### Strengths and limitations of the SSCHA

###### how dows it compare with other approaches?

by Lorenzo Monacelli

### Self-Consistent Harmonic Approximation

• A method to compute the Free energy of the system
• $$F = \min_{\hat \rho}\left\{ \braket{H}_{\hat \rho} - T S[\hat \rho]\right\}$$
• Restrict $\hat\rho$ to solutions of harmonic $\hat{\mathcal H}_{\mathcal{R}, \Phi}$
• $$\hat\rho_{\mathcal{R},\Phi} = \frac{e^{-\beta \hat{\mathcal{H}}_{\mathcal{R}, \Phi}}}{\mathcal{Z}}$$

### Self-Consistent Harmonic Approximation

• A method to compute the Free energy of the system
• $$F = \min_{\hat \rho}\left\{ \braket{H}_{\hat \rho} - T S[\hat \rho]\right\}$$
• Restrict $\hat\rho$ to solutions of harmonic $\hat{\mathcal H}_{\mathcal{R}, \Phi}$
• $$\hat{\mathcal{H}}_{\mathcal{R}, \Phi} = \sum_\alpha \frac{ {{\hat P}_\alpha}^2}{2m_\alpha} +\underbrace{\sum_{\alpha\beta}(\hat R_\alpha - {\color{blue}\mathcal{R}_\alpha}) {\color{darkgreen}\Phi_{\alpha\beta}}(\hat R_\beta - {\color{blue}\mathcal{R}_\beta})}_{\mathcal V(R)}$$
• Minimize $F$ with respect to $\mathcal{R}$ and $\Phi$ $$F \le F[\hat\rho_{\mathcal R, \Phi}] = F_{\mathcal H} + \left< V(R) - \mathcal V(R) \right>_{\hat\rho_{\mathcal R, \Phi}}$$
• ### Self-Consistent Harmonic Approximation

• Minimize $F$ with respect to $\mathcal{R}$ and $\Phi$ $$F \le F[\hat\rho_{\mathcal R, \Phi}] = F_{\mathcal H} + \left< V(R) - \mathcal V(R) \right>_{\hat\rho_{\mathcal R, \Phi}}$$
• ### SCHA is a first-principles theory

• $\Phi$ are not the phonons!
• Phase-stability through linear response theory
$$\frac{d^2 F}{d{\mathcal R}_a d{\mathcal R}_b} = \Phi_{ab} + \stackrel{(3)}{\Phi}_{acd}\Lambda_{cdef}\stackrel{(3)}{\Phi}_{efb} + \ldots$$
• Dynamics solves a time-dependent SchrÃ¶dinger equation:
$$i\frac{d}{dt} \hat \rho(t) = \left[\hat{\mathcal H}[\hat\rho(t)], \hat\rho(t)\right]$$
• Phonons are dynamical perturbations
$$G_{ab}(t) =\sqrt{m_a m_b}\braket{\left[R_a(t) - {\mathcal R}_a\right] \left[R_b(0) - {\mathcal R}_b\right]}$$

### Strengths of SSCHA

• Exploit symmetries.
$S^\dagger \Phi S = \Phi \;\;\;\; S\mathcal R = \mathcal R$
• Analytic expression for the entropy.
$$F = F_{\mathcal H} + \left< V(R) - \mathcal V(R)\right>$$
• Interpolation on q-mesh: simulate infinite supercells.
• Rigorous linear-response properties:

### Weaknesses of SSCHA

• The probability distribution is a Gaussian.
• Fails in systems where atoms displace significantly.
• Ionic diffusion
• Ionic movements deviating from straight lines
• Bad scaling in absence of symmetries

#### What we cannot do with the SSCHA?

• Liquids
• Rotating molecules
• Glasses

### Comparison with different methods

#### SSCHA

• Quantum zero-point motion
• Symmetries
• Space group constraint
• Analytic free energy
• Approximated distribution
• Dynamical Properties
• $N^3$ scaling

#### Molecular Dynamics

• Classical sampling
• No symmetries
• No space group constraint
• No entropy
• Exact distribution
• Dynamical Properties
• Linear scaling

### Comparison with different methods

#### SSCHA

• Quantum zero-point motion
• Symmetries
• Space group constraint
• Analytic free energy
• Approximated distribution
• Dynamical Properties
• $N^3$ scaling

#### Path-Integral Molecular dynamics

• Quantum sampling
• No symmetries
• No space group constraint
• No entropy
• Exact distribution
• No dynamical properties
• $PN$ linear scaling

### Use the right tool at the right time

• Is $T \lesssim \frac{\hbar\omega_\text{max}}{k_b}$?
• Phase-diagram?
• Dynamical properties?
• Liquid, amorphous, rotations?
• Anharmonicity and high T?
• PIMD, SSCHA, Harmonic
• SSCHA, Harmonic
• SSCHA, MD
• PIMD, MD
• PIMD, MD, SSCHA

### Alternatives to the SSCHA

• Self-consistent ab initio lattice dynamics (SCAILD)
• Self-consistent phonons (SCP)
• Temperature-dependent effective potential (TDEP)
• Many others ....

### Alternatives to the SSCHA

The name Self-Consistent Harmonic comes from the equations $$\Phi_{ab} = \left< \frac{\partial^2V}{\partial R_a \partial R_b} \right>_{\rho_{{\mathcal R}, \Phi}} \qquad 0 = \left < \frac{\partial V}{\partial R_a} \right>_{\rho_{{\mathcal R}, \Phi}}$$ that are solved self-consistently
• SCAILD and SCP solve the same equations
• TDEP computes $\Phi_{ab}$ averaging over MD trajectories

### Self-Consistent Ab Initio Lattice Dynamics (SCAILD)

• Displace atoms along the polarization vectors of $\Phi_{ab}$ $$R_a^\mu = \mathcal{R}_a + \sum_b e_\mu^b \sqrt{\frac{\hbar\left(2n_\mu + 1\right)}{2\omega_\mu m_b}}$$
• Compute the ab-initio force on the displaced atoms.
• Compute the new $\Phi_{ab}$ from the forces $$f_a(R_a) = -\sum_b \Phi_{ab} (R_b - \mathcal{R}_b)$$
Souvatzis et al, Phys. Rev. Lett. 100, 095901 (2008)

### Self-Consistent Ab Initio Lattice Dynamics (SCAILD)

#### but...

• This would coincide with the SSCHA if:
• The lattice vectors $\mathcal R$ do not change
• The polarization vectors of $\Phi_{ab}$ do not change
• The free energy contains only the harmonic part: $$F_{\text{scaild}} \neq F_{\text{sscha}}$$ $$F_{\text{scaild}} = F_{\mathcal H} + V({\mathcal R}) \qquad F_{\text{sscha}} = F_{\mathcal H} + \braket{V - \mathcal V}_{\rho_{\mathcal R, \Phi}}$$
• SCAILD is not variational!
Souvatzis et al, Phys. Rev. Lett. 100, 095901 (2008)

### Temperature Dependent Energy Potential (TDEP)

• Model the potential as a Taylor expansion with temperature dependent coefficients fitted from AIMD trajectories
$${\mathcal V}({R}, T) = \sum_{ab} \frac{1}{2} \stackrel{(2)}{\Phi}_{ab}(T) (R_a - \mathcal{R}_a) (R_b - \mathcal{R}_b) +$$
$$+ \sum_{abc} \frac{1}{6} \stackrel{(3)}{\Phi}_{abc}(T) (R_a - \mathcal{R}_a) (R_b - \mathcal{R}_b) (R_c - \mathcal{R}_c) + \dots$$
• The free energy is evaluated from the harmonic part as: $$F_{\text{tdep}} = F_{\Phi(T)} + \braket{V - \mathcal V}_{\rho_{\text{MD}}}$$
• TDEP is not variational!
Hellman et al, Phys. Rev. B 84, 180301(R) (2011)

### The auxiliary force constant matrix $\Phi(T)$

• $\Phi(T)$ have no physical meaning.
• It does not indicate the phase stability. $$\frac{d^2F}{d{\mathcal R}_a d{\mathcal R}_b} \neq \Phi$$
• It does not describe the anharmonic phonon energy.
• The spectral function is not physical.
• In TDEP, $\Phi$ depends on the order $n$ of the fit of the potential.
• $\Phi$ is the harmonic force constant matrix if $n\rightarrow\infty$ .

### Self-Consistent Phonons (SCP)

• Model the potential as a Taylor expansion
$$V({R}) = V(\mathcal{R}) + \sum_{ab} \frac{1}{2} \stackrel{(2)}{\Phi}_{ab} (R_a - \mathcal{R}_a) (R_b - \mathcal{R}_b) +$$
$$+ \sum_{abc} \frac{1}{6} \stackrel{(3)}{\Phi}_{abc} (R_a - \mathcal{R}_a) (R_b - \mathcal{R}_b) (R_c - \mathcal{R}_c) + \dots$$
• Compressed sensing technique to fit $\stackrel{(2)}{\Phi}$, $\stackrel{(3)}{\Phi}$ and $\stackrel{(4)}{\Phi}$, ...
• The SSCHA integrals can be solved analytically $$\Phi_{ab} = \left< \frac{\partial^2 V}{\partial R_a \partial R_b} \right>_{\rho_{{\mathcal R}, \Phi}} \qquad 0 = \left < \frac{\partial V}{\partial R_a} \right>_{\rho_{{\mathcal R}, \Phi}}$$
Tadano et al, Phys. Rev. B 92, 054301 (2015)

### Temperature Dependent Effective Potential (TDEP)

• Model the potential as a Taylor expansion
$$V({R}) = V(\mathcal{R}) + \sum_{ab} \frac{1}{2} \stackrel{(2)}{\Phi}_{ab} (R_a - \mathcal{R}_a) (R_b - \mathcal{R}_b) +$$
$$+ \sum_{abc} \frac{1}{6} \stackrel{(3)}{\Phi}_{abc} (R_a - \mathcal{R}_a) (R_b - \mathcal{R}_b) (R_c - \mathcal{R}_c) + \dots$$
• But in TDEP the force-constants are temperature dependent
$$\stackrel{(2)}{\Phi}(T) \qquad \stackrel{(3)}{\Phi}(T) \qquad \stackrel{(4)}{\Phi}(T) \qquad \dots$$
Tadano et al, Phys. Rev. B 92, 054301 (2015)

### Comparison

SSCHA SCAILD SCP
TDEP
Not Empirical
Variational
Computationally cheap
Relax lattice
Phase stability
Phase diagram
Quantum effects

### Comparison

SSCHA SCAILD SCP
TDEP(SCP)
Not Empirical
Variational
Computationally cheap
Relax lattice
Phase stability
Phase diagram
Quantum effects