þ Æc; ð4Þ 0 0 On the basis of the many-body perturbation theory beyond Æx ¼ iGð1; 1 ÞUð1; 1 Þ; ð5Þ the framework of the density functional theory, we consider Æ ¼ iGð1; 10Þ½Wð1þ; 10ÞUð1; 10Þ; ð6Þ the T-matrix which is related to the electron–electron two- c particle Green’s function G2 via where H is a field operator in the Heisenberg representation and P is the polarizability function within the random phase Tð1; 2j30; 40ÞGð3; 30ÞGð4; 40Þ¼Uð1; 2ÞG ð1; 2j3; 4Þ; ð1Þ 2 approximation.9) The self-energy operator ÆGWA is evaluated where G is the one-particle Green’s function, U is the bare by using the wave functions and the eigenvalues of the local Coulomb interaction, and T is the T-matrix. More details of density approximation (LDA) in the density functional G and T are described in the following section. We evaluate theory. Æx and Æc represent the Fock exchange term and the pole of the electron–electron two-particle Green’s the correlation term, respectively. Here the generalized function, which is equal to the pole of the T-matrix. plasmon-pole model1) is used to bypass the !-integration in In the present paper, we formulate the T-matrix theory Æc. from first principles and apply it to the calculation of the double electron affinity. In particular, we discuss the 2.2 Two-particle picture instability of the dianions of Li2,Na2, and K2. Since the Coulomb repulsive interaction between two particles plays an important role, the double electron affinity 2. Calculation cannot be treated within a one-particle picture. Therefore, we consider the interaction in a new formulation based on a two- In the present calculation, the GWA is selected as a particle picture. 1104 Y. Noguchi, S. Ishii and K. Ohno 1 3 11331′ be positive within the GWA demonstrates the electronic stability of these anions. The resulting electron affinity is in good agreement with the experiments.11,12) The same level = + GWA calculation has already been performed by Ishii et al. 224 4 2 2′ 4 in Refs. 4, 5. In the present calculation, however, the spherical cutoff of the Coulomb interaction is also introduced Fig. 1 Bethe–Salpater for T-matrix(square): The dotted and solid lines in the LDA calculation. Moreover the renormalization of the with arrow represent the bare Coulomb interaction U and the Green’s quasiparticle energy (eq. (4) in Ref. 4) is not considered in the function G, respectively. present GWA. In the case of LUMO level, which is rather close to the vaccuum level within the GWA, it is known that GWA LDA 0 The Bethe–Salpeter equation of the T-matrix theory is the off-diagonal elements of hnjÆ À xc jn i might constructed with the non-interacting two-particle Green’s become important. However, we do not consider such function, Kð1; 2j10; 20Þ¼iGð10; 1ÞGð20; 2Þ, as follows:10) elements in the present calculation. Tð1; 2j3; 4Þ¼Uð1; 2Þð1 À 3Þð2 À 4ÞþUð1; 2Þ The ‘‘G1’’ in the double electron affinity represents twice Z of the electron affinity obtained by the GWA and all of these ð7Þ Â d10d20Kð1; 2j10; 20ÞTð10; 20j3; 4Þ: values are positive. Since the interaction between two electrons is not considered at all in deriving these values of Diagrammatically, the T-matrix is a sum of the ladder ‘‘G1’’, there appears an unphysical result that the dianions of diagrams to the infinite order and is represented in Fig. 1. The Li2,Na2, and K2 are stable. On the other hand, when the T-matrix is sandwiched by the LDA wave functions and interaction between two electrons is taken into account by satisfies a following matrix equation, solving the Bethe-Salpeter equation for the T-matrix (see the X result shown in the column of ‘‘T-matrix’’), all these values À U KT ¼ U : ð8Þ are negative. The negative values in the column denoted by K ‘‘ T-matrix’’ mean that the dianions of Li2,Na2, and K2 are In eq. (8), all indices ( ) are confined to be empty electronically unstable. This conclusion within the present T- states only, although the real calculation is performed matrix theory is also physically acceptable. The effective without this constraint by introducing an additional factor. Coulomb interaction between the two electrons, estimated Then we solve the eigenvalue problem. The lowest eigen- from the difference between the values given in the columns value, which is a pole of the T-matrix, gives directly the denoted by ‘‘T-matrix’’ and ‘‘G1’’, is almost the same for double electron affinity. three alkali-metal clusters and amount to be about À1:8 eV In the calculation of Æx and U, the cutoff energies of G for Li2, À1:9 eV for Na2, and À1:8 eV for K2. vectors are 12.5 Ry for Li2, 28.1 Ry for Na2, and 18.0 Ry for The matrix elements of the on-site Coulomb interaction, K2, respectively. We take into account 600 levels and use the U , where is the LUMO level, is presented in the column same cutoff energy as the PW’s also for the GðG0Þ vectors of ‘‘Coulomb’’. We note that it has much larger value than the needed for the Æcð!Þ calculation. difference between ‘‘T-matrix’’ and ‘‘G1’’. This U is the matrix element which carries the largest contribution to the 3. Results double electron affinity. In the case of Li2, two attached electrons are more closely populated compared to the cases Table 1 represents the energy gain for attaching one or two of Na2 and K2 (due to the shortest bond length among three electrons to the neutral Li2,Na2, and K2, i.e., the electron alkali-metal clusters), and jU j takes the largest value. Its affinity or the double electron affinity. In the column of the absolute value tends to be smaller as the atomic number of electron affinity, the absolute values of the LUMO level the constituent atoms increases. This tendency is, however, enery obtained by the LDA and the GWA are shown together not seen either in the effective Coulomb interaction discussed with the corresponding experimental values.
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