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Author: abdonrd

data/variational-problems/hamiltonians/4fe_4s/README.md

@@ -2,4 +2,4 @@
 
 [4Fe–4S] cluster as considered in [_Low-energy spectrum of iron–sulfur clusters directly from many-particle quantum mechanics_](https://www.nature.com/articles/nchem.2041).
 
-The active spaces are obtained from [_Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems_](https://pubs.acs.org/doi/abs/10.1021/acs.jctc.7b00270), which are spanned by Fe[3d] and S[3p] orbitals, derived from a localized Density Functional Theory calculation with BP86 functional, is a TZP-DKH basis, and sf-X2C (spin-free exact two-component) Hamiltonian to include scalar relativistic effects. 
+The active spaces are obtained from [_Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems_](https://pubs.acs.org/doi/abs/10.1021/acs.jctc.7b00270), which are spanned by Fe[3d] and S[3p] orbitals, derived from a localized Density Functional Theory calculation with BP86 functional, is a TZP-DKH basis, and sf-X2C (spin-free exact two-component) Hamiltonian to include scalar relativistic effects.

data/variational-problems/hamiltonians/anderson_impurity_model/README.md

@@ -12,22 +12,22 @@ $$
 H_\textrm{imp.} = \sum_{l,l' = 1,  \sigma \in [ \uparrow, \downarrow]}^L t_{ll'} d^\dagger_{l\sigma} d_{l'\sigma} + U \sum_{ l =1}^L d^\dagger_{l\uparrow}d_{l\uparrow} d^\dagger_{l\downarrow} d_{l\downarrow},
 $$
 
-where $d^\dagger_{l\sigma}/d_{k\sigma}$ are the creation/annihilation operators for impurity mode $l$ with spin $\sigma$. The symmetric matrix with $t_{ll'}$ elements describes the hopping amplitudes between impurities, and its diagonal part is a chemical potential, which is set to $t_{ll} = U/2$.  
+where $d^\dagger_{l\sigma}/d_{k\sigma}$ are the creation/annihilation operators for impurity mode $l$ with spin $\sigma$. The symmetric matrix with $t_{ll'}$ elements describes the hopping amplitudes between impurities, and its diagonal part is a chemical potential, which is set to $t_{ll} = U/2$.
 
 $H_\textrm{bath}$ describes a number $K$ of non-interacting fermionic modes per impurity. Given its non-interacting nature, the bath can always be written in the single-particle basis where it is diagonal:
 
 $$
 H_\textrm{bath} = \sum_{l = 1,  \sigma \in [ \uparrow, \downarrow] }^L \sum_{\mathbf{k} = 1}^K\varepsilon_\mathbf{k} c^\dagger_{\mathbf{k}l\sigma} c_{\mathbf{k}l\sigma}
 $$
 
-where $c^\dagger_{\mathbf{k}l\sigma}/c_{\mathbf{k}l\sigma}$ are the creation/annihilation operators for mode $k$ and spin $\sigma$, associated to impurity $l$. $\varepsilon_\mathbf{k}$ is the energy of each bath mode. 
+where $c^\dagger_{\mathbf{k}l\sigma}/c_{\mathbf{k}l\sigma}$ are the creation/annihilation operators for mode $k$ and spin $\sigma$, associated to impurity $l$. $\varepsilon_\mathbf{k}$ is the energy of each bath mode.
 
 The hybridization term describes the hopping of electrons between impurities and their corresponding bath sites:
 
 $$
 H_\textrm{hyb.} = \sum_{ l = 1, \sigma \in [ \uparrow, \downarrow] }^L \sum_{\mathbf{k} = 1}^K V_\mathbf{k} \left(c^\dagger_{\mathbf{k}l\sigma} d_{l\sigma} + d^\dagger_{l\sigma} c_{\mathbf{k}l\sigma} \right).
 $$
 
-where $V_{\mathbf{k}}$ is the so-called hybridization function. Given band-width of the bath $D = \max_\mathbf{k}(\varepsilon_\mathbf{k})-\min_\mathbf{k}(\varepsilon_\mathbf{k})$, a semicircle-like hybridization function is considered $V_{\mathbf{k}} = V\sqrt{(D/2)^2 - \varepsilon_\mathbf{k}^2}$, with $V$ a parameter that controls the hybridization amplitude. 
+where $V_{\mathbf{k}}$ is the so-called hybridization function. Given band-width of the bath $D = \max_\mathbf{k}(\varepsilon_\mathbf{k})-\min_\mathbf{k}(\varepsilon_\mathbf{k})$, a semicircle-like hybridization function is considered $V_{\mathbf{k}} = V\sqrt{(D/2)^2 - \varepsilon_\mathbf{k}^2}$, with $V$ a parameter that controls the hybridization amplitude.
 
-The parameters considered in this Hamiltnian consist on $L = 4$ impurities and $K = 7$ bath sites per impurity. Half occupation and zero total magnitization are considered. Correspondingly, the number of spinfull orbitals is $32$ and the number of electrons is $32$. The impurity modes are arranged in a square geometry, with hopping amplitudes  $t_{l, l+1} = t = -1$ and $t_{l, l+2} = t'= -0.5$, and a value of $U = 10$. The values of $\varepsilon_\mathbf{k}$ are sampled from a uniform distribution with $\max_\mathbf{k}(\varepsilon_\mathbf{k}) = 2$ and $\min_\mathbf{k}(\varepsilon_\mathbf{k}) = -2$. The hybridization amplitude $V = 0.16$.
+The parameters considered in this Hamiltnian consist on $L = 4$ impurities and $K = 7$ bath sites per impurity. Half occupation and zero total magnitization are considered. Correspondingly, the number of spinfull orbitals is $32$ and the number of electrons is $32$. The impurity modes are arranged in a square geometry, with hopping amplitudes $t_{l, l+1} = t = -1$ and $t_{l, l+2} = t'= -0.5$, and a value of $U = 10$. The values of $\varepsilon_\mathbf{k}$ are sampled from a uniform distribution with $\max_\mathbf{k}(\varepsilon_\mathbf{k}) = 2$ and $\min_\mathbf{k}(\varepsilon_\mathbf{k}) = -2$. The hybridization amplitude $V = 0.16$.