Table 1. Overview of Different Topics and Applications of ML-FFs Discussed in This Section^a.

category	ref	ML-FF	max. Natoms	reference theory
electronic effects	Sauceda et al.70	sGDML	21	CCSD(T), CCSD
	Sauceda et al.214	sGDML	21	CCSD(T), CCSD
	Sauceda et al.69	sGDML	21	PBE, CCSD(T), CCSD
thermodynamics	Morawietz et al.260	BP-NNP	6912	RPBE, BLYP
	Andrade et al.261	DeepMD	426	SCAN
	Deringer et al.262	GAP	1000	LDA
	Behler et al.263	BP-NNP	64	LDA
	Bartok et al.264	GAP	23 496	PW91
	Deringer et al.265	GAP	4096	PW91
	Bonati et al.266	DeepMD	680	SCAN
	Brickel et al.213	PhysNet	6	MP2
reactions	Unke et al.267	RKHS	3	UCCSD(T)
	Denis et al.268	RKHS	3	UCCSD(T)-F12a
	Lu et al.269	PIP-NN	7	UCCSD(T)-F12a
	Sweeny et al.270	PhysNet	7	MP2
	Käser et al.271	PhysNet	7	MP2
	Rivero et al.272	PhysNet	19	M06-2X
	Liu et al.273	BP-NNP	38	RPBE
nuclear quantum effects	Chmiela et al.105	GDML	21	PBE
	Chmiela et al.69	sGDML	21	CCSD, CCSD(T)
	Schütt et al.274	SchNet	20	PBE
	Sauceda et al.161	sGDML	21	CCSD, CCSD(T)
	Hellström et al.275	BP-NNP	1700	RPBE
excited states	Chen et al.276	HDNN	5	CASSCF
	Westermayr et al.277	NN	6	MR-CISD
	Westermayr et al.278	SchNet	6	MR-CISD, CASSCF
spectroscopy	Gastegger et al.71	BP-NNP	209	BLYP, BP86, B2PLYP
	Yao et al.195	BP-NNP	60	ωB97X-D
	Raimbault et al.279	SOAP	80	PBE
	Sommers et al.280	DeepMD	512	SCAN

^aIn all cases, the type of employed ML-FF is given along with the number of atoms of the largest system used to study the respective phenomenon. The basic level of reference theory (neglecting basis sets and dispersion corrections for clarity) is also reported.