Author Correction: From a quantum-electrodynamical light–matter description to novel spectroscopies

Nature Reviews Chemistry (2018) https://doi.org/10.1038/s41570-018-0118 Published online 07 March 2018

Equation 1 in the original version of the article (pdf and online) should read:

$$\begin{array}{l}{\hat{H}}_{{\rm{PF}}}\left(t\right)=\sum _{l=1}^{{N}_{e}}\frac{1}{2m}{\left[{{\boldsymbol{\sigma }}}_{l}\cdot \left(-i\hslash {\nabla }_{{{\bf{r}}}_{l}}+\frac{\left|e\right|}{c}{\widehat{{\bf{A}}}}_{\perp }^{{\rm{tot}}}({{\bf{r}}}_{l},t)\right)\right]}^{2}\,\,\,\,+\sum _{l=1}^{{N}_{n}}\left\{\frac{1}{2{M}_{l}}{\left(-i\hslash {\nabla }_{{{\bf{R}}}_{l}}-\frac{{Z}_{l}\left|e\right|}{c}{\widehat{{\bf{A}}}}_{\perp }^{{\rm{tot}}}({{\bf{R}}}_{l},t)\right)}^{2}\right.\\ \left.\,\,\,\,-\frac{{Z}_{l}\left|e\right|\hslash }{2{M}_{l}c}{{\bf{S}}}_{l}^{({n}_{l}/2)}\cdot \left({\nabla }_{{{\bf{R}}}_{l}}\times {\widehat{{\bf{A}}}}_{\perp }^{{\rm{tot}}}({{\bf{R}}}_{l},t)\right)\right\}\\ \,\,\,\,+\frac{1}{2}\sum _{l\ne m}^{{N}_{e}}w\left(\left|{{\bf{r}}}_{l}-{{\bf{r}}}_{m}\right|\right)+\frac{1}{2}\sum _{l\ne m}^{{N}_{n}}{Z}_{l}{Z}_{m}w\left(\left|{{\bf{R}}}_{l}-{{\bf{R}}}_{m}\right|\right)\\ \,\,\,\,-\sum _{l}^{{N}_{e}}\sum _{m}^{{N}_{n}}{Z}_{m}w\left(\left|{{\bf{r}}}_{l}-{{\bf{R}}}_{m}\right|\right)+\sum _{{\bf{k}},{\rm{\lambda }}}\hslash {\omega }_{{\bf{k}}}{\hat{a}}_{{\bf{k}},{\rm{\lambda }}}^{\dagger }{\hat{a}}_{{\bf{k}},{\rm{\lambda }}}\end{array}$$

Pauli Hamiltonian for N_e electrons and N_p nuclei in Box 1 (pdf and online) should read:

$$\begin{array}{l}{\hat{H}}_{{\rm{P}}}\left(t\right)=\sum _{l=1}^{{N}_{e}}\frac{1}{2m}{\left[{{\boldsymbol{\sigma }}}_{l}\cdot \left(-i\hslash {\nabla }_{{{\bf{r}}}_{l}}+\frac{\left|e\right|}{c}{{\bf{A}}}_{\perp }^{{\rm{tot}}}({{\bf{r}}}_{l},t)\right)\right]}^{2}\\ \,\,\,\,+\sum _{l=1}^{{N}_{n}}\left\{\frac{1}{2{M}_{l}}{\left(-i\hslash {\nabla }_{{{\bf{R}}}_{l}}-\frac{{Z}_{l}\left|e\right|}{c}{{\bf{A}}}_{\perp }^{{\rm{tot}}}({{\bf{R}}}_{l},t)\right)}^{2}\right.\\ \,\,\,\,\left.-\frac{{Z}_{l}\left|e\right|\hslash }{2{M}_{l}c}{{\bf{S}}}_{l}^{({n}_{l}/2)}\cdot \left({\nabla }_{{{\bf{R}}}_{l}}\times {{\bf{A}}}_{\perp }^{{\rm{tot}}}({{\bf{R}}}_{l},t)\right)\right\}\\ \,\,\,\,+\frac{1}{2}\sum _{l\ne m}^{{N}_{e}}w\left(\left|{{\bf{r}}}_{l}-{{\bf{r}}}_{m}\right|\right)+\frac{1}{2}\sum _{l\ne m}^{{N}_{n}}{Z}_{l}{Z}_{m}w\left(\left|{{\bf{R}}}_{l}-{{\bf{R}}}_{m}\right|\right)\\ \,\,\,\,-\sum _{l}^{{N}_{e}}\sum _{m}^{{N}_{n}}{Z}_{m}w\left(\left|{{\bf{r}}}_{l}-{{\bf{R}}}_{m}\right|\right)\end{array}$$

The total transversal vector potential in Box 1 (pdf and online) needs to be expressed as:

$${{\bf{A}}}_{\perp }^{{\rm{tot}}}\left({\bf{r}},t\right)={{\bf{A}}}_{\perp }({\bf{r}},t)+{{\bf{A}}}^{{\rm{ext}}}({\bf{r}},t)$$

Maxwell–Kohn–Sham Hamiltonian in Box 2 (pdf and online) should read:

$$\begin{array}{l}{\hat{H}}_{{\rm{MKS}}}\left(t\right)=\sum _{l=1}^{{N}_{e}}\frac{1}{2m}{\left[{{\boldsymbol{\sigma }}}_{l}\cdot \left(-i\hslash {\nabla }_{{{\bf{r}}}_{l}}+\frac{\left|e\right|}{c}({{\bf{A}}}_{\perp }^{{\rm{tot}}}({{\bf{r}}}_{l},t)+{{\bf{A}}}^{{\rm{xc}}}({{\bf{r}}}_{l},t))\right)\right]}^{2}\\ \,\,\,\,+\sum _{l=1}^{{N}_{n}}\left\{\frac{1}{2{M}_{l}}{\left[-i\hslash {\nabla }_{{{\bf{R}}}_{l}}-\frac{{Z}_{l}\left|e\right|}{c}({{\bf{A}}}_{\perp }^{{\rm{tot}}}({{\bf{R}}}_{l},t)+{{\bf{A}}}^{{\rm{xc}}}({{\bf{R}}}_{l},t))\right]}^{2}\right.\\ \,\,\,\,\left.-\frac{{Z}_{l}\left|e\right|\hslash }{2{M}_{l}c}{{\bf{S}}}_{l}^{({n}_{l}/2)}\cdot \left[{\nabla }_{{{\bf{R}}}_{l}}\times ({{\bf{A}}}_{\perp }^{{\rm{tot}}}({{\bf{R}}}_{l},t)+{{\bf{A}}}^{{\rm{xc}}}({{\bf{R}}}_{l},t))\right]\right\}\end{array}$$

Density–current response function in Box 3 (online only) should read as:

$${\chi }_{n,{\bf{J}}}^{\left(1\right)}\left({\bf{r}},t;{{\bf{r}}}^{^{\prime} },{t}^{^{\prime} }\right)=-i\theta (t-{t}^{^{\prime} })\langle \left[{\widehat{n}}_{I}\left({\bf{r}},t\right);{\widehat{{\bf{J}}}}_{I}({{\bf{r}}}^{^{\prime} },{t}^{^{\prime} })\right]\rangle $$

In Box 4 (pdf only), the splitting of the excited electronic states observed in recent molecular experiment has been found proportional to:

$${{\rm{\Omega }}}_{{\bf{R}}}/{\omega }_{k}\approx 0.25$$

The induced density response function in Box 5 (online only) should read:

$${\chi }_{n,{\bf{A}}}\left({\bf{r}},t;{{\bf{r}}}^{^{\prime} },{t}^{^{\prime} }\right)=-i\theta (t-{t}^{^{\prime} })\langle \left[{\widehat{n}}_{I}\left({\bf{r}},t\right);\,{\widehat{{\bf{A}}}}_{I}({{\bf{r}}}^{^{\prime} },{t}^{^{\prime} })\right]\rangle $$

and in Box 5 (online only) the above mode-resolved response function leads to:

$${\chi }_{n,q}\left({\bf{r}},t;\alpha ,{t}^{^{\prime} }\right)=-i\theta (t-{t}^{^{\prime} })\langle [{\widehat{n}}_{I}\left({\bf{r}},t\right);\,{\widehat{q}}_{\alpha I}({t}^{^{\prime} })]\rangle $$