J. Phys. Chem. 1996, 100, 6057-6060
6057
Binding Energy of PsCH3 System by Quantum Monte Carlo and ab Initio Molecular Orbital Calculations Tatsuo Saito, Masanori Tachikawa,* Chikaomi Ohe, and Kaoru Iguchi Department of Chemistry, School of Science and Engineering, Waseda UniVersity, Shinjuku-ku, Tokyo, Japan
Kazunari Suzuki Department of Commerce, Faculty of Commerce, Takachiho UniVersity, Suginami-ku, Tokyo, Japan ReceiVed: August 29, 1995; In Final Form: February 7, 1996X
The simplest positron-organic compound, the PsCH3 system, has been calculated by quantum Monte Carlo (QMC) and ab initio molecular orbital (MO) methods. In the QMC method the Jastrow factor is added to the trial function to satisfy the cusp condition, and parameters of trial function are optimized by the variational Monte Carlo method. In the ab initio MO method the large basis set based on the conventional 6-31G is used and the Møller-Plesset second-order energy is included. The positron affinity is calculated to be 6.92 eV by QMC and 5.15 eV by ab initio MO. Using the value of 6.92 eV, we have found the binding energy between CH3 and Ps to be 0.19 eV, and using the value of 5.15 eV, we find the binding energy to be -1.58 eV. However, the possibility of getting a larger positron affinity by the ab initio MO method can be expected with a larger basis set, so the stable existence of the PsCH3 system may be predicted by this QMC calculation.
1. Introduction
and
Recently, a system containing antiparticles has attracted the interest of many experimental and theoretical researchers1-3 of physical chemistry. Ore4 performed a variational calculation on PsH using Hylleraas-type functions, and Clary5 computed PsHe by the Hylleraas configuration interaction method. Cade et al.6 performed calculations on the PsX systems, where Ps is a positronium (a bound state of a positron and an electron) and X a halogen atom, using the numerical restricted Hartree-Fock method. Kurtz et al.7,8 performed LCAO Hartree-Fock calculations on PsF and PsCN ([CN-; e+]). Kao et al.9 performed calculations on systems of di- and triatomic negative ions attached by a positron. Moreover, some molecular orbital computations were made by considering the correlations between electrons or between a positron and electrons.10,11 On the other hand, Schrader et al.12,13 performed quantum Monte Carlo (QMC) calculations on PsX systems. In this study we have chosen the PsCH3 system. The first reason is that this system is the simplest organic compound with a methyl radical, which is the most conventional functional group. If the experiment on positron compounds becomes easy in the future, this system will be the most popular target of researchers. The second reason is that the stability of PsCH3 is promising because it is only a positron substitution of a proton in CH4, and we may be able to avoid the futile effort to build a tower on loose sand. The third reason is that the probable reaction process to produce PsCH3 was already proposed by Schrader et al.14 in the gas phase experiment. The subsidiary reaction process effective in liquid phase, accompanying the generation of a proton, is
B(PsCH3) ) PA(CH3-) + EA(CH3) - E(Ps)
CH4 + e+ f Ps + CH3 + H+ + 11.31 eV +
+
CH4 + e f PsCH3 + H + 11.31 eV - B(PsCH3)
(1) (2)
In this reaction there exists a path (1) generating Ps and CH3 and a path (2) generating PsCH3, and the dominance between them is fixed by the value of the binding energy B(PsCH3), X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6057$12.00/0
) -E(PsCH3) + E(CH3-) + EA(CH3) - E(Ps) (3) where PA(CH3-) is the positron affinity of CH3- and is given by the difference between the energies of PsCH3 and CH3-. Also, EA(CH3) is the electron affinity of CH3 ()0.08 eV),15 and E(Ps) is the dissociation energy of Ps ()1/4 hartree ) 0.681 eV).1 In this study we compute the energy difference between PsCH3 and CH3- by both the QMC method and the ab initio MO method with consideration of the second-order perturbation by Møller-Plesset, and examine the possibility of a stable existence of PsCH3. We also compute the binding energy B(PsCH3) and examine if the creation of PsCH3 is dominant in the competing processes in path (1) and path (2). 2. Theory The Hamiltonian of a system containing M nuclei, N electrons, and a positron is assumed to be N
N
H ) ∑he(a) + ∑ a)1
1
a>brab
N
+ hp(p) - ∑
1
a)1 rpa
(4)
where M Z R 1 he(a) ) - ∇2a - ∑ 2 R raR M Z R 1 hp(p) ) - ∇2p + ∑ 2 R rpR
rab ) |ra - rb| rpa ) |rp - ra| and ZR is the charge of the Rth nucleus. The raR is the distance © 1996 American Chemical Society
6058 J. Phys. Chem., Vol. 100, No. 15, 1996
Saito et al.
of the ath electron from the Rth nucleus, and rpR is that of the positron from the same nucleus. The rab is the distance between the ath and bth electrons, and rpa is that between the positron and ath electron. The wave function for eq 4 is assumed to be
Ψ ) ΦΦp
m
e φa ) ∑Cai χi
(5)
Φ is the antisymmetrized Slater determinant for an N-electron closed shell system, Φp is a positronic spin orbital. We denote the set of spatial orbitals of electrons and the positron as {φa} (a ) 1, ..., N/2) and φp, respectively. 2.1. Quantum Monte Carlo Method. The Monte Carlo calculations in physical problems can be divided into two methods: the variational Monte Carlo method (VMC) in which the energy of a trial function can be calculated rather easily and the quantum Monte Carlo method (QMC), which requires much more computing time and an extrapolation but is able to calculate physical quantities more exactly than the trial function does by itself. QMC is a method for locating the stationary state solution of the imaginary time-dependent Schro¨dinger equation such as16
-
operator. The molecular orbitals are expanded by a set of m basis functions {χi}:
∂Ψ(R, t) ) (H - ET)Ψ(R, t) ∂t
m
φp ) ∑Cipχi
(13)
i)1
and the Roothaan equations are obtained for the electrons and the positron. The energy of this system is finally given as m
EHF ) ∑Rjie (hije + Fije ) + i,j
1
m
∑Rjip(hijp + Fijp) 2 i,j
(14)
where Rije and Rijp are density matrices. To include the correlation of electrons and the positron, we employ the Møller-Plesset scheme of perturbation ν. N/2
H0 ) 2∑f e(a) + f p(p)
(15)
ν ) H - H0
(16)
a)1
) [-D∇2 + V(R) - ET]Ψ(R, t)
(6)
Then the second-order energy E(2) is given by
where D ) 1/2 and N M
V ) ∑∑ a)1 R
-ZR raR
M
ZR
N
1
N
1
+∑ +∑ -∑ R rpR a>brab a)1rpa
(7)
-∂f(R, t) ) -D∇2f + [EL(R) - ET]f + D∇·[fFQ] ∂t
E
(2)
(0) 2 |〈Ψ(0) 0 |ν|Ψn 〉|
) ∑′ n
Here, ET is the trial energy, and we assume a value that is as appropriate as possible, and R represents whole coordinates of electrons and the positron. By employment of a trial function ΨT (importance sampling), eq 6 is deformed to a diffusion equation for the distribution function f(R, t).
(8)
where
(17) (0) E(0) 0 - En
where the prime on ∑ means the exclusion of n ) 0. By addition of these second-order energies to EHF in eq 14, the energy of the system E(2) is finally obtained. 3. Numerical Computations 3.1. Computation by QMC. At first we assume a trial function of the form in eq 5 and define the electronic parts and positronic part of the wave functions as
φ1 ) χC1s
f(R, t) ) ΨT(R)Ψ(R, t)
φ2 ) cos θaχC2s + sin θa(cos θbχC2pz + sin θbχHz1s)
EL(R) ) HΨT(R)/ΨT(R)
φ3 ) -sin θaχC2s + cos θa(cos θbχC2pz + sin θbχHz1s)
FQ(R) ) 2∇ΨT(R)/ΨT(R) The time-evolution of f(R, t) can be written, using a Green function G(R, R′, τ), as
φ4 ) cos θcχC2px + sin θcχHx1s
f(R′, t + τ) ) ∫f(R, t)G(R, R′, τ) dR
φ5 ) cos θcχC2py + sin θcχHy1s
(9)
To solve this equation, short-time and fixed node approximations were used. 2.2. Ab Initio Method. The Fock operators for an electron and a positron10 are, respectively,
f (1) ) h (1) + ∑{2Jb(1) - Kb(1)} - Jp(1) e
(10)
b
N/2
f p(p) ) hp(p) - ∑2Jb(p)
φp ) χCs+ + χCp+ + χHAs+ + χHBs+ + χHCs+ where χHi1s (i ) x, y, z) are the linear combinations of AO’s of the three hydrogens:
χHz1s ) χHA1s + χHB1s + χHC1s
N/2
e
(12)
i)1
(11)
b
where Jb(1) is the Coulomb operator and Kb(1) is the exchange
χHx1s ) 2χHA1s - χHB1s - χHC1s χHy1s ) χHB1s - χHC1s Here, the orbital exponents of AO’s (Slater-type orbitals) and angles θa, θb, θc are optimized by VMC, and the values of the
Binding Energy of PsCH3
J. Phys. Chem., Vol. 100, No. 15, 1996 6059
TABLE 1: Optimized Parameters Used in QMC Method χC1s χC2s χC2p χH1s χCs+ χCp+ χHs+
exponent for AO
Jastrow factor B C-H radius H-C-H angle
5.4681 1.2095 1.2691 0.6029 1.0929 1.1504 0.6066 2.00 2.1920 au 116.46°
TABLE 2: Exponents of Added Diffuse and Polarization Functions Used in ab Initio MO Method type
C atom
H atom
s p d f
0.0438, 0.0150, 0.0050 0.0438, 0.0150, 0.0050 0.8000, 0.2700, 0.0800 0.8000, 0.2700
0.0360, 0.0100, 0.0030 1.1000, 0.3600, 0.1000 1.1000, 0.3600
Figure 1. Energy level diagram from CH3 to PsCH3. All energy values in eV.
4. Result and Discussion
the 6-31G basis set. For a system without a positron the stabilization energy due to MP2 is larger than that due to the diffuse function, and the need for a diffuse function seems to be small. In contrast, for a system with a positron the stabilization energy due to the diffuse function is larger than that due to MP2, so the addition of a diffuse function is very important. Secondly, we show the results by QMC in Table 3. The energy values, in which the correlations between electrons and between the positron and electrons are taken into account through the Jastrow factor, are smaller than those without the Jastrow factor. We also see that QMC gives an energy lower than that by the ab initio method with the electronic correlation effect. One of the reasons is that the Jastrow factor is more effective than MP2 for treating these correlations considered here. The relation between the total energies and binding energies of PsCH3 and CH3- is shown in Figure 1. The binding energy of positronium (Ps) is known to be 6.81 eV:1 e- + e+ ) Ps + 6.81 eV. For the stable existence of the positroncontaining system PsCH3, the sum of the electron affinity (EA) of CH3 and PA must be larger than 6.81 eV. Otherwise, the state dissociated into Ps and CH3 would become more stable. Actually, the EA of CH3 is 0.08 eV.15 If we use the value of 6.92 eV (0.2542 au) obtained by QMC for PA, the sum of EA and PA is 7.00 eV and the binding energy of Ps and CH3 is 0.19 eV. On the other hand, if the value 5.15 eV (0.1893 au) obtained by the ab initio method is used, the binding energy becomes -1.58 eV. However, considering that generally QMC gives a more exact value compared to the ab initio MO method and that we may get an improvement of the ab initio result with a much larger size basis set and a higher-order correlation correction in future, we will adopt the QMC results. As a conclusion, the stable existence of the PsCH3 system is expected.
Results of the computation are shown in Table 3. Firstly, we show the results obtained by the ab initio MO method. A system with a positron has energy equivalent to that of an electronic system without a positron, as calculated by the conventional 6-31G basis set and the HF method. With the inclusion of Møller-Plesset second-order perturbation (MP2), a system with a positron becomes a little stable even with only
Acknowledgment. The authors express their thanks to the Centre for Informatics of Waseda University for our utilization of the VP2200/10 supercomputer system. The authors feel it a great honor to contribute this paper to the special issue of The Journal of Physical Chemistry dedicated to Professors Samuel Francis Boys and Isaiah Shavitt.
exponents are shown in Table 1. The orbital exponents for a positron are smaller than those for electrons except for the hydrogenic 1s AO. Thus, we see that the positron is spread wider than the electrons and is pulling them outward. Moreover, we have employed another type of trial function in which the Jastrow factor Uab is multiplied by eq 5:
[
Uab ) exp
1
Aabrab
AaRraR +∑ + a,R 1 + BraR ab Aaprap ApRrpR ∑a 1 + Br + ∑R 1 + Br (18) ap pR
∑ 2a,b*a 1 + Br
]
The values of Aab, etc. are determined by the cusp condition, and B is optimized by VMC. Also, the geometry of CH3- is optimized with this type of trial function. These parameters are shown in Table 1. 3.2. Computation by ab Initio Method. The employed Gaussian-type basis set is (13s7p3d2f/7s3p2d) contracted to [6s5p3d2f/5s3p2d]. Because of the repulsion between the positron and the nucleus, the positron orbital has a tendency to spread out in comparison to the occupied electron orbitals. We have therefore added diffuse and polarization functions to the conventional 6-31G basis set. The exponents of these added functions are shown in Table 2. The GAUSSIAN 9017 ab initio program packages were used for the calculation of the atomic integrals, orbital energies, and MO coefficients.
TABLE 3: Total Energies of CH3- and PsCH3a method ab initio 6-31G our basis QMC with Jastrow a
All energy values in au.
EHF E(2) EHF E(2)
CH3-
PsCH3
PA
-39.4485 -39.5443 -39.5123 -39.7257 -39.6846 ( 0.0342 -39.8155 ( 0.0609
-39.4485 -39.5473 -39.6675 -39.9150 -39.9260 ( 0.0478 -40.0697 ( 0.0421
0.0000 0.0030 0.1552 0.1893 0.2414 ( 0.0410 0.2542 ( 0.0515
6060 J. Phys. Chem., Vol. 100, No. 15, 1996 References and Notes (1) Cartier, E.; Heinrich, F.; Kiess, H.; Wieners, G.; Monkenbusch, M. Positron Annihilation; Jain, P. C., Singru, R. M., Gopinathan, K. P., Eds.; World Scientific: Singapore, 1985. (2) Atomic Physics with Positrons; Humberston, J. W., Armous, E. A. G., Eds.; NATO ASI Series Vol. 169; Plenum Press: New York and London, 1987. (3) Positron and Positronium Chemistry; Schrader, D. M., Jean, Y. C., Eds.; NATO ASI Series Vol. 169; Elsevier Science Publishers: New York, 1988. (4) Ore, A. Phys.ReV. 1951, 83, 665. (5) Clary, D. C. J. Phys. 1976, B9, 3115. (6) Cade, P. E.; Farazdel, A. J. Chem. Phys. 1977, 66, 2598. (7) Kurtz, H. A.; Jordan, K. D. Int. J. Quantum Chem. 1978, 14, 747. (8) Kurtz, H. A.; Jordan, K. D. J. Chem. Phys. 1981, 75, 1876. (9) Kao, C. M.; Cade, P. E. J. Chem. Phys. 1984, 80, 3234. (10) Tachikawa, M.; Sainowo, H.; Iguchi, K.; Suzuki, K. J. Chem. Phys. 1994, 101, 5925.
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