17 Theoretical Analysis of Phase-Transition Temperature
321
crystals, the phase transition temperature upon changing the deuteron concentration is investigated using the mixed K 3 H 1−x D x (SO 4 ) 2 and (H 1−x D x ) 2 SQ crystals with random substitution of protons with deuterons [73, 74]. In the mixed
K 3 H 1−x D x (SO 4 ) 2 crystal, the phase transition appearance was reported with increasing deuterium concentration. The deuterium concentration dependent behavior
of T c was described by simple transverse Ising Hamiltonian. However, it is impossible to express the geometrical and electronical changes as well as the T c between
the H- and D-compounds. Therefore, the first-principle calculation is necessary to
theoretically analyze the origin of isotope effect for phase transition including geometrical changes. In the mixed (H 1−x D x ) 2 SQ crystal, the critical temperature T c
for the dielectric phase transition was found to increase linearly with the deuterium
concentration. Since, the phase transition of H 2 SQ and D 2 SQ crystals has large geometrical change, conventional model calculations are not enough to theoretically
investigate the relation between phase transition and geometrical changes.
In addition, tritium behaves as an isotope of hydrogen since the mass is three
times of hydrogen. Tritium is also well known as a radioactive nuclide that emits
β-rays, even a trace amount of tritiated products can be detected with high sensitivity [75]. Tritium is often used to measure the kinetic isotope effect on enzyme catalyzed hydrogen transfer [76, 77]. To date, there is no experimental report about the
tritiated hydrogen-bonded dielectric materials. It is a great and important challenge
to theoretically elucidate the T c of tritiated hydrogen-bonded dielectric materials.
In this chapter, the deuterium concentration dependence of phase transition temperature and geometrical changes of mixed K 3 H 1−x D x (SO 4 ) 2 and (H 1−x D x ) 2 SQ
crystal in random substitution of protons with deuterons was theoretically investigated. Furthermore, the phase transition temperature of TKHS and T 2 SQ crystals
that is the tritium substitutions from theoretical prediction was estimated.
In order to describe the proton-deuteron mixed crystal, the mass of proton is
adopted as the 1836.59, 2293.59, 2750.59, 3207.58, and 3664.58 a.u. for x = 0.00,
0.25, 0.50, 0.75, and 1.00, respectively. These masses are indicated as the average
of H 1−x D x in each K 3 H 1−x D x (SO 4 ) 2 and (H 1−x D x ) 2 SQ crystal at the random substitution of protons with deuterons.
In the MC_MO calculation which directly takes into account the quantum effect,
the proton, deuteron, and other mixed particles (these particles are called the “light
nuclei” in this chapter) are treated as quantum waves, as well as electrons under the
field of S, C, and O nuclear point charges. The positions of these point charges are
determined by conventional optimization procedures using analytical gradient [70].
The single s-type gaussian type function (GTF), exp{−α(r − R) 2 }, was employed
for each light nuclear basis function in which the GTF variational parameters were
optimized, simultaneously. The standard [3s1p]/(4s1p) electronic basis set for hydrogen and Pople’s 3-21G ∗ basis set [56, 78] for S, C, and O were used. The centers
of electronic GTFs are fixed on each nucleus. All calculations were carried out at the
Hartree-Fock level using modified versions of Gaussian/98 program packages [59].
Furthermore, in order to predict the phase transition temperature and geometries of TKHS and T 2 SQ in which the triton is substituted from proton, the cluster model was calculated using the MC_MO method by the same procedure of the
321
crystals, the phase transition temperature upon changing the deuteron concentration is investigated using the mixed K 3 H 1−x D x (SO 4 ) 2 and (H 1−x D x ) 2 SQ crystals with random substitution of protons with deuterons [73, 74]. In the mixed
K 3 H 1−x D x (SO 4 ) 2 crystal, the phase transition appearance was reported with increasing deuterium concentration. The deuterium concentration dependent behavior
of T c was described by simple transverse Ising Hamiltonian. However, it is impossible to express the geometrical and electronical changes as well as the T c between
the H- and D-compounds. Therefore, the first-principle calculation is necessary to
theoretically analyze the origin of isotope effect for phase transition including geometrical changes. In the mixed (H 1−x D x ) 2 SQ crystal, the critical temperature T c
for the dielectric phase transition was found to increase linearly with the deuterium
concentration. Since, the phase transition of H 2 SQ and D 2 SQ crystals has large geometrical change, conventional model calculations are not enough to theoretically
investigate the relation between phase transition and geometrical changes.
In addition, tritium behaves as an isotope of hydrogen since the mass is three
times of hydrogen. Tritium is also well known as a radioactive nuclide that emits
β-rays, even a trace amount of tritiated products can be detected with high sensitivity [75]. Tritium is often used to measure the kinetic isotope effect on enzyme catalyzed hydrogen transfer [76, 77]. To date, there is no experimental report about the
tritiated hydrogen-bonded dielectric materials. It is a great and important challenge
to theoretically elucidate the T c of tritiated hydrogen-bonded dielectric materials.
In this chapter, the deuterium concentration dependence of phase transition temperature and geometrical changes of mixed K 3 H 1−x D x (SO 4 ) 2 and (H 1−x D x ) 2 SQ
crystal in random substitution of protons with deuterons was theoretically investigated. Furthermore, the phase transition temperature of TKHS and T 2 SQ crystals
that is the tritium substitutions from theoretical prediction was estimated.
In order to describe the proton-deuteron mixed crystal, the mass of proton is
adopted as the 1836.59, 2293.59, 2750.59, 3207.58, and 3664.58 a.u. for x = 0.00,
0.25, 0.50, 0.75, and 1.00, respectively. These masses are indicated as the average
of H 1−x D x in each K 3 H 1−x D x (SO 4 ) 2 and (H 1−x D x ) 2 SQ crystal at the random substitution of protons with deuterons.
In the MC_MO calculation which directly takes into account the quantum effect,
the proton, deuteron, and other mixed particles (these particles are called the “light
nuclei” in this chapter) are treated as quantum waves, as well as electrons under the
field of S, C, and O nuclear point charges. The positions of these point charges are
determined by conventional optimization procedures using analytical gradient [70].
The single s-type gaussian type function (GTF), exp{−α(r − R) 2 }, was employed
for each light nuclear basis function in which the GTF variational parameters were
optimized, simultaneously. The standard [3s1p]/(4s1p) electronic basis set for hydrogen and Pople’s 3-21G ∗ basis set [56, 78] for S, C, and O were used. The centers
of electronic GTFs are fixed on each nucleus. All calculations were carried out at the
Hartree-Fock level using modified versions of Gaussian/98 program packages [59].
Furthermore, in order to predict the phase transition temperature and geometries of TKHS and T 2 SQ in which the triton is substituted from proton, the cluster model was calculated using the MC_MO method by the same procedure of the