The Gaussion Hartree Approximated Quantum Mechanics Framework 76

Chapter V: Gaussion Hartree Approximated Quantum Mechanics for Large-

5.3 The Gaussion Hartree Approximated Quantum Mechanics Framework 76

by the sum of pair-wise interaction energies of elections, and 2) the kinetic en- ergy component dominates the energy change caused by antisymmetrization of the wavefunction (the Pauli energy). The functional form comes from mixing

Eu= hΨ_Slater| − 1

2∇²_i − 1

2∇²_j|Ψ_Slateri − hΨ_Hatree| − 1

2∇²_i − 1

2∇²_j|Ψ_Hartreei and a correlation energy

Eg =hΨ_VB| −1

2∇_i²− 1

2∇²_j|Ψ_VBi − hΨ_Hatree| − 1

2∇²_i − 1

2∇²_j|Ψ_Hartreei, where

Ψ_VB = 1 q

2+2S_{i j}²

(φiφj+φjφi). (5.13) Klakow (23) used E(↑↑) = Eu andE(↑↓) = 0 in kinetic-energy-based Pauli poten- tials. To reduce the likelihood of coalescence for both same spin and opposite spin electrons, eFF chose

E(↑↑) = Eu−(1− ρ)E_g

E(↑↓) = −ρE_g. (5.14)

This is discussed in detail in (22) by Su.

5.3 The Gaussion Hartree Approximated Quantum Mechanics Framework

77 in which ∆Ti j is defined the same as Equation 5.7, Si j is defined the same as Equation 5.6, and

s¯i j = sisj

q s²_i +s²_j

. (5.18)

E_Pauli^base(↑↓)i j accounts for electron correlation and the form is inspired by the Wigner correlation functional, andpk, k =0, ...9 are parameters fitted to singlet H2bonding curve calculated using B3LYP/FSG.

∆C_{ee,i j} and∆C_{ne,i j} come from the electron-electron and electron-nucleus Coulomb

terms in the total Hamiltonian of an electron pair Hˆ = − X

k=1,2

1

2∇²_k+ 1

r₁₂ −X

n

X

k=1,2

Zn

rnk

. (5.19)

Similar to the derivation of∆T₁₂in Equation 5.7,

∆Cee,12 = hΨ_Slater| 1

r₁₂|Ψ_Slateri − hΨ_Hatree| 1

r₁₂|Ψ_Hartreei

= O₁₂(J₁₂−K₁₂/S₁₂² ), (5.20) where

O₁₂= S₁₂²

1−S₁₂² , (5.21)

J₁₂ =h12| 1

r₁₂|12i= 1 R₁₂Erf*

. . ,

√ 2R₁₂ q

s²₁+s²₂ + / / -

, (5.22)

and the exchange energy

K12 = h12| 1

r₁₂|21i= r2

π q

s²₁+s²₂

s₁s₂ S₁₂² . (5.23) For the nucleus-electron Coulomb interaction contribution,

∆Cne,12 = hΨ_Slater| − Zn

rn1

− Zn

rn2

|Ψ_Slateri − hΨ_Hatree| − Zn

rn1

− Zn

rn2

|Ψ_Hartreei

= O12(j11+ j22−2j12/S₁₂² ), (5.24) where

jk k =hk| − Zn

rnk

|ki= −Zn

Erf ^√

2 s_k Rnk

Rnk

(k =1,2) (5.25)

j12 = h1| − Zn

r_n1|2i= −Zn

¯ s₁₂

Erf rR²_n1

s²₁ + ^R_s²ⁿ²2 2

− ^R

2 12

s²₁+s²₂

!

rR_n1²

s²₁ + ^R_s²ⁿ²2 2

− ^R

2 12

s²₁+s²₂

S12. (5.26)

The full forms of the scaling factors in Equation 5.15 are

F(↑↑)i j = F(↑↑)i j,sym(∆T,∆Cee,∆Cne)F(↑↑)i j,asym(∆T,∆Cee,∆Cne) (5.27) and

F(↑↓)i j = F(↑↓)i j,symF(↑↓)i j,asym. (5.28)

Define

X

Si j = 1 2*

, X

k

Sik+X

k0

Sik0+ -

(5.29) X

S_{i j}² = 1 2*

, X

k

S_ik² +X

k0

S_ik0² + -

. (5.30)

For same spin electrons that each has exactly the same environment, the scaling factors for∆T and∆Cnecan be derived based onD3hquartet H₃orTd quintet H₄to give

F(↑↑)i j,sym(∆T,∆Cne) = 1+Si j

1+P

S. (5.31)

Therefore, we separate the scaling factor into two terms multiplying each other, F_sym andF_asym, which account for the "symmetric" environment and "asymmetric"

environment, respectively.

The forms of the other components of the same spin electron scaling factors are F(↑↑)i j,sym(∆Cee) = p_se0*

,

pse1+Si j

p_se1+P Si j

+

PSi j −Si j

p_se1Si j+P Si j

+ -

p_se2

+(1− p_se0)* . ,

p_se3+S_{i j}² p_se3+P

S_{i j}² +

PS_{i j}² −S_{i j}² p_se3S_{i j}² +P

S_{i j}² + / -

p_se4

,(5.32) where the parameterspsek,k = 0, ...,4 are fitted with the exact quantum mechanics [unrestricted Hartree Fock (UHF)/FSG] results ofD_3hquartet H₃andTdquintet H₄ symmetric stretching, and

F(↑↑)_{i j,asym}(∆T,∆Cee,∆Cne) = 1+p_a0

Si j

XSi j −X S_{i j}²

+p_a1

Si j

XSi j −X S_{i j}²

2

+p_a2

Si j

XSi j −X S_{i j}²

3

, (5.33)

79 wherepak,k = 0,1,2 are parameters whose values depend on whether it is for∆T,

∆Cee or ∆Cne, and fitted with D3h to C2v transition of quartet H3 andTd to C3v

transition of quintet H₄.

For opposite spin scaling factors, the forms are F(↑↓)i j,sym = p_os0*

,

1+pos1Si j

1+p_os1P Si j

+ -

p_os2

+(1−p_os0)* . ,

1+p_os3S_{i j}² 1+p_os3P

S_{i j}² + / -

p_os4

, (5.34) where the parameters posk,k = 0, ...,4 are fitted with D∞h doublet H₃ and D4h

singlet H₄symmetric stretching, and

F(↑↓)i j,asym = 1+p_oa0* . ,

Si jP

Si j −P S_{i j}² Si jP

Si j +P S_{i j}²

+ / - +poa1*

. ,

Si jP

Si j −P S_{i j}² Si jP

Si j +P S_{i j}²

+ / -

2

+p_oa2* . ,

Si jP

Si j −P S_{i j}² Si jP

Si j +P S_{i j}²

+ / -

3

, (5.35)

where the parameterspoak,k = 0,1,2 are fitted withD∞hdoublet H₃andD_4hsinglet H₄asymmetric stretching.

In summary, the total Pauli potential for same spin electron pairs is E_Pauli(↑↑)i j = F(↑↑)_{i j,sym}(∆T)F(↑↑)_{i j,asym}(∆T)∆Ti j

+F(↑↑)i j,sym(∆Cee)F(↑↑)i j,asym(∆Cee)∆Cee,i j

+X

n

F(↑↑)i j,sym(∆Cne)F(↑↑)i j,asym(∆Cne)∆Cne,i j (5.36) and the total Pauli potential for opposite spin pairs is

E_Pauli(↑↓)i j = F(↑↓)i j,symF(↑↓)i j,asymE_Pauli^base(↑↓)i j. (5.37) The above has been discussed in more detail in (18).

5.4 Improvements on GHA-QM

The GHA-QM framework in the previous section produces QM quality bonding curve for singlet and triplet H₂, and gives good results for symmetric and asymmetric stretching of H₃ and H₄ molecules in mild conditions. However, when we tested the system more substantially and tried to simulate hydrogen systems with more electrons in high density, we found certain problems. This section discusses several strategies that were taken to fix these problems.

New Opposite Spin Pauli Potential

We found that the size of the electron in the center of linear H₃ sometimes goes to zero. This is because in Equation 5.17 wherep₄is negative and all other parameters are positive, one electron having zero size ( ¯sgoing to zero) meansE_Pauli^base(↑↓)becomes negative infinite and energetically favored. To avoid this problem, we introduced a positive shiftp₁₀ of electron size in Equation 5.17. The new form is

E_Pauli^base(↑↓)i j =−

p₀S_{i j}^p1 +p₂S_{i j}^p3(s¯i j +p₁₀)^p4

1+p₅S_{i j}^p6+p₇S_{i j}^p8s¯_{i j}^p9 . (5.38) We refitted the parameters against the exact QM results of singlet H₂ bonding curve.(24) The new parameters arep₀=0.439387,p₁= 2.914263,p₂=8.180823,p₃= 6.100496,p₄ = −18.088005,p₅ = 0.698750,p₆ = 3.305666,p₇ = 2.031224,p₈ = 7.359878,p₉ = 5.552337,p₁₀ = 8.345865. The resulting GHA-QM H₂ bonding curve is nearly exact in the bonding regions, and performs better than B3LYP/6- 311++G** at intermediate bond lengths (Figure 5.1). Because the base energy of the

-125 -75 -25 25 75 125

0 0.5 1 1.5 2 2.5 3

E (kcal/mol)

H-H Distance (Å)

QM

B3LYP/6-311++G**

GHA eFF

(Exact)

Figure 5.1: Performance of GHA-QM on single H₂.

opposite spin Pauli potential has changed, we refitted the parameters in the opposite spin scaling factors too. We used genetic algorithm to train the parameters for both symmetric and asymmetric scaling factors at the same time. The best parameters are p_os0 = 0.968658,p_os1 = 1.021811,p_os2 = 2.720616,p_os3 = 15.859202,p_os4 = 20.201212 and p_oa0 = 2.376509,p_oa1 = 18.590298,p_oa2 = 13.245066. These pa- rameters give much improved energy for H₃ reaction path (Figure 5.2), and overall

81 a more accurate H₃ potential energy surface (Figure 5.4) than eFF (Figure 5.5) comparing toab initioH3potential energy surface (Figure 5.5).(25)

Transition state geometry:

QM: r₁ = r₂ = 0.930 Å B3LYP: r₁ = r₂ = 0.931 Å GHA: r₁ = r₂ = 0.975 Å eFF: r₁ = r₂ = 1.035 Å

r₁ r₂ Fix r₁ and relax r₂

0 2 4 6 8 10 12 14 16 18 20

0.8 1 1.2 1.4 1.6 1.8 2

E (kcal/mol)

r₁ (Å)

QM

B3LYP/6-311++G**

GHA eFF

(CI)

Figure 5.2: Performance of GHA-QM on H₃reaction path. The QM configuration interaction (CI) data is from (26).

Preventing Same Spin Electron Coalescence

When we tried to simulate the equation of state (EOS) of a dense H₂liquid (Wigner radiusrs = 2.2 bohr), we find the same spin electrons are energetically favorable to diffuse and cluster at some configurations. The following paragraphs are devoted to attempts at solving this problem.

A New Form of∆T We’d like to shift the numerator in Equation 5.7 by a small amountdd so that∆T goes to infinity whenS→1. For convenience, the subscripts of∆T,S,Oare omitted.

The new form is

∆T = S²(t11+t22−2t12/S)+dd 1−S²

= S² 3

2 1 s²₁ + ³_2s¹2

2

− ⁶

s²₁+s²₂ + _(s^4R2 ²¹² 1+s²₂)²

+dd 1−S²

= O·T + dd

1−S², (5.39)

QM r

₁

r

₂

E (kcal/mol)

r₁ (Å) r2 (Å)

Figure 5.3: Analytical H₃potential energy surface. The data is from (25).

GHA

E (kcal/mol)

r2 (Å)

r₁ (Å)

Figure 5.4: GHA-QM H₃ potential energy surface. r₁ and r₂ are defined in Fig- ure 5.3.

83

eFF E (kcal/mol)

r

2

( Å )

r

₁

(Å)

Figure 5.5: eFF H₃potential energy surface. r₁andr₂are defined in Figure 5.3.

where

T ≡ 3 2

1 s²₁ + 3

2 1

s²₂ − 6

s²₁+s²₂ + 4R₁₂² (s²₁+s²₂)². We chosedd =1.0×10⁻⁴.

To avoid numerical instability when S → 1, s₁ → s₂ and R₁₂ → 0, we did series expansion of∆T at this condition (Appendix D). In its practical implementation, when 1−S² ≤ 0.001, instead of the form in Equation 5.39, we take

∆T =* ,

1

s₁s₂ − R²₁₂

2s²₁s²₂ + R⁴₁₂ 12s³₁s³₂+

- +dd*

, s₁s₂

R₁₂² + 1

2 + R₁₂²

12s₁s₂ − R₁₂⁶ 720s³₁s³₂+

-

. (5.40)

Although this scheme can prevent the same spin electrons to have the same coordi- nates, it does not prevent same spin electrons of similar sizes to become unphysically close (overlap S → 1). As can be seen in Figure 5.6, the energy penalty from the new form of∆T is not big enough for most configurations the dynamics simulation can reach. Nevertheless, we kept this form because it does prevent the same spin electrons to have the same coordinates and enforce the Pauli exclusion principle.

New Same Spin Scaling Factors The form of Equation 5.33 may cause serious issues during simulation because it is unbounded. In addition, the sign of the

0.9 0.92 0.94 0.96 0.98 1 0

0.2 0.4 0.6 0.8 1

S

∆ T (hartree)

s1 = s

2 = 5.3 bohr

old ∆ T new ∆ T

Figure 5.6: Comparison of ∆T values from the old and new forms for diffused electrons of the same size and spin.

function may change due to the odd powers and the possibility for(Si jP

Si j−P S_{i j}²) to become negative in some cases. To make the function bounded, we adopt the following form instead:

F(↑↑)i j,asym(∆T,∆Cee,∆Cne) =1+p_a0Di j +p_a1D²_{i j}+ p_a2D_{i j}³, (5.41) where Di j = _S^{Si jPSi j−PSi j2}

i jPS_{i j}+PS_{i j}²+d

!

. The small constant d = 1.0× 10⁻⁶ is to prevent the denominator from going to zero when all the overlap S → 0. In fact, this shift has been added to the opposite spin asymmetric scaling factor too so that now same spin and opposite spin asymmetric scaling factors adopt the same form, with different parameters. The parameters for same spin asymmetric scaling factors are summarized in Table 5.1.

For the fitting to cover the whole range of S values from 0 to 1, we fitted against C_2v quartet H₃, with one H-H distance fixed at 0.1 bohr and 1.7 bohr. The size of the electrons were kept to 1.5 bohr. Figures 5.7 shows that the whole range of S has been covered, especially for largeSvalues that were not considered in previous fitting.

85

∆T ∆Cee ∆Cne

p_a0 1.31546298328185 0.727335080043498 0.468486643682531 p_a1 3.39853157886649 1.311794439120800 13.20277753109270 pa2 2.33306859558464 0.584459359077299 12.98429088741020 Table 5.1: New parameters for same spin asymmetric scaling factors F(↑↑

)i j,asym(∆T,∆Cee,∆Cne).

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0 0.5 1.0

Energy (hartree)

S₁₃

C_2v Quartet H₃

UHF/FSG dT

eFF (GHA unscaled) dT GHA sym scaled dT GHA asym scaled dT

1 2

3

d₁₂ = 0.1 bohr

F_symΔT F_symF_asymΔT

ΔT ΔT

Figure 5.7: Performance of GHA-QM forC2v quartet H₃ with one H-H distance fixed at 0.1 bohr.

Figure 5.8 shows the GHA-QM same spin symmetric scaling factor does not work for C_2v quartet H₃ when one H-H distance is fixed at 1.7 bohr and the other H-H distances are much smaller than 1.7 bohr. This suggests that the symmetric scaling factor is not likely able to deal with linear H₃. Since the same spin symmetric scaling factor for∆T is exact for D3h H₃, we hypothesized that the scaling factors are highly symmetry-dependent and the F(↑↑)_sym derived from D3h H₃ cannot be universally applied to all molecules with other symmetry. Interestingly, eFF seems to go to the correct limit as the system approaches being linear. To further explore this issue, we developed a scaling factor F(↑↑)_linear to replace F(↑↑)_sym in linear molecules:

F(↑↑)i j,linear = 1+ pl0v+pl1v²+pl2v³, (5.42)

0.1 0.3 0.5 0.7 0.9 1.1

0.5 2.5 4.5 6.5

Energy (hartree)

d₁₃ (bohr)

C_2v Quartet H₃

UHF/FSG dT

eFF (GHA unscaled) dT GHA sym scaled dT GHA asym scaled dT

1 2

3

d₁₂ = 1.7 bohr

1.7

F_symΔT F_symF_asymΔT

ΔT

ΔT

Figure 5.8: Performance of GHA-QM forC_2v quartet H₃ with one H-H distance fixed at 1.7 bohr.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0 1 2 3 4

Energy (hartree)

d₁₃ (bohr)

Symmetric stretching of linear quartet H3

UHF/FSG dT eFF dT

GHA sym scaled dT GHA asym scaled dT GHA asym,l scaled dT

1 3 2

F_symΔT F_symF_asymΔT (GHA unscaled) ΔT

ΔT

F_linearF_asymΔT

Figure 5.9: Comparison of the effect of different scaling factors in the performance of symmetric stretching of linear quartet H₃.

where

v =* ,

(Si jP

Si j −P S_{i j}²)² (p_l3Si jP

Si j −P

S_{i j}²)²+dl

+ -

, (5.43)

we chosedl =1.0×10⁻⁷and fitted the parameterspl0=0.0098195829828946,pl1 = 0.0758036465880992,pl2 =0.440549994711866,pl3=0.999965221776706.

Intuitively, when the H-H distances become very small, linear H₃andD_3hH₃energy should go to the same limit. However, this is not the case as seen from Figure 5.9.

87 This confirms that the scaling factors should be symmetry-dependent or geometry- dependent. However, in practice it is likely impossible to implement. Since our goal is to simulate condensed matters, there will be too many electrons in the system for the program to classify the configuration to determine suitable scaling factors.

Removing the Same Spin Scaling Factors The analysis in the previous subsection suggests it might be a good idea to remove the same spin scaling factors completely (i.e. make them take the value of 1). In addition, minimization of the dense H₂liquid (rs =1.76 bohr or denser) shows that same spin scaling factors for∆T and∆Cnecan scale these positive energy contributions down to∼ ₁₀¹ times the base values while maintaining the negative energy contribution from∆Cee similar to the base value.

This energetically favors the same spin electrons to cluster, even when we keep the electron sizes fixed at 1.55 bohr (the electron size in H2 optimized by GHA-QM).

The different behaviors of the scaling factors for different energy components may be due to the empirical nature of some of the functional forms of the scaling factors, and may also be due to the symmetry issue discussed in the previous subsection.

To see how much removing same spin scaling factors could affect molecular interac- tions, we calculated the H₂-H₂association curve with GHA-QM without same spin scaling factors. The results look reasonable as they are close to QM results (27) in Figure 5.10. They are also an improvement on eFF.

r₁ r₂

r₁ = 0.74 Å

-300 -200 -100 0 100 200 300 400

0 2 4 6 8 10

E (kcal/mol)

r₂ (Å)

D_2h H₂-H₂ Association

QM GHA eFF

Figure 5.10: D_2hH₂-H₂association without same spin scaling factors. The energy values areE(H₄)−4E(H). The QM data is from (27).

Therefore, with the same spin scaling factors turned off, we went on to calculate the liquid H2EOS and compared it to QM results. A cubic box of 108 H2molecules has been used with minimum image on each side of the box as the periodic boundary condition. We took me = mH so that we can use a relatively large time step 0.02 femtosecond. A total of 2 ps dynamics was carried out for each temperature, and the first 0.5 ps was discarded when calculating the average pressure. During the dynamics, for each time step we optimized the electron sizes based on the x,y,z coordinates of the electrons and nuclei. For simplicity, we approximated the optimization procedure by carrying out only the first step of the Newton-Raphson method and considering the second derivative of only the electronic kinetic energy.

Figure 5.11 shows both GHA-QM and eFF are quite close to QM results, with both eFF and GHA-QM having a tendency to underestimate the pressure at low density.

At high density, GHA-QM has a tendency to underestimate the pressure, while eFF tends to overestimate the pressure. In terms of the energy-volume relationship, GHA-QM is clearly an improvement upon eFF (Figure 5.12).

-50 50 150 250 350 450

2.5 7.5 12.5

P (GPa)

V (ų) / molecule

exp. 0K GHA 0K GHA at 300K GHA at 2000K GHA at 3700K eFF 0K

QM corrected at 300 K+ZP

Figure 5.11: Pressure-volume diagram of H₂ liquid. The experimental values and QM extrapolation are from (28).