- Email: [email protected]
Pseudorotating Na4 at finite temperatures: tight-binding molecular dynamics studies
25 August 2000
Chemical Physics Letters 326 Ž2000. 461–467 www.elsevier.nlrlocatercplett
Pseudorotating Na 4 at finite temperatures: tight-binding molecular dynamics studies F. Wang a
a,b
, F.S. Zhang
a,b,)
, Y. Abe
c
Center of Theoretical Nuclear Physics, National Laboratory of HeaÕy Ion Accelerator, Lanzhou 730000, China b Institute of Modern Physics, the Chinese Academy of Sciences, P.O. Box 31, Lanzhou 730000, China c Yukawa Institute for Theoretical Physics, Kyoto UniÕersity, Kyoto 606, Japan Received 6 March 2000; in final form 16 June 2000
Abstract We present the results of distance-dependent tight-binding molecular dynamics studies of Na 4 at finite temperatures, and especially discuss the pseudorotating pattern obtained at temperatures larger than 200 K. A pseudorotating barrier height of 0.023 eV is extracted. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The study of the dynamical aspect of microclusters is still a developing domain. Thorough works have so far been done only to microaggregates of argon w1,2x based on classical molecular dynamics ŽMD. calculations using Lennard-Jones potentials. These detailed investigations have brought about the knowledge of existence of many phases, ‘rigid’, ‘non-rigid’, melting, and so on. The extension of this type of studies to metallic and semiconductor clusters is being made. By using the ab initio molecular dynamics Žab initio MD. w3x Ro¨ thlisberger and Andreoni w4x reported pseudorotation of Si 10 at the temperature of 500 K. Pseudorotation implies a breaking of several bonds and a formation of several
)
Corresponding author: Institute of Modern Physics, Chinese Academy of Sciences, PO Box 31, Lanzhou 730000, China. Fax: q86-931-8272100; e-mail: [email protected]
new bonds leading to the same structure rotated. In this way, all the atoms change their positions without rotating collectively. For alkali metallic clusters which serves as a prototype model system to understand the dynamics of polyatomic metallic clusters, they w5x investigated the structure and the electronic properties of Na n Ž n s 2–20. at low and high temperatures based on the simulations of the ab initio MD. Bulgac and Kusnezov w6x studied the properties of Na 7 – 9 microclusters at a wide range of temperatures in the framework of their improved molecular dynamics approach. They found that sodium microclusters undergo two ‘phase transitions’, one around 100 K from a crystal to a glassy or a molten state, and the other around 800 K, to a fluid state of the cluster. Interestingly, Kawai et al. w7x, and Gibson and Carter w8x reported the pseudorotating pattern of Li 5 obtained by the simulations of ab initio MD, respectively. Up to now, there has been no report on the pseudorotating pattern of sodium clusters, although it is intriguing whether the pseudorotating pattern exists also in sodium clusters.
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 8 0 6 - X
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
462
In the present Letter, we first provide some evidence for the pseudorotation of Na 4 at temperature T ) 200 K with the barrier height for the pseudorotation being 0.023 eV by using the distance-dependent tight-binding molecular dynamics ŽDDTB-MD. simulations.
m i r¨ i s y=r i Ep ,
2. Theoretical framework The model we employ is the DDTB-MD, which has been used to study the structure properties of sodium clusters in the range of n s 2–40, producing results in good agreement with those obtained by the ab initio kind of calculations w4,9,10x. Briefly, the potential energy is computed as the sum of the one-electron eigenvalues of the Hamiltonian which is recapitulated below w11,12x: hˆ s Ý h i j aq i aj , i, j Ž2. h i i s hŽ0. ii q hii
s
Ý k/i
½
rss Ž R i k . y
ts2s Ž R i k .
e3 p y e3s
5
,
Ž2. h i j s hŽ0. i j q hi j
s tss Ž R i j . y
Ý
R ik P R jk N R ik NN R jk N
½ 5
ts s Ž R i k . ts s Ž R jk .
e3 p y e3s ,
Ž 1.
where aq and a j denote creation and annihilation i operators corresponding to s-orbitals on sites i and j, respectively. The matrix elements h i j which are distance-dependent include perturbatively the effect of p-orbitals. The three functions rss Ž R ., tssŽ R ., and ts s Ž R . represent the ion–ion repulsion, the s–s, and the s–p s transfer integrals at interatomic distance R, respectively. This parametrization is accomplished 3 q by fitting precisely 1 Sq g and S u potential curves of Na 2 Ž R e s 5.8 Bohr and De s 0.74 eV in agreement with experiment. and the dissociation energy of Na 4 Ž De s 0.41 eV into Na 2 q Na 2 . as obtained by the ab initio CI calculations. The stable structure of Na n is obtained by Monte Carlo simulated-annealing
Ž 2.
where Ep is the Born–Oppenheimer potential energy, m i and r i are the mass and coordinate of the ith atom. The Hellmann–Feynman theorem is used to calculate forces acting on each ion, namely derivatives of the potential energy as =r i Ep s
² w i < =r hˆ < w i : , i
Ý
Ž 3.
igocc
where w i is the molecular orbital wavefunction. The cluster Na 4 is prepared with a set of temperatures of 0.02, 105, 212 and 224 K. The MD equations are solved with a time step of 1 fs, which was checked to ensure proper total momentum and energy conservation. The temperature is determined as usual in the microcanonical ensemble by the time average of the kinetic energy per particle ²Ek: Ts
k/i , j
=
technique. In the present investigation, we just confine ourselves to Na 4 cluster with a symmetry of D 2 h with the total energy y1.883 eV. The dynamics of the ions constituting the clusters is described by a standard classical molecular dynamics based on the potential energy. Ions hence follow Newton’s equations of motion w13,14x:
2n
² Ek :
3n y 6 k B
.
Ž 4.
where ²Ek: is averaging over all particles and all trajectories, k B is the Boltzmann constant. After a thermalization time of 2000 fs, each trajectory is propagated during 35 000 fs in the present calculations.
3. Results and discussions In order to characterize the pseudorotating pattern of Na 4 , we select the potential energy of system E p as an energy probe, while the acute angle between major axis of rhombus Na 4 and x-axis u as a reaction coordinate to diagnose the geometric properties in pseudorotating Na 4 . Of course, a direct pursuit of them during the time evolution provides powerful means of analysis. Let us now turn to the results obtained at a set of temperatures of T s 0.02, 105, 212 and 224 K.
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
First, we check the time evolutions of the potential energy Ep during 1–35 000 fs, those of the acute angle, and those of the piling-up of atomic positions, respectively for T s 0.02 K. We immediately see that the potential energy Ep and the angle u for T s 0.02 K are constant of y1.883 eV and 08, respectively, for any time. While for the case of T s 105 K, as is shown in Fig. 1, there are thermal fluctuations in the time evolutions of Ep and u , and the piling-up of atom positions during 1–35 000 fs also shows the corresponding fluctuations around the equilibrium positions. Therefore, at lower temperatures our results are consistent with the knowledge that the structure of Na 4 does not vary and each atom consisting the cluster only oscillates around its equilibrium position. After comparing the case of the temperature T s 105 K as is shown in Fig. 1 to that of the temperature T s 0.02 K, one concludes that although the thermal fluctuations in the potential energy Ep and the angle u increases as temperature
463
increases, the cluster still stays in so-called ‘rigid’ state, whose eigenmode w14x frequencies provide a sensitive probe for distinguishing cluster isomer. Fig. 2 shows the case of T s 212 K. Fig. 2a shows the time evolution of the potential energy, which exhibits obvious fluctuations between y1.870 and y1.766 eV. Inspecting Fig. 2a more in detail, we notice an existence of several characteristic narrow regions where their amplitudes of the fluctuation are much smaller than those in the rest, i.e., than those in the most region of the time evolutions. They are around 3750, 24 000, 26 000, 27 300 and 29 000 fs, respectively. Their other characteristic is high average potential energies of about y1.792 eV, while the average of the rest is y1.814 eV. The physical meaning of this behavior is clearly understood by looking at the corresponding time evolution of the angle u , which is displayed in Fig. 2b. There, we clearly see that the rest region is divided into two distinct regions with respect to the values of the
Fig. 1. The potential energy Ep Ža., the acute angle between major axis of rhombus Na 4 and x-axis u Žb. versus time, and the piling-up of atom positions during 1–35 000 fs Žc. for temperature T s 105 K.
464
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
Fig. 2. Same as Fig. 1 but for temperature T s 212 K.
angle u , i.e., u ( 08 and 458. The former is from 0 to 3750 fs, 24 000 to 26 000 fs, and 27 300 to 29 000 fs, while the latter from 3750 to 24 000 fs, 26 000 to 27 300 fs, and 29 000 to 35 000 fs. This means that the several narrow characteristic regions correspond to a kind of ‘transition’ between the above two regions of the rest. Since the two regions in the rest have the same values of the average potential energy and its fluctuation, the ‘transition’ would be between states with the same intrinsic structure. This is confirmed by the piling-up of atoms positions Žsee Fig. 2c., which indicates intuitively an existence of rotated configuration, which will be shown below to correspond not to a real rotation of the system but to a so-called pseudorotation. This means that the pseudorotating pattern occurs in Na4 . Fig. 3 shows the results for higher temperature T s 224 K. In particular, from Fig. 3d we can see that the pseudorotating pattern has perfect symmetry, which implies the pseudorotating pathway is a closed
loop on the Born–Oppenheimer potential energy surface. Fig. 3b shows the potential averaging values over different pseudorotating pathways ŽFig. 3a. by distinguishing whether the motion of the system is inside the well or not. Now, as shown in Fig. 3c, we can divide the time evolutions of the angle u into five parts and sample ‘characteristic observable windows’ for each part as follows: Ž1. 3000–3500 fs; Ž2. 10 000–10 500 fs; Ž3. 20 000–20 500 fs; Ž4. 26 000– 26 500 fs; Ž5. 33 000–33 500 fs. Fig. 4 shows the piling-up of marked atom positions during 1–35 000 fs during each ‘characteristic observable windows’. One immediately finds that the two atoms on major axis have became ones of same side among pseudorotating pathway. The fact that there is a pseudorotating pattern only at higher T ŽT ) 200 K. in Na 4 implies an existence of energy barrier, which separates the pseudorotating pattern from the vibrational states. Starting from the burgeon of the pseudorotating pattern at T f 212 K,
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
465
Fig. 3. The potential energy Ep Ža., the mean value of the potential energy over saddle and well pathway on the Born–Oppenheimer potential energy surface Ep Žb., respectively, the cross acute angle u Žc. versus time, and the piling-up of atom positions during 1–35 000 fs Žd. for temperature T s 224 K.
one observes that the piling-up of atom positions during 1–35 000 fs turns out to be perfect symmetry for higher temperature T f 224 K. From the piling-up of atom positions of the characteristic time windows we conclude that an atom can stand on any geometry site of equilibrium structures, which results from the pseudorotating pattern of Na 4 . Ergodicity of atoms on geometry sites of equilibrium structures and finite
number of equilibrium structures show the pseudorotating pathway is a closed loop on the Born–Oppenheimer potential energy surface. As displayed in Fig. 5, the pseudorotating barrier hight of V0 s 0.023 eV is extracted by averaging the potential energy over all well and the saddle pseudorotationg pathway. The standard deviations of the barrier hight Ž dV0 s 0.019 eV. and the angle
466
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
Fig. 4. The piling-up of atom positions during each ‘characteristic time window’ Ž1. 3000–3500 fs, Ž2. 10 000–10 500 fs, Ž3. 20 000–20 500 fs, Ž4. 26 000–26 500 fs, and Ž5. 33 000–33 500 fs with distinguishing different atom.
Ž du s 1.7 0 . are also calculated. From the statistical point of view, an existence of a degree of freedom gives a new mode of collective motion to cluster, and must has contributions to thermodynamical partition function, then influences the physical variables such as heat capacity, free energy, entropy and so on. Formally, we can treat pseudorotation as follows. Ž1. For k B T 4 V0 Ž V0 s 0.023 eV, the pseudorotating barrier height., the pseudorotation is free, therefore the pseudorotating partition function can be obtained by using classical energy partition-function in phase space. Ž2. For k B T f V0 , so-called ‘hobbling pseudorotation’, there is no simple pseudorotating partition function, and numerical solution of
Schrodinger equation is required. Ž3. For k B T < V0 , ¨ the pseudorotating degree of freedom degenerates into the vibrational modes. Compared to the pseudorotation of the Li 5 , for which Kawai et al. w7x pointed out there is pseudorotation around the D5h symmetry point, we find that the rhombus Na 4 seems to pseudorotate around the D 8 h symmetric point. However, a dynamics pseudorotation path does not sample all of the extremum states around the D 8 h symmetric point as illustrated in Fig. 4. It is worth to emphasize that Kawai et al. assumed the system is pseudorotating by quantummechanical tunneling and zero-point energy plays a crucial role, since Lithium is the third lightest ele-
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
467
4. Conclusions
Fig. 5. The potential energy V0 s 0.023 eV versus the cross acute u by averaging over all pseudorotating pathways.
ment, while our DDTB-MD is a classical simulation for the sodium ions and thus includes no tunneling effects. For Na 4 , our calculations suggest that it is not necessary to invoke tunneling to produce pseudorotation, compatible with the results by Gibson et al. w8x. One should mention that the present work just pay attention to a specific temperature range 0–224 K where the pseudorotating is observed. Actually, the calculations are made for a wider temperature region 0–400 K in our studies by using the tight-binding molecular dynamics model. It is interest that one finds the Na 4 behaves like a crystal with the atoms oscillating with very small amplitudes around their equilibrium positions at lower temperatures ŽT ( 50 K.. At higher temperatures ŽT f 300 K., one finds the melted sodium atoms of Na 4 which are in agreement with the findings of Rothlisberger and An¨ dreoni w4x, Bulgac and Kusnezov w6x. The thermodynamical properties of sodium clusters for a wider temperature range will be published elsewhere.
In summary, for Na 4 with symmetry of D 2 h , one finds that, each atom only vibrates around its equilibrium position, and the vibrational amplitude increases as heating the system in lower temperatures ŽT - 200 K.. There is a new mode of collective motion, namely the pseudorotating pattern, with a height of barrier V0 f 0.023 eV for the temperatures 212 and 224 K. We suggest that the pseudorotation is a general phenomenon of finite system at finite temperatures, depending on governing specific atom–atom interaction with a few degrees of freedom of the system. References w1x R.S. Berry, Chem. Rev. 93 Ž1993. 2379. w2x T.L. Beck, J. Jellinek, R.S. Berry, J. Chem. Phys. 87 Ž1987. 545. w3x R. Car, M. Parrinello, Phys. Rev. Lett. 55 Ž1985. 2471. w4x U. Rothlishberger, W. Andreoni, Z. Phys. D 20 Ž1991. 243. ¨ w5x U. Rothlisberger, W. Andreoni, J. Chem. Phys. 94 Ž1991. ¨ 8129. w6x A. Bulgac, D. Kusnezov, Phys. Rev. Lett. 68 Ž1992. 1335. w7x R. Kawai, J.F. Tombrello, J.H. Weare, Phys. Rev. A 49 Ž1994. 4236. w8x D.A. Gibson, E.A. Carter, Chem. Phys. Lett. 271 Ž1997. 266. w9x V. Bonacic-Koutecky, P. Fantacci, J. Koutecky, Phys. Rev. B 37 Ž1988. 4369. w10x V. Bonacic-Koutecky, P. Fantucci, J. Koutecky, Chem. Rev. 91 Ž1991. 1035. w11x R. Poteau, F. Spiegelmann, Phys. Rev. B 48 Ž1992. 1878. w12x R. Poteau, F. Spiegelmann, P. Labaste, Z. Phys. D 30 Ž1994. 57. w13x F.S. Zhang, F. Spiegelmann, E. Suraud, V. Fraysse, ´ R. Poteau, R. Glowinski, F. Chatelin, Phys. Lett. A 193 Ž1994. 75. w14x F.S. Zhang, E. Suraud, F. Calvo, F. Spiegelmann, Chem. Phys. Lett. 300 Ž1999. 595.
Chemical Physics Letters 326 Ž2000. 461–467 www.elsevier.nlrlocatercplett
Pseudorotating Na 4 at finite temperatures: tight-binding molecular dynamics studies F. Wang a
a,b
, F.S. Zhang
a,b,)
, Y. Abe
c
Center of Theoretical Nuclear Physics, National Laboratory of HeaÕy Ion Accelerator, Lanzhou 730000, China b Institute of Modern Physics, the Chinese Academy of Sciences, P.O. Box 31, Lanzhou 730000, China c Yukawa Institute for Theoretical Physics, Kyoto UniÕersity, Kyoto 606, Japan Received 6 March 2000; in final form 16 June 2000
Abstract We present the results of distance-dependent tight-binding molecular dynamics studies of Na 4 at finite temperatures, and especially discuss the pseudorotating pattern obtained at temperatures larger than 200 K. A pseudorotating barrier height of 0.023 eV is extracted. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The study of the dynamical aspect of microclusters is still a developing domain. Thorough works have so far been done only to microaggregates of argon w1,2x based on classical molecular dynamics ŽMD. calculations using Lennard-Jones potentials. These detailed investigations have brought about the knowledge of existence of many phases, ‘rigid’, ‘non-rigid’, melting, and so on. The extension of this type of studies to metallic and semiconductor clusters is being made. By using the ab initio molecular dynamics Žab initio MD. w3x Ro¨ thlisberger and Andreoni w4x reported pseudorotation of Si 10 at the temperature of 500 K. Pseudorotation implies a breaking of several bonds and a formation of several
)
Corresponding author: Institute of Modern Physics, Chinese Academy of Sciences, PO Box 31, Lanzhou 730000, China. Fax: q86-931-8272100; e-mail: [email protected]
new bonds leading to the same structure rotated. In this way, all the atoms change their positions without rotating collectively. For alkali metallic clusters which serves as a prototype model system to understand the dynamics of polyatomic metallic clusters, they w5x investigated the structure and the electronic properties of Na n Ž n s 2–20. at low and high temperatures based on the simulations of the ab initio MD. Bulgac and Kusnezov w6x studied the properties of Na 7 – 9 microclusters at a wide range of temperatures in the framework of their improved molecular dynamics approach. They found that sodium microclusters undergo two ‘phase transitions’, one around 100 K from a crystal to a glassy or a molten state, and the other around 800 K, to a fluid state of the cluster. Interestingly, Kawai et al. w7x, and Gibson and Carter w8x reported the pseudorotating pattern of Li 5 obtained by the simulations of ab initio MD, respectively. Up to now, there has been no report on the pseudorotating pattern of sodium clusters, although it is intriguing whether the pseudorotating pattern exists also in sodium clusters.
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 8 0 6 - X
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
462
In the present Letter, we first provide some evidence for the pseudorotation of Na 4 at temperature T ) 200 K with the barrier height for the pseudorotation being 0.023 eV by using the distance-dependent tight-binding molecular dynamics ŽDDTB-MD. simulations.
m i r¨ i s y=r i Ep ,
2. Theoretical framework The model we employ is the DDTB-MD, which has been used to study the structure properties of sodium clusters in the range of n s 2–40, producing results in good agreement with those obtained by the ab initio kind of calculations w4,9,10x. Briefly, the potential energy is computed as the sum of the one-electron eigenvalues of the Hamiltonian which is recapitulated below w11,12x: hˆ s Ý h i j aq i aj , i, j Ž2. h i i s hŽ0. ii q hii
s
Ý k/i
½
rss Ž R i k . y
ts2s Ž R i k .
e3 p y e3s
5
,
Ž2. h i j s hŽ0. i j q hi j
s tss Ž R i j . y
Ý
R ik P R jk N R ik NN R jk N
½ 5
ts s Ž R i k . ts s Ž R jk .
e3 p y e3s ,
Ž 1.
where aq and a j denote creation and annihilation i operators corresponding to s-orbitals on sites i and j, respectively. The matrix elements h i j which are distance-dependent include perturbatively the effect of p-orbitals. The three functions rss Ž R ., tssŽ R ., and ts s Ž R . represent the ion–ion repulsion, the s–s, and the s–p s transfer integrals at interatomic distance R, respectively. This parametrization is accomplished 3 q by fitting precisely 1 Sq g and S u potential curves of Na 2 Ž R e s 5.8 Bohr and De s 0.74 eV in agreement with experiment. and the dissociation energy of Na 4 Ž De s 0.41 eV into Na 2 q Na 2 . as obtained by the ab initio CI calculations. The stable structure of Na n is obtained by Monte Carlo simulated-annealing
Ž 2.
where Ep is the Born–Oppenheimer potential energy, m i and r i are the mass and coordinate of the ith atom. The Hellmann–Feynman theorem is used to calculate forces acting on each ion, namely derivatives of the potential energy as =r i Ep s
² w i < =r hˆ < w i : , i
Ý
Ž 3.
igocc
where w i is the molecular orbital wavefunction. The cluster Na 4 is prepared with a set of temperatures of 0.02, 105, 212 and 224 K. The MD equations are solved with a time step of 1 fs, which was checked to ensure proper total momentum and energy conservation. The temperature is determined as usual in the microcanonical ensemble by the time average of the kinetic energy per particle ²Ek: Ts
k/i , j
=
technique. In the present investigation, we just confine ourselves to Na 4 cluster with a symmetry of D 2 h with the total energy y1.883 eV. The dynamics of the ions constituting the clusters is described by a standard classical molecular dynamics based on the potential energy. Ions hence follow Newton’s equations of motion w13,14x:
2n
² Ek :
3n y 6 k B
.
Ž 4.
where ²Ek: is averaging over all particles and all trajectories, k B is the Boltzmann constant. After a thermalization time of 2000 fs, each trajectory is propagated during 35 000 fs in the present calculations.
3. Results and discussions In order to characterize the pseudorotating pattern of Na 4 , we select the potential energy of system E p as an energy probe, while the acute angle between major axis of rhombus Na 4 and x-axis u as a reaction coordinate to diagnose the geometric properties in pseudorotating Na 4 . Of course, a direct pursuit of them during the time evolution provides powerful means of analysis. Let us now turn to the results obtained at a set of temperatures of T s 0.02, 105, 212 and 224 K.
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
First, we check the time evolutions of the potential energy Ep during 1–35 000 fs, those of the acute angle, and those of the piling-up of atomic positions, respectively for T s 0.02 K. We immediately see that the potential energy Ep and the angle u for T s 0.02 K are constant of y1.883 eV and 08, respectively, for any time. While for the case of T s 105 K, as is shown in Fig. 1, there are thermal fluctuations in the time evolutions of Ep and u , and the piling-up of atom positions during 1–35 000 fs also shows the corresponding fluctuations around the equilibrium positions. Therefore, at lower temperatures our results are consistent with the knowledge that the structure of Na 4 does not vary and each atom consisting the cluster only oscillates around its equilibrium position. After comparing the case of the temperature T s 105 K as is shown in Fig. 1 to that of the temperature T s 0.02 K, one concludes that although the thermal fluctuations in the potential energy Ep and the angle u increases as temperature
463
increases, the cluster still stays in so-called ‘rigid’ state, whose eigenmode w14x frequencies provide a sensitive probe for distinguishing cluster isomer. Fig. 2 shows the case of T s 212 K. Fig. 2a shows the time evolution of the potential energy, which exhibits obvious fluctuations between y1.870 and y1.766 eV. Inspecting Fig. 2a more in detail, we notice an existence of several characteristic narrow regions where their amplitudes of the fluctuation are much smaller than those in the rest, i.e., than those in the most region of the time evolutions. They are around 3750, 24 000, 26 000, 27 300 and 29 000 fs, respectively. Their other characteristic is high average potential energies of about y1.792 eV, while the average of the rest is y1.814 eV. The physical meaning of this behavior is clearly understood by looking at the corresponding time evolution of the angle u , which is displayed in Fig. 2b. There, we clearly see that the rest region is divided into two distinct regions with respect to the values of the
Fig. 1. The potential energy Ep Ža., the acute angle between major axis of rhombus Na 4 and x-axis u Žb. versus time, and the piling-up of atom positions during 1–35 000 fs Žc. for temperature T s 105 K.
464
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
Fig. 2. Same as Fig. 1 but for temperature T s 212 K.
angle u , i.e., u ( 08 and 458. The former is from 0 to 3750 fs, 24 000 to 26 000 fs, and 27 300 to 29 000 fs, while the latter from 3750 to 24 000 fs, 26 000 to 27 300 fs, and 29 000 to 35 000 fs. This means that the several narrow characteristic regions correspond to a kind of ‘transition’ between the above two regions of the rest. Since the two regions in the rest have the same values of the average potential energy and its fluctuation, the ‘transition’ would be between states with the same intrinsic structure. This is confirmed by the piling-up of atoms positions Žsee Fig. 2c., which indicates intuitively an existence of rotated configuration, which will be shown below to correspond not to a real rotation of the system but to a so-called pseudorotation. This means that the pseudorotating pattern occurs in Na4 . Fig. 3 shows the results for higher temperature T s 224 K. In particular, from Fig. 3d we can see that the pseudorotating pattern has perfect symmetry, which implies the pseudorotating pathway is a closed
loop on the Born–Oppenheimer potential energy surface. Fig. 3b shows the potential averaging values over different pseudorotating pathways ŽFig. 3a. by distinguishing whether the motion of the system is inside the well or not. Now, as shown in Fig. 3c, we can divide the time evolutions of the angle u into five parts and sample ‘characteristic observable windows’ for each part as follows: Ž1. 3000–3500 fs; Ž2. 10 000–10 500 fs; Ž3. 20 000–20 500 fs; Ž4. 26 000– 26 500 fs; Ž5. 33 000–33 500 fs. Fig. 4 shows the piling-up of marked atom positions during 1–35 000 fs during each ‘characteristic observable windows’. One immediately finds that the two atoms on major axis have became ones of same side among pseudorotating pathway. The fact that there is a pseudorotating pattern only at higher T ŽT ) 200 K. in Na 4 implies an existence of energy barrier, which separates the pseudorotating pattern from the vibrational states. Starting from the burgeon of the pseudorotating pattern at T f 212 K,
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
465
Fig. 3. The potential energy Ep Ža., the mean value of the potential energy over saddle and well pathway on the Born–Oppenheimer potential energy surface Ep Žb., respectively, the cross acute angle u Žc. versus time, and the piling-up of atom positions during 1–35 000 fs Žd. for temperature T s 224 K.
one observes that the piling-up of atom positions during 1–35 000 fs turns out to be perfect symmetry for higher temperature T f 224 K. From the piling-up of atom positions of the characteristic time windows we conclude that an atom can stand on any geometry site of equilibrium structures, which results from the pseudorotating pattern of Na 4 . Ergodicity of atoms on geometry sites of equilibrium structures and finite
number of equilibrium structures show the pseudorotating pathway is a closed loop on the Born–Oppenheimer potential energy surface. As displayed in Fig. 5, the pseudorotating barrier hight of V0 s 0.023 eV is extracted by averaging the potential energy over all well and the saddle pseudorotationg pathway. The standard deviations of the barrier hight Ž dV0 s 0.019 eV. and the angle
466
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
Fig. 4. The piling-up of atom positions during each ‘characteristic time window’ Ž1. 3000–3500 fs, Ž2. 10 000–10 500 fs, Ž3. 20 000–20 500 fs, Ž4. 26 000–26 500 fs, and Ž5. 33 000–33 500 fs with distinguishing different atom.
Ž du s 1.7 0 . are also calculated. From the statistical point of view, an existence of a degree of freedom gives a new mode of collective motion to cluster, and must has contributions to thermodynamical partition function, then influences the physical variables such as heat capacity, free energy, entropy and so on. Formally, we can treat pseudorotation as follows. Ž1. For k B T 4 V0 Ž V0 s 0.023 eV, the pseudorotating barrier height., the pseudorotation is free, therefore the pseudorotating partition function can be obtained by using classical energy partition-function in phase space. Ž2. For k B T f V0 , so-called ‘hobbling pseudorotation’, there is no simple pseudorotating partition function, and numerical solution of
Schrodinger equation is required. Ž3. For k B T < V0 , ¨ the pseudorotating degree of freedom degenerates into the vibrational modes. Compared to the pseudorotation of the Li 5 , for which Kawai et al. w7x pointed out there is pseudorotation around the D5h symmetry point, we find that the rhombus Na 4 seems to pseudorotate around the D 8 h symmetric point. However, a dynamics pseudorotation path does not sample all of the extremum states around the D 8 h symmetric point as illustrated in Fig. 4. It is worth to emphasize that Kawai et al. assumed the system is pseudorotating by quantummechanical tunneling and zero-point energy plays a crucial role, since Lithium is the third lightest ele-
F. Wang et al.r Chemical Physics Letters 326 (2000) 461–467
467
4. Conclusions
Fig. 5. The potential energy V0 s 0.023 eV versus the cross acute u by averaging over all pseudorotating pathways.
ment, while our DDTB-MD is a classical simulation for the sodium ions and thus includes no tunneling effects. For Na 4 , our calculations suggest that it is not necessary to invoke tunneling to produce pseudorotation, compatible with the results by Gibson et al. w8x. One should mention that the present work just pay attention to a specific temperature range 0–224 K where the pseudorotating is observed. Actually, the calculations are made for a wider temperature region 0–400 K in our studies by using the tight-binding molecular dynamics model. It is interest that one finds the Na 4 behaves like a crystal with the atoms oscillating with very small amplitudes around their equilibrium positions at lower temperatures ŽT ( 50 K.. At higher temperatures ŽT f 300 K., one finds the melted sodium atoms of Na 4 which are in agreement with the findings of Rothlisberger and An¨ dreoni w4x, Bulgac and Kusnezov w6x. The thermodynamical properties of sodium clusters for a wider temperature range will be published elsewhere.
In summary, for Na 4 with symmetry of D 2 h , one finds that, each atom only vibrates around its equilibrium position, and the vibrational amplitude increases as heating the system in lower temperatures ŽT - 200 K.. There is a new mode of collective motion, namely the pseudorotating pattern, with a height of barrier V0 f 0.023 eV for the temperatures 212 and 224 K. We suggest that the pseudorotation is a general phenomenon of finite system at finite temperatures, depending on governing specific atom–atom interaction with a few degrees of freedom of the system. References w1x R.S. Berry, Chem. Rev. 93 Ž1993. 2379. w2x T.L. Beck, J. Jellinek, R.S. Berry, J. Chem. Phys. 87 Ž1987. 545. w3x R. Car, M. Parrinello, Phys. Rev. Lett. 55 Ž1985. 2471. w4x U. Rothlishberger, W. Andreoni, Z. Phys. D 20 Ž1991. 243. ¨ w5x U. Rothlisberger, W. Andreoni, J. Chem. Phys. 94 Ž1991. ¨ 8129. w6x A. Bulgac, D. Kusnezov, Phys. Rev. Lett. 68 Ž1992. 1335. w7x R. Kawai, J.F. Tombrello, J.H. Weare, Phys. Rev. A 49 Ž1994. 4236. w8x D.A. Gibson, E.A. Carter, Chem. Phys. Lett. 271 Ž1997. 266. w9x V. Bonacic-Koutecky, P. Fantacci, J. Koutecky, Phys. Rev. B 37 Ž1988. 4369. w10x V. Bonacic-Koutecky, P. Fantucci, J. Koutecky, Chem. Rev. 91 Ž1991. 1035. w11x R. Poteau, F. Spiegelmann, Phys. Rev. B 48 Ž1992. 1878. w12x R. Poteau, F. Spiegelmann, P. Labaste, Z. Phys. D 30 Ž1994. 57. w13x F.S. Zhang, F. Spiegelmann, E. Suraud, V. Fraysse, ´ R. Poteau, R. Glowinski, F. Chatelin, Phys. Lett. A 193 Ž1994. 75. w14x F.S. Zhang, E. Suraud, F. Calvo, F. Spiegelmann, Chem. Phys. Lett. 300 Ž1999. 595.