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Hydrogen is the lightest and most abundant element in the universe. Ever since Wigner and Huntington's prediction that pressure induced metallization might happen in solid hydrogen, understanding the hydrogen phase diagram has become one of the greatest challenges in condensed matter and high pressure physics. The light mass of hydrogen means that the nuclear quantum effects could be important in describing this phase diagram under high pressures. Numerical evaluations of their contributions to the structural, vibrational, and energetic properties, however, are difficult and up to now most of the theoretical simulations still remain classical. This is particularly true for the energetic properties. When the free-energies of different phases are compared in determining the ground state structure of the system at a given pressure and temperature, most of the theoretical simulations remain classical. When nuclear quantum effects must be taken into account, one often resorts to the harmonic approximation. In the very rare case, the anharmonic contributions from the nuclear statistical effects are considered by using a combination of the thermodynamic integration and the at initio molecular dynamics methods, which helps to include the classical nuclear anharmonic effects. Quantum nuclear anharmonic effects, however, are completely untouched. Here, using a self-developed combination of the thermodynamic integration and the at initio path-integral molecular dynamics methods, we calculated the free-energies of the high pressure hydrogen at 100 K from 200 GPa to 300 GPa. The harmonic lattice was taken as the reference and the Cmca phase of the solid hydrogen was chosen. When the bead number of the path-integral (P) equals one, our approach reaches the so-called classical limit. Upon increasing P until the results are converged, our approach reaches the limit when both classical and quantum nuclear anharmonic effects are included. Therefore, by comparing the free-energy of the harmonic lattice and the thermodynamic integration results at P equals one, we isolate the classical nuclear anharmonic effects. By comparing the thermodynamic integration results at P equals one and with those when they are converged with respect to P, we isolate the quantum nuclear anharmonic effects in a very clean manner. Our calculations show that the classical nuclear anharmonic contributions to the free-energy are negligible at this low temperature. Those contributions from the quantum nuclear anharmonic effects, however, are as large as ～15 meV per atom. This value also increases with pressure. This study presents an algorithm to quantitatively calculate the quantum contribution of the nuclear motion to free-energy beyond the often used harmonic approximation. The large numbers we got obtained also indicate that such quantum nuclear anharmonic effects are important in describing the phase diagram of hydrogen, at/above the pressures studied.
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[4] Xu G, Ming W, Yao Y, Dai X, Zhang S C, Fang Z 2008 EPL 82 67002
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[6] Pickard C J, Needs R J 2007 Nat. Phys. 3 473
[7] Li X Z, Wang E G 2014 Computer Simulations of Molecules and Condensed Matters: From Electronic Structures to Molecular Dynamics (Beijing: Peking University Press) pp134-140
[8] Frenkel D, Lekkerkerker H N W, Stroobants A 1988 Nature 332 822
[9] Meijer E J, Frenkel D 1991 J. Chem. Phys. 94 2269
[10] Alfé D, Gillan M J, Price G D 1999 Nature 401 462
[11] Alfé D, Price G D, Gillan M J 2001 Phys. Rev. B 64 045123
[12] Wigner E, Huntington H B 1935 J. Chem. Phys. 3 764
[13] Babaev E, Sudbo A, Ashcroft N W 2004 Nature 431 666
[14] Bonev S A, Schwegler E, Ogitsu T, Galli G 2004 Nature 431 669
[15] Deemyad S, Silvera I F 2008 Phys. Rev. Lett. 100 155701
[16] Li X Z, Walker B, Probert M I J, Pickard C J, Needs R J, Michaelides A 2013 J. Phys.: Condens. Matter 25 085402
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[18] Mao H K, Hemley R J 1994 Rev. Mod. Phys. 66 671
[19] McMahon J M, Morales M A, Pierleoni C, Ceperley D M 2012 Rev. Mod. Phys. 84 1607
[20] Zha C S, Liu Z X, Hemley R J 2012 Phys. Rev. Lett. 108 146402
[21] Liu H Y, Zhu L, Cui W W, Ma Y M 2012 J. Chem. Phys. 137 074501
[22] Perez A, von Lilienfeld O A 2011 J. Chem. Theory Comput. 7 2358
[23] Habershon S, Manolopoulos D E 2011 J. Chem. Phys. 135 224111
[24] Kresse G, Furthmller J 1996 Phys. Rev. B 54 11169
[25] Feng Y X, Chen J, Alfè D, Li X Z, Wang E G 2015 J. Chem. Phys. 142 064506
[26] Alfé D 2009 Comput. Phys. Commun. 180 2622
[27] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
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[1] Mermin N D 1985 Phys. Rev. A 137 1441
[2] Gillan M J 1989 J. Phys.: Condens. Matter 1 689
[3] Wentzcovitch R M, Martins J L, Allen P B 1992 Phys. Rev. B 45 11372
[4] Xu G, Ming W, Yao Y, Dai X, Zhang S C, Fang Z 2008 EPL 82 67002
[5] Lee P A, Nagaosa N, Wen X G 2006 Rev. Mod. Phys. 78 17
[6] Pickard C J, Needs R J 2007 Nat. Phys. 3 473
[7] Li X Z, Wang E G 2014 Computer Simulations of Molecules and Condensed Matters: From Electronic Structures to Molecular Dynamics (Beijing: Peking University Press) pp134-140
[8] Frenkel D, Lekkerkerker H N W, Stroobants A 1988 Nature 332 822
[9] Meijer E J, Frenkel D 1991 J. Chem. Phys. 94 2269
[10] Alfé D, Gillan M J, Price G D 1999 Nature 401 462
[11] Alfé D, Price G D, Gillan M J 2001 Phys. Rev. B 64 045123
[12] Wigner E, Huntington H B 1935 J. Chem. Phys. 3 764
[13] Babaev E, Sudbo A, Ashcroft N W 2004 Nature 431 666
[14] Bonev S A, Schwegler E, Ogitsu T, Galli G 2004 Nature 431 669
[15] Deemyad S, Silvera I F 2008 Phys. Rev. Lett. 100 155701
[16] Li X Z, Walker B, Probert M I J, Pickard C J, Needs R J, Michaelides A 2013 J. Phys.: Condens. Matter 25 085402
[17] Chen J, Li X Z, Zhang Q F, Probert M I J, Pickard C J, Needs R J, Michaelides A, Wang E G 2013 Nat. Commun. 4 2064
[18] Mao H K, Hemley R J 1994 Rev. Mod. Phys. 66 671
[19] McMahon J M, Morales M A, Pierleoni C, Ceperley D M 2012 Rev. Mod. Phys. 84 1607
[20] Zha C S, Liu Z X, Hemley R J 2012 Phys. Rev. Lett. 108 146402
[21] Liu H Y, Zhu L, Cui W W, Ma Y M 2012 J. Chem. Phys. 137 074501
[22] Perez A, von Lilienfeld O A 2011 J. Chem. Theory Comput. 7 2358
[23] Habershon S, Manolopoulos D E 2011 J. Chem. Phys. 135 224111
[24] Kresse G, Furthmller J 1996 Phys. Rev. B 54 11169
[25] Feng Y X, Chen J, Alfè D, Li X Z, Wang E G 2015 J. Chem. Phys. 142 064506
[26] Alfé D 2009 Comput. Phys. Commun. 180 2622
[27] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
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