Combined QM/MM Modeling Methods
Interactions in the QM/MM coupling
$$E_{total} = E_{QM} + E_{MM} + E_{QM/MM}$$
$$
E_{QM/MM} = E_{ES}(QM/MM) + E_{vdW}(QM/MM) + E_{bonded}(QM/MM)
$$
The effective Hamiltonian
$$\hat{\mathcal{H}}_{eff} = \hat{\mathcal{H}}_{QM} + \hat{\mathcal{H}}_{ES}(QM/MM)$$
Average Solvent Potential in Mean-Field Theory
The mutual solute–solvent polarization
The ASEP/MD Hamiltonian
$$
\hat{\mathcal{H}} = \hat{\mathcal{H}}_{QM} + \hat{\mathcal{H}}_{class} + \hat{\mathcal{H}}_{int}
$$
The effective Schrodinger equation
$$
(\hat{\mathcal{H}}_{QM}+\hat{\mathcal{H}}_{int}) \left | \Psi \right \rangle = E \left | \Psi \right \rangle
$$
The interaction Hamiltonian associated to the electrostatic and van der Waals contributions
$$
\hat{\mathcal{H}}_{int} = \hat{\mathcal{H}}^{elect}_{int} + \hat{\mathcal{H}}^{vdw}_{int}
$$
The MFA electrostatic interaction(a statistical average over configurations)
$$
\left \langle \hat{\mathcal{H}}^{elect}_{int} \right \rangle = \int {\hat{\rho} \cdot \left \langle V_{S}(r) \right \rangle } \, dr
$$
- $\hat{\rho}$ : the solute charge density operator
- $\left \langle V_{S}(r) \right \rangle$ : the average electrostatic potential by the solvent
The effective Schrodinger equation considering the MFA energy $$ (\hat{\mathcal{H}}_{QM} + \left \langle \hat{\mathcal{H}}_{int} \right \rangle) \left | \Psi \right \rangle = E \left | \Psi \right \rangle $$
- $\left \langle V_{S}(r) \right \rangle$ : the average electrostatic potential by the solvent
The effective Schrodinger equation considering the MFA energy $$ (\hat{\mathcal{H}}_{QM} + \left \langle \hat{\mathcal{H}}_{int} \right \rangle) \left | \Psi \right \rangle = E \left | \Psi \right \rangle $$
Location of critical points on free energy surfaces
The force on the Helmholtz free energy surface
$$
G = -k_{B}T\ln{Z_{NVT}}
$$
$$
\left \langle F(R) \right \rangle
= - \frac{\partial G(R)}{\partial R}
= - \left \langle \frac{\partial E}{\partial R} \right \rangle
= - \left \langle \frac{\partial E_{QM}}{\partial R} \right \rangle - \left \langle \frac{\partial E_{int}}{\partial R} \right \rangle
$$
Calculation of free energy differences
Within the ASEP/MD methodology, the free energy difference in solution between two given states
$$
\Delta G_{s} = \Delta E_{solute} + \Delta G_{int} + \Delta ZPE_{solute}
$$
$$
\Delta E_{solute} = E_{B} - E_{A} = \left \langle \Psi_{B} | \hat{\mathcal{H}}^{0}_{B} | \Psi_{B} \right \rangle - \left \langle \Psi_{A} | \hat{\mathcal{H}}^{0}_{A} | \Psi_{A} \right \rangle
$$
The Electronic Structure
Statistical mechanics sampling for many-body interacting systems in condensed phases
The Hamiltonian of the system
$$\hat{\mathcal{H}} = -\frac{1}{2}\sum_{A=1}^{M}{\frac{1}{M_{A}}\nabla^{2}_{M}} -\frac{1}{2}\sum_{i=1}^{N}{\nabla^{2}_{N}} + V(\vec{r_{i}};\vec{R_{A}},\vec{X})$$
Electronic spectra in QM/MM
A QM solute(M) and MM solvent molecules(I) system
The QM/MM coupling schemes
$$E_{QM/MM}[M] = E[M] = E_{QM}(M) + E_{MM}(\{ I \}) + E_{int}$$
$$E_{QM}(M) = \left \langle \Psi(\vec{r};\vec{R};\vec{X})| \hat{\mathcal{H}} | \Psi(\vec{r};\vec{R};\vec{X}) \right \rangle$$
Mechanical coupling
$$E_{int}(el) = \sum_{A\alpha}\frac{q_{A}q_{\alpha}}{r_{A\alpha}} $$
Electrostatic coupling : Mechanical coupling with QM polarization considering potential for the solvent
$$E_{int}(el) = \left \langle \Psi(r_{i};R_{A};q_{\alpha}) | \sum_{\alpha}\frac{q_{\alpha}}{r_{i\alpha}} | \Psi(r_{i};R_{A};q_{\alpha}) \right \rangle + \sum_{A\alpha}\frac{q_{A}q_{\alpha}}{r_{A\alpha}} $$
The wavefunctions for both ground and excited states
$$\begin{align*}
\Psi^{0} &= \Phi^{0}_{(0)} + \Phi^{0}_{(1)} + \Phi^{0}_{(2)} + \cdots \\
\Psi^{*} &= \Phi^{*}_{(0)} + \Phi^{*}_{(1)} + \Phi^{*}_{(2)} + \cdots \\
\end{align*}$$
QM Expectation for the excitation energy
$$\begin{align*}
\omega_{(0)} &= \left \langle \Phi^{*}_{(0)} | \hat{\mathcal{H}} | \Phi^{*}_{(0)} \right \rangle - \left \langle \Phi^{0}_{(0)} | \hat{\mathcal{H}} | \Phi^{0}_{(0)} \right \rangle \\
\omega_{(1)} &= \left \langle \Phi^{*}_{(0)} | \frac{q_{\alpha}}{r_{i\alpha}} | \Phi^{*}_{(0)} \right \rangle - \left \langle \Phi^{0}_{(0)} | \frac{q_{\alpha}}{r_{i\alpha}} | \Phi^{0}_{(0)} \right \rangle \\
\omega_{(2)} &= \left \langle \Phi^{*}_{(0)} | \frac{q_{\alpha}}{r_{i\alpha}} | \Phi^{*}_{(1)} \right \rangle - \left \langle \Phi^{0}_{(0)} | \frac{q_{\alpha}}{r_{i\alpha}} | \Phi^{0}_{(1)} \right \rangle \\
&\vdots \\
\omega_{(n)} &= \left \langle \Phi^{*}_{(0)} | \frac{q_{\alpha}}{r_{i\alpha}} | \Phi^{*}_{(n-1)} \right \rangle - \left \langle \Phi^{0}_{(0)} | \frac{q_{\alpha}}{r_{i\alpha}} | \Phi^{0}_{(n-1)} \right \rangle \\
\end{align*}$$
Many body system expansion
The QM/MM coupling schemes
$$E_{QM/MM} = E[M] = \left \langle \Psi | \hat{\mathcal{H}} +\sum_{\alpha}\frac{q_{\alpha}}{r_{i}r_{\alpha}} | \Psi \right \rangle + E_{MM}(solvent) + E_{int}(vdW) $$
The total system energy expression making a distinction between the solute M and the solvent molecules {I}
$$E = E[M] + \sum_{I \neq M}{E[I]} + \sum_{I \neq M}{(E[MI]-E[M]-E[I])} + \sum_{I>J\ \& \ i,j \neq M}{(E[IJ]-E[I]-E[J])}$$
The energy difference between the ground($M^{0}$) and a specific excited($M^{*}$) state
$$\begin{align*}
\Delta E = &E[M^{*}] - E[M^{0}] \\
&+ \sum_{I \neq M}{\Delta E[I]} \\
&+ \sum_{I \neq M}{(E[M^{*}I]-E[M^{0}I] -E[M^{*}]+E[M^{0}] -\Delta E[I])} \\
&+ \sum_{I>J\ \& \ i,j \neq M}{(\Delta E[IJ] -\Delta E[I] -\Delta E[J])} \\
\end{align*}$$
$$\text{where } \Delta E[I] = E[I]_{M^{*}} - E[I]_{M^{0}}$$
QM methods for the calculation of electronic spectra
SCC-DFTB : The self-consistent charge-density functional tight-binding method
Density functional tight-binding method
Self-consistent charge–density functional tight-binding
A posteriori treatment for London dispersion in SCC–DFTB
The Effect of the Protein Environment
Born–Oppenheimer approximation
Conical intersections
Excited state molecular dynamics
Diabatic surface hopping
Mixed quantum classical molecular dynamics
The potential energy of the system by MM force-field
$$
V_{MM} = \sum_{i}^{N_{bonds}}V^{bond}_{i}
+ \sum_{j}^{N_{angles}} V^{angle}_{j}
+ \sum_{l}^{N_{torsions}} V^{torsion}_{l}
+ \sum_{i}^{N_{MM}}\sum_{j>i}^{N_{MM}} V^{Coulomb}_{ij}
+ \sum_{i}^{N_{MM}}\sum_{j>i}^{N_{MM}} V^{LJ}_{ij}
$$
QM/MM Methodology and Applications
Energy Expression Schemes
- Subtractive scheme
- Additive scheme
Free Energy Perturbation(FEP)
$$\Delta F(A\rightarrow B) = -k_{B}T\ln{\left < e^{-\frac{E_{B}-E_{A}}{k_{B}T}} \right >_{A}}$$
Thermodynamic Integration(TI)
Potential energy function
$$U(\lambda) = \lambda U_{A} + (1-\lambda)U_{B}$$
The partition function of the system in the canonical ensemble
$$Q(N,V,T,\lambda) = \sum_{s} e^{-\frac{-U_{s}(\lambda)}{k_{B}T}}$$
The free energy of this system
$$F(N,V,T,\lambda) = -k_{B}T\ln{Q(N,V,T,\lambda)}$$
The ensemble average of the derivative of potential energy with respect to $\lambda$
$$\Delta F(A\rightarrow B) = \int_{0}^{1} {\frac{\partial F(\lambda)}{\partial \lambda}}\, d\lambda = \int_{0}^{1} {\frac{\partial U(\lambda)}{\partial \lambda}}\, d\lambda$$
Enhanced sampling
- Multiple time-step sampling
- Umbrella sampling
- Replica Exchange
- Reaction coordinate-driven methods
- Transition path sampling
Reference
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