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Atomistic Simulation of Solid-State Actinide Materials

ThUL School 2014 in Actinide Chemistry, June 2-6, KIT, Karlsruhe

Matthias Krack Outline

● Electronic structure methods

● Force field methods

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 2 Electronic structure versus force field methods

Electronic structure methods Force field methods

Motion of nuclei and electrons Motion of atoms

No a priori knowledge of the pre-defined fitted interaction potentials interatomic interactions (empirical potentials)

Dynamic (re)bonding processes Bonding (topology) pre-defined

Predictive Only limited predictive

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 3 Electronic structure versus force field methods

Electronic structure methods Force field methods

Motion of nuclei and electrons Motion of atoms

No a priori knowledge of the pre-defined fitted interaction potentials interatomic interactions (empirical potentials)

Dynamic (re)bonding processes Bonding (topology) pre-defined

Predictive Only limited predictive

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 4 Ab initio methods

First-principle or ab initio methods: ● Directly derived from established laws (first principles) of physics ● No ad-hoc assumptions ● No fitting of model parameters to experimental data

Example: ● Electronic structure methods based on the Schrödinger equation that do not include any fitting parameter, e.g. to experimental data

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 5 Electronic structure theory

● Schrödinger equation: H | = |

� Ψ 〉 𝐸𝐸 Ψ 〉

● Wavefunction of a system of nuclei and electrons: , , … , ; , , … , 𝑁𝑁 𝑛𝑛 Ψ → Ψ 𝒓𝒓1 𝒓𝒓2 𝒓𝒓𝑛𝑛 𝑹𝑹1 𝑹𝑹2 𝑹𝑹𝑁𝑁

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 6 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 7 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 8 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 9 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 10 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 11 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 12 Electronic structure theory

● Hamilton operator in atomic units [a.u.]: H = T + V + T + V + V

� �n �nn �e �ee �ext = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � 𝑗𝑗 𝑖𝑖 − � � 𝐴𝐴 𝑖𝑖 ⋯ ● Atomic units𝑖𝑖 [a.u.]: 𝑖𝑖= 𝑗𝑗>𝑖𝑖 =𝒓𝒓 −=𝒓𝒓 =𝑖𝑖 1𝐴𝐴 𝑹𝑹 − 𝒓𝒓 1 ℏ 𝑚𝑚e 𝑒𝑒 4𝜋𝜋𝜖𝜖0 ( ) ● Laplacian of ( ): ( ) = ( ) = ( ) = 2 2 𝜕𝜕 𝑓𝑓 𝑥𝑥 2 𝑓𝑓 𝑥𝑥 𝛥𝛥𝛥𝛥 𝑥𝑥 𝛻𝛻 𝑓𝑓 𝑥𝑥 𝛻𝛻 ⋅ 𝛻𝛻𝛻𝛻 𝑥𝑥 𝜕𝜕𝑥𝑥

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 13 Controlled approximations

● Ignore further terms (e.g. no external fields) V = V

● Decouple the motion of nuclei and electrons →due� extto their� nelarge mass difference (Born-Oppenheimer approximation): H = H + H = T + V + T + V + V

� �nuc �elec �n �nn �e �ee �ne = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Treat H classically𝑖𝑖 𝑖𝑖 𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 �nuc

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 14 Controlled approximations

● Ignore further terms (e.g. no external fields) V = V

● Decouple the motion of nuclei and electrons →due� extto their� nelarge mass difference (Born-Oppenheimer approximation): H = H + H = T + V + T + V + V

� �nuc �elec �n �nn �e �ee �ne = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Treat H classically𝑖𝑖 𝑖𝑖 𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 �nuc

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 15 Controlled approximations

● Ignore further terms (e.g. no external fields) V = V

● Decouple the motion of nuclei and electrons →due� extto their� nelarge mass difference (Born-Oppenheimer approximation): H = H + H = T + V + T + V + V

� �nuc �elec �n �nn �e �ee �ne = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Treat H classically𝑖𝑖 𝑖𝑖 𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 �nuc

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 16 Controlled approximations

● Ignore further terms (e.g. no external fields) V = V

● Decouple the motion of nuclei and electrons →due� extto their� nelarge mass difference (Born-Oppenheimer approximation): H = H + H = T + V + T + V + V

� �nuc �elec �n �nn �e �ee �ne = + 𝑁𝑁 2 2 𝑁𝑁 𝑁𝑁 𝛻𝛻𝐴𝐴 𝑍𝑍𝐴𝐴𝑍𝑍𝐵𝐵 − � 𝐴𝐴 � � 𝐵𝐵 𝐴𝐴 𝐴𝐴 𝑀𝑀 𝐴𝐴 𝐵𝐵>𝐴𝐴1 𝑹𝑹 − 𝑹𝑹 + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Treat H classically𝑖𝑖 𝑖𝑖 𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 �nuc

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 17 Controlled approximations

● Thus we are left with electronic Hamiltonian: H = T + V + V

�elec �e �ee �ne 1 = + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Calculation of T 𝑖𝑖 and V is𝑖𝑖 straightforwardly𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 possible𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 ● We are still ab �initioe due�ne to physically justified approximations ● Many-body problem: correlated motion of the electrons ● Further accurate and efficient approximations are needed

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 18 Controlled approximations

● Thus we are left with electronic Hamiltonian: H = T + V + V

�elec �e �ee �ne 1 = + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Calculation of T 𝑖𝑖 and V is𝑖𝑖 straightforwardly𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 possible𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 ● We are still ab �initioe due�ne to physically justified approximations ● Many-body problem: correlated motion of the electrons ● Further accurate and efficient approximations are needed

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 19 Controlled approximations

● Thus we are left with electronic Hamiltonian: H = T + V + V

�elec �e �ee �ne 1 = + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Calculation of T 𝑖𝑖 and V is𝑖𝑖 straightforwardly𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 possible𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 ● We are still ab �initioe due�ne to physically justified approximations ● Many-body problem: correlated motion of the electrons ● Further accurate and efficient approximations are needed

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 20 Controlled approximations

● Thus we are left with electronic Hamiltonian: H = T + V + V

�elec �e �ee �ne 1 = + 𝑛𝑛 22 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑁𝑁 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 − � � � − � � 𝐴𝐴 𝑖𝑖 ● Calculation of T 𝑖𝑖 and V is𝑖𝑖 straightforwardly𝑗𝑗>𝑖𝑖 𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 possible𝑖𝑖 𝐴𝐴 𝑹𝑹 − 𝒓𝒓 ● We are still ab �initioe due�ne to physically justified approximations ● Many-body problem: correlated motion of the electrons ● Further accurate and efficient approximations are needed

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 21 Pseudopotential approximation

Physical justification: ● Chemistry is mostly determined by the valence electrons ● Core (inner shell) electrons are chemically rather inert (tightly bound) ● The inner shells are completely filled with electrons spherically symmetric charge distribution (Unsöld theorem)

Core→ electrons can be represented by an effective potential which exhibits the same scattering properties as the (spherical) ⇒ potential of the core electrons

● Actinides are heavy elements with a large number of core electrons

(e.g. a single actinide atom is equivalent to about 10 H2O molecules) ● Their inner core electrons are not tractable with plane waves methods

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 22 Pseudopotential approximation

Advantages: ● Electronic degrees of freedom are significantly reduced ● Less (occupied) electronic wavefunctions (states) ● Size of Hamiltonian matrix is reduced ● Eigenvalue spectrum less spread smaller matrix condition number

● Reduction of the basis set size → ● Faster energy minimisation, i.e. wavefunction optimisation ● Implicit inclusion of relativistic effects

Significant computational speedup while keeping the accuracy

Matthias⇒ Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 23 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Core𝑠𝑠 electrons:𝑓𝑓 𝑑𝑑 86 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 ● Valence𝑠𝑠 𝑠𝑠 electrons:𝑝𝑝 𝑠𝑠 𝑝𝑝6 electrons𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 7 5 6 2 3 1 ● Pseudo𝑠𝑠 𝑓𝑓 atom𝑑𝑑 configuration: 86 + 6 electrons Rn 7 5 6 2 3 1 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 24 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Core𝑠𝑠 electrons:𝑓𝑓 𝑑𝑑 86 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 ● Valence𝑠𝑠 𝑠𝑠 electrons:𝑝𝑝 𝑠𝑠 𝑝𝑝6 electrons𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 7 5 6 2 3 1 ● Pseudo𝑠𝑠 𝑓𝑓 atom𝑑𝑑 configuration: 86 + 6 electrons Rn 7 5 6 2 3 1 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 25 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Core𝑠𝑠 electrons:𝑓𝑓 𝑑𝑑 86 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 ● Valence𝑠𝑠 𝑠𝑠 electrons:𝑝𝑝 𝑠𝑠 𝑝𝑝6 electrons𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 7 5 6 2 3 1 ● Pseudo𝑠𝑠 𝑓𝑓 atom𝑑𝑑 configuration: 86 + 6 electrons Rn 7 5 6 2 3 1 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 26 Electronic energy levels of the uranium atom

All-electron Valence electrons Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 27 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Small𝑠𝑠 -𝑓𝑓core𝑑𝑑 pseudo atom configuration: 60 + 32 electrons Kr 4 4 5 5 5 6 6 7 5 6 10 14 2 6 10 2 6 2 3 1 ● Medium𝑑𝑑 -core𝑓𝑓 pseudo𝑠𝑠 atom𝑝𝑝 𝑑𝑑 configuration:𝑠𝑠 𝑝𝑝 𝑠𝑠78 +𝑓𝑓 14 𝑑𝑑electrons Xe 4 5 6 6 7 5 6 14 10 2 6 2 3 1 𝑓𝑓 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 28 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Small𝑠𝑠 -𝑓𝑓core𝑑𝑑 pseudo atom configuration: 60 + 32 electrons Kr 4 4 5 5 5 6 6 7 5 6 10 14 2 6 10 2 6 2 3 1 ● Medium𝑑𝑑 -core𝑓𝑓 pseudo𝑠𝑠 atom𝑝𝑝 𝑑𝑑 configuration:𝑠𝑠 𝑝𝑝 𝑠𝑠78 +𝑓𝑓 14 𝑑𝑑electrons Xe 4 5 6 6 7 5 6 14 10 2 6 2 3 1 𝑓𝑓 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 29 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Small𝑠𝑠 -𝑓𝑓core𝑑𝑑 pseudo atom configuration: 60 + 32 electrons Kr 4 4 5 5 5 6 6 7 5 6 10 14 2 6 10 2 6 2 3 1 ● Medium𝑑𝑑 -core𝑓𝑓 pseudo𝑠𝑠 atom𝑝𝑝 𝑑𝑑 configuration:𝑠𝑠 𝑝𝑝 𝑠𝑠78 +𝑓𝑓 14 𝑑𝑑electrons Xe 4 5 6 6 7 5 6 14 10 2 6 2 3 1 𝑓𝑓 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 30 Electronic energy levels of the uranium atom

Small-core Medium-core Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 31 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Small𝑠𝑠 -𝑓𝑓core𝑑𝑑 pseudo atom configuration: 60 + 32 electrons Kr 4 4 5 5 5 6 6 7 5 6 10 14 2 6 10 2 6 2 3 1 ● Medium𝑑𝑑 -core𝑓𝑓 pseudo𝑠𝑠 atom𝑝𝑝 𝑑𝑑 configuration:𝑠𝑠 𝑝𝑝 𝑠𝑠78 +𝑓𝑓 14 𝑑𝑑electrons Xe 4 5 6 6 7 5 6 14 10 2 6 2 3 1 𝑓𝑓 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 32 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Small𝑠𝑠 -𝑓𝑓core𝑑𝑑 pseudo atom configuration: 60 + 32 electrons Kr 4 4 5 5 5 6 6 7 5 6 10 14 2 6 10 2 6 2 3 1 ● Medium𝑑𝑑 -core𝑓𝑓 pseudo𝑠𝑠 atom𝑝𝑝 𝑑𝑑 configuration:𝑠𝑠 𝑝𝑝 𝑠𝑠78 +𝑓𝑓 14 𝑑𝑑electrons Xe 4 5 6 6 7 5 6 14 10 2 6 2 3 1 𝑓𝑓 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 33 Pseudopotential approximation

Example: Uranium atom ● All-electron configuration: 92 electrons 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 2 5 2 6 6 2 6 10 2 6 10 14 2 6 10 2 6 𝑠𝑠2 𝑠𝑠 3 𝑝𝑝 1 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑓𝑓 𝑠𝑠 𝑝𝑝 𝑑𝑑 𝑠𝑠 𝑝𝑝 ● Small𝑠𝑠 -𝑓𝑓core𝑑𝑑 pseudo atom configuration: 60 + 32 electrons Kr 4 4 5 5 5 6 6 7 5 6 10 14 2 6 10 2 6 2 3 1 ● Medium𝑑𝑑 -core𝑓𝑓 pseudo𝑠𝑠 atom𝑝𝑝 𝑑𝑑 configuration:𝑠𝑠 𝑝𝑝 𝑠𝑠78 +𝑓𝑓 14 𝑑𝑑electrons Xe 4 5 6 6 7 5 6 14 10 2 6 2 3 1 𝑓𝑓 𝑑𝑑 𝑠𝑠 𝑝𝑝 𝑠𝑠 𝑓𝑓 𝑑𝑑

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 34 Projector augmented-wave (PAW) method

● The PAW method provides a generalisation of the pseudopotential method ● Pseudo wavefunction coincides with the all-electron wavefunction only beyond a cut-off radius

● Smaller values result𝑟𝑟cut in harder pseudopotentials ● Correct nodal𝑟𝑟cut behaviour of the valence electron wavefunctions is retained ● Inclusion of semi-core states into the valence part is straightforwardly possible ● Orthogonality between core and valence states is ensured ● Frozen-core all-electron method

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 35 Projector augmented-wave (PAW) method

● A decomposition of the all-electron density in hard and soft contributions is performed ( is the radiusall−electron of the circles): 𝜌𝜌

𝑟𝑟cut

= 𝑁𝑁 + 𝑁𝑁 all−electron soft soft hard 𝜌𝜌 𝜌𝜌 − � 𝜌𝜌𝐴𝐴 � 𝜌𝜌𝐴𝐴 𝐴𝐴 𝐴𝐴

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 36 Example: Medium-core pseudopotential for uranium

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 37 Solving the Schrödinger equation

● Expand the wavefunctions into a basis

● Recast Schrödinger’s differential𝜓𝜓𝑖𝑖 equation in matrix formulation as an eigenvalue equation: =

: Wavefunction coefficients𝐇𝐇 𝐂𝐂 𝐒𝐒 𝐂𝐂 𝜖𝜖 𝐂𝐂: Overlap matrix ( = for an orthogonal basis) 𝐒𝐒: Energy eigenvalues𝐒𝐒 𝐈𝐈 ● Solve𝜖𝜖 the eigenvalue problem iteratively using diagonalisation or a direct minimiser method ● Self Consistent Field (SCF) method: ( )

𝐇𝐇 → 𝐇𝐇 𝐂𝐂 Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 38 Solving the Schrödinger equation

1) Atomic configuration: isolated or (1,2,3)D-periodic system 2) Define an initial guess for the electron density 3) Calculate the Hamiltonian matrix 4) Solve the Schrödinger equation to obtain the new wavefunction 𝐇𝐇 coefficients 5) Calculate the new electron density from the wavefunction coefficients 𝐂𝐂 6) Check convergence criterion (e.g. change of energy or density): ● Converged = no return to (3) ● Converged = yes exit cycle and proceed with (7) → 7) Post-processing: Optionally→ calculate various properties and/or the forces on the atoms

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 39 Solving the Schrödinger equation

Given the basis functions :

𝜙𝜙𝑖𝑖 T + V = | + | 𝑛𝑛 22 𝑁𝑁 | | 𝛻𝛻𝑖𝑖 𝑍𝑍𝐴𝐴 𝑖𝑖 �e �ext 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝜙𝜙 𝜙𝜙 − �⟨𝜙𝜙 � 𝐴𝐴 𝑖𝑖 𝜙𝜙 〉 𝑖𝑖 𝐴𝐴 𝑅𝑅 − 𝑟𝑟 Possible pathways for the calculation of :

● Wavefunction based methods 𝑉𝑉ee . Ab initio (from first principles) . Empirical (parameterised) ● Density based methods

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 40 Wavefunction based methods (ab initio)

● Hartree-Fock methods (HF): Single determinant approximation ● Møller-Plesset perturbation theory (MPn): Perturbative treatment of the electron correlation starting from the HF orbitals (e.g. MP2) ● Configuration interaction methods (CI): Linear combination of different electronic configurations (e.g. CISD) ● Coupled cluster theory (CC): Exponential approach providing size consistency and extensivity, = (e.g. CCSD) ● Multi-reference methods (MR): Multiple𝑇𝑇 reference determinants (states) Ψ 𝑒𝑒 ΨHF are employed (e.g. MR-CISD or MR-CCSD) ● n-electron valence state perturbation theory (NEVPT): MP2 type expansion based on a CASSCF reference (e.g. CASPT2) ● R12 methods ( Geminals): Beyond the HF approximation using two-electron instead of one-electron functions (e.g. CCSD-R12) Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 41 Wavefunction based methods (empirical)

● Extended Hückel theory (EHT) ● Semiempirical methods: The zero differential overlap (ZDO) approximation is applied to different degree: CNDO, INDO, NDDO (AM1, MNDO, PM3) ● Empirical tight-binding methods (TB): Approximate wavefunctions derived from a set of atomic orbitals, parameterised (hopping) integrals ● However, mostly applicable for main group element chemistry

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 42 Density Functional Theory (DFT)

● The positive definite electron density of a system is given by

= … = , , … , … 𝑛𝑛 2 2 𝜌𝜌 𝒓𝒓reduction𝑛𝑛 � ofΨ the𝒓𝒓 problem𝑑𝑑𝒓𝒓2 complexity𝑑𝑑𝒓𝒓𝑛𝑛 𝑛𝑛 � fromΨ 𝒓𝒓3 𝒓𝒓 2to 3 degrees𝒓𝒓𝑛𝑛 𝑑𝑑𝒓𝒓 of2 freedom𝑑𝑑𝒓𝒓𝑛𝑛

● ⇒It provides the key connections 𝑛𝑛 electron wavefunction ( ) 𝑛𝑛 𝑛𝑛 Ψ 𝒓𝒓 Hohenberg-Kohn (HK) one-electron density ( ) ⇕ 𝜌𝜌 𝒓𝒓 Kohn-Sham (KS) one-electron spin orbitals ⇕( ) 2 𝑖𝑖 Matthias Krack𝑛𝑛 (PSI) ThUL School 2014 in Actinide� Chemistry𝜓𝜓 (June 2-6,𝒓𝒓 2014, KIT, Karlsruhe) 43 𝑖𝑖 Pros and Cons of the methods

● Wavefunction based methods like Hartree-Fock, CI, CC ● accuracy can be improved systematically ● accurate, but often computationally demanding ● Density based methods like Kohn-Sham density functional theory ● efficient and most of the times sufficiently accurate ● accuracy improvement is less straightforward

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 44 Density Functional Theory (DFT)

Hohenberg-Kohn Theorem 1: ● P. Hohenberg and W. Kohn, Phys. Rev. B 136, B864-B871 (1964)

● Theorem 1: The external potential ( ), and hence the total energy of a system, is univocally determined by the electron density ( ) ext within a trivial additive constant. 𝑉𝑉 𝒓𝒓 𝜌𝜌 𝒓𝒓 One-to-one correspondence: ( ) ( )

→ 𝑉𝑉ext 𝒓𝒓 ⇔ 𝜌𝜌 𝒓𝒓

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 45 Density Functional Theory (DFT)

Hohenberg-Kohn Theorem 2: ● P. Hohenberg and W. Kohn, Phys. Rev. B 136, B864-B871 (1964)

● Theorem 2: The energy [ ] given by a trial positive definite electron density (normalised to the number of electrons ) is an upper bound to the exact ground𝐸𝐸 state𝜌𝜌 𝒓𝒓 energy , provided that the exact functional is used.𝜌𝜌 𝑛𝑛 0 𝐸𝐸 Variational principle: = < with ( ) → 𝐸𝐸0 𝐸𝐸 𝜌𝜌0 𝒓𝒓 𝐸𝐸 𝜌𝜌 𝒓𝒓 𝜌𝜌 𝒓𝒓 ≠ 𝜌𝜌0 𝒓𝒓

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 47 Density Functional Theory (DFT)

Kohn-Sham principle: ● The intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential: H = T + V

● Solve the so-called Kohn�R-Sham𝒓𝒓 equations�R �R of𝒓𝒓 this auxiliary non- interacting reference system H =

● where the local potential�R has𝒓𝒓 𝜓𝜓to𝑖𝑖 be𝒓𝒓 chosen𝜖𝜖𝑖𝑖 𝜓𝜓 such𝑖𝑖 𝒓𝒓 that

= 𝑛𝑛 ( ) 2 R 𝑖𝑖 Matthias Krack (PSI) 𝜌𝜌 ThUL 𝒓𝒓School 2014 in Actinide� Chemistry𝜓𝜓 (June𝒓𝒓 2-6, 2014,≡ KIT, Karlsruhe)𝜌𝜌 𝒓𝒓 49 𝑖𝑖 Exchange-correlation functionals

● DFT’s “Jacob’s ladder” of approximation to the heaven of chemical accuracy, i.e. John Perdew’s view of the development of exchange- correlation functionals:

Rung Explicit dependence on Functional type 5 unoccupied orbitals generalised RPA 4 occupied orbitals hyper-GGA, hybrid-GGA 3 kinetic energy density, Laplacian meta-GGA 2 gradient of the electron density GGA 1 local density only LDA, LSDA 0 Hartree world

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 50 Exchange-correlation functionals (rung 1)

● DFT’s “Jacob’s ladder” of approximation to the heaven of chemical accuracy, i.e. John Perdew’s view of the development of exchange- correlation functionals:

Rung Explicit dependence on Functional type 5 unoccupied orbitals generalised RPA 4 occupied orbitals hyper-GGA, hybrid-GGA 3 kinetic energy density, Laplacian meta-GGA 2 gradient of the electron density GGA 1 local density only LDA, LSDA 0 Hartree world

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 51 Exchange-correlation functionals (rung 1)

Local density approximation (LDA): ● = ● Examples:LDA Vosko-Wilk-Nusair (VWN), Perdew-Zunger (PZ) 𝐸𝐸xc 𝜌𝜌 ∫ 𝜖𝜖xc 𝜌𝜌 𝒓𝒓 𝜌𝜌 𝒓𝒓 𝑑𝑑𝒓𝒓 ● Molecular structures ● Vibrational frequencies ● Phonons in solids ● Computationally cheaper than Hartree-Fock ● Binding energies are too large (overbinding) ● Lattice constants are too small ● Reaction barriers are underestimated ● Poor performance for electron lone pairs and multiple bonds

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 52 Exchange-correlation functionals (rung 2)

● DFT’s “Jacob’s ladder” of approximation to the heaven of chemical accuracy, i.e. John Perdew’s view of the development of exchange- correlation functionals:

Rung Explicit dependence on Functional type 5 unoccupied orbitals generalised RPA 4 occupied orbitals hyper-GGA, hybrid-GGA 3 kinetic energy density, Laplacian meta-GGA 2 gradient of the electron density GGA 1 local density only LDA, LSDA 0 Hartree world

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 53 Exchange-correlation functionals (rung 2)

Generalised gradient approximation (GGA): ● = , ( ) ● Examples:GGA Becke-Lee-Yang-Parr (BLYP) 𝐸𝐸xc 𝜌𝜌 ∫ 𝜖𝜖xc 𝜌𝜌 𝒓𝒓 𝐹𝐹xc 𝜌𝜌 𝒓𝒓 𝛻𝛻𝜌𝜌 𝒓𝒓 𝜌𝜌 𝒓𝒓 𝑑𝑑𝒓𝒓 Perdew-Burke-Ernzerhof (PBE, revPBE) ● Computationally only slightly more expensive than LDA ● Improvements with respect to LDA: . Bond lengths and lattice constants . Binding energies and atomic energies . Vibrational frequencies are of similar quality ● Binding energies are still too large ● Band gaps are only slightly improved ● Strongly correlated systems like actinide compounds

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 54 Exchange-correlation functionals (rung 3)

● DFT’s “Jacob’s ladder” of approximation to the heaven of chemical accuracy, i.e. John Perdew’s view of the development of exchange- correlation functionals:

Rung Explicit dependence on Functional type 5 unoccupied orbitals generalised RPA 4 occupied orbitals hyper-GGA, hybrid-GGA 3 kinetic energy density, Laplacian meta-GGA 2 gradient of the electron density GGA 1 local density only LDA, LSDA 0 Hartree world

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 55 Exchange-correlation functionals (rung 3)

Meta-GGA functionals: ● In addition to the gradient of the electron density its Laplacian ( ) and/or the kinetic energy density ( ) is considered: 2 𝛻𝛻 𝜌𝜌 𝒓𝒓 = 𝜏𝜏 𝒓𝒓, , , ( ) mGGA 2 ●𝐸𝐸Example:xc 𝜌𝜌 Tao�-Perdew𝜖𝜖xc 𝜌𝜌-Staroverov𝒓𝒓 𝐹𝐹xc 𝜌𝜌 -𝒓𝒓Scuseria𝛻𝛻𝜌𝜌 𝒓𝒓 (TPSS,𝛻𝛻 𝜌𝜌 𝒓𝒓revTPSS𝜏𝜏 𝒓𝒓 ),𝜌𝜌 M06𝒓𝒓 -𝑑𝑑L𝒓𝒓 ● Often only minor improvements compared to plain GGA ● Larger computational cost, since ( ) and/or are required 2 𝛻𝛻 𝜌𝜌 𝒓𝒓 𝜏𝜏 𝒓𝒓

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 56 Exchange-correlation functionals (rung 4)

● DFT’s “Jacob’s ladder” of approximation to the heaven of chemical accuracy, i.e. John Perdew’s view of the development of exchange- correlation functionals:

Rung Explicit dependence on Functional type 5 unoccupied orbitals generalised RPA 4 occupied orbitals hyper-GGA, hybrid-GGA 3 kinetic energy density, Laplacian meta-GGA 2 gradient of the electron density GGA 1 local density only LDA, LSDA 0 Hartree world

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 57 Exchange-correlation functionals (rung 4)

Hybrid functionals: ● A portion of the exact Hartree-Fock exchange is mixed in: = + + B3LYP LDA HF LDA + GGA LDA 𝐸𝐸xc 𝐸𝐸xc 𝑎𝑎0 𝐸𝐸x − 𝐸𝐸x 𝑎𝑎𝑥𝑥 𝐸𝐸x − 𝐸𝐸x ● Examples: Becke-Lee-Yang-Parr (B3LYP) GGA LDA 𝑎𝑎𝑐𝑐 𝐸𝐸c − 𝐸𝐸c Perdew-Burke-Ernzerhof (PBE0) Minnesota functionals (e.g. M05, M06-HF) ● Self-interaction error (SIE) is reduced ● Improved bond lengths and atomisation energies ● Band gaps are significantly improved ● Mixing parameters for the exact exchange required ● Significantly more expensive than GGA (10-100 times)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 58 Exchange-correlation functionals (rung 4-5)

Double-hybrid density functionals (DHDF): ● Portions of the exact Hartree-Fock exchange and the MP2 correlation are mixed in: = 1 + + 1 + ● ExamplesDHDF: B2-PLYP, mPW2-PLYP (HFGrimme) MP2 𝐸𝐸xc − 𝑎𝑎x 𝐸𝐸x 𝑎𝑎x𝐸𝐸x − 𝑎𝑎c 𝐸𝐸c 𝑎𝑎c𝐸𝐸c ● Improves non-covalent interaction energies, e.g. intramolecular dispersion effects ● MP2 is available for condensed phase systems ● Mixing parameters required ● Computationally even more expensive than hybrid functionals ● MP2 needs larger basis sets

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 59 Exchange-correlation functionals (rung 5)

● DFT’s “Jacob’s ladder” of approximation to the heaven of chemical accuracy, i.e. John Perdew’s view of the development of exchange- correlation functionals:

Rung Explicit dependence on Functional type 5 unoccupied orbitals generalised RPA 4 occupied orbitals hyper-GGA, hybrid-GGA 3 kinetic energy density, Laplacian meta-GGA 2 gradient of the electron density GGA 1 local density only LDA, LSDA 0 Hartree world

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 60 Exchange-correlation functionals

Pending deficiencies: ● Wrong asymptotic behaviour (no Rydberg states) ● Band gaps and excitation energies are unsatisfactory ● Insufficient self-interaction correction (SIC) ● Description of van der Waals interactions ● Bad description of strongly correlated system: . d orbitals of transition metals . f orbitals of lanthanides and actinides

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 61 Dispersion corrections

● DFT-D: Inclusion of a classical (empirical) dispersion term based on atomic pair potentials (e.g. term) . Computationally cheap 𝐶𝐶6 . Density of water is significantly improved . Empirical fit is functional dependent ● Non-local dispersion correction: van der Waals density functionals (vdW-DF) methods: . = + E + E . DionvdW-RydbergLDA-SchröderGGA-Langrethnon-−Lundqvistlocal (DRSLL) 𝐸𝐸xc 𝐸𝐸x c c . Lee-Murray-Kong-Lundqvist-Langreth (LMKLL) . Klimes-Bowler-Michaelides (optB88) . Parameter-free, correct asymptotic behaviour . Further improvement not straightforwardly feasible

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 62 DFT+U correction

● Hubbard correction term is added to the energy functional as an atomic on-site term ● Explicit correction applied for the and/or orbitals of a specific atomic kind (e.g. cobalt or uranium) 𝑑𝑑 𝑓𝑓 ● Computationally cheap ● The parameter U is usually treated as an empirical fitting parameter ● Fixed U parameter does not adapt to the chemical environment (e.g. change of oxidation state)

On-going development, e.g. dynamical mean field theory (DFT-DMFT)

⇒

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 63 Dynamical mean field theory (DMFT)

● Solver for the Anderson impurity model required ● Computationally very expensive ● First DFT-DMFT calculations have been performed for actinides

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 64 Periodic boundary conditions (PBC)

The wavefunction of an electron in an external periodic potential = ( + )

can be written as the product𝑣𝑣 of𝒓𝒓 a function𝑣𝑣 𝒓𝒓 with𝑳𝑳 the same periodicity = ( ) 𝑖𝑖𝒌𝒌⋅𝒓𝒓 with 𝜓𝜓𝒌𝒌 𝒓𝒓 𝑒𝑒 𝑢𝑢𝒌𝒌 𝒓𝒓 = ( + )

From this follows Bloch’s theorem𝑢𝑢𝒌𝒌 𝒓𝒓 𝑢𝑢𝒌𝒌 𝒓𝒓 𝑳𝑳 + = ( ) 𝑖𝑖𝒌𝒌⋅𝒂𝒂𝑖𝑖 where is a vector in the𝜓𝜓𝒌𝒌 first𝒓𝒓 Brillouin𝑳𝑳 𝑒𝑒zone. 𝜓𝜓𝒌𝒌 𝒓𝒓 𝒌𝒌

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 65 Periodic boundary conditions (PBC)

The periodically repeated electronic density = 2 ( ) 2 The wavefunction is not periodic𝜌𝜌𝒌𝒌 𝒓𝒓 in the unit𝜓𝜓𝒌𝒌 cell,𝒓𝒓 unless = , but it is periodic in a larger supercell 𝒌𝒌 𝟎𝟎 + = ( ) 𝑖𝑖𝑖𝑖𝒌𝒌⋅𝒂𝒂𝑖𝑖 From this a general expression𝜓𝜓𝒌𝒌 𝒓𝒓 𝑛𝑛 for𝒂𝒂𝑖𝑖 the electronic𝑒𝑒 𝜓𝜓 density𝒌𝒌 𝒓𝒓

= ( ) 2 𝜌𝜌 𝒓𝒓 � 𝜔𝜔𝒌𝒌 𝜓𝜓𝒌𝒌 𝒓𝒓 with the symmetry weight factors 𝒌𝒌∈BZ can be obtained

𝜔𝜔𝒌𝒌

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 66 Basis function types

● Numerical atomic orbital functions (NAO) ● Slater-type atomic orbital functions (STO) ● Gaussian-type atomic orbital functions (GTO) ● Plane waves (PW) ● Wavelets ● Real-space grids

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 67 Gaussian-type orbital functions

● Primitive Cartesian Gaussian function | at atom :

= , , , = 𝒂𝒂〉 𝐴𝐴 2 𝑎𝑎𝑥𝑥 𝑎𝑎𝑦𝑦 𝑎𝑎𝑧𝑧 −𝜁𝜁𝑎𝑎 𝒓𝒓−𝑨𝑨 Angular𝒂𝒂 𝜒𝜒momentum𝑎𝑎 𝒓𝒓 𝜁𝜁𝑎𝑎 𝒂𝒂 indices:𝑨𝑨 𝑁𝑁 c 𝑥𝑥 − 𝐴𝐴=𝑥𝑥 ,𝑦𝑦 −, 𝐴𝐴𝑥𝑥 with𝑧𝑧 −=𝐴𝐴𝑧𝑧 + 𝑒𝑒 +

Electronic coordinates: 𝒂𝒂 = 𝑎𝑎,𝑥𝑥 ,𝑎𝑎𝑦𝑦 𝑎𝑎𝑧𝑧 𝑙𝑙 𝑎𝑎𝑥𝑥 𝑎𝑎𝑦𝑦 𝑎𝑎𝑧𝑧 Nuclear coordinates of atom : 𝒓𝒓 = (𝑥𝑥 𝑦𝑦, 𝑧𝑧 , ) 𝑥𝑥 𝑦𝑦 𝑧𝑧 ● Differential relations: 𝐴𝐴 𝑨𝑨 𝐴𝐴 𝐴𝐴 𝐴𝐴

= 2 + = 𝜕𝜕 𝜕𝜕 𝒂𝒂 𝜁𝜁𝑎𝑎 𝒂𝒂 𝟏𝟏𝑖𝑖 − 𝑁𝑁𝑖𝑖 𝒂𝒂 𝒂𝒂 − 𝟏𝟏𝑖𝑖 − 𝒂𝒂 with = ( 𝜕𝜕𝐴𝐴,𝑖𝑖 , ) and = , , 𝜕𝜕𝑟𝑟𝑖𝑖

𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ● Horizontal𝟏𝟏 𝛿𝛿 and𝛿𝛿 vertical𝛿𝛿 recurrence𝑖𝑖 𝑥𝑥 𝑦𝑦 𝑧𝑧 relations (e.g. see Obara and Saika)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 68 Gaussian-type orbital functions: Pros and Cons

● Good results already for small basis sets ● Correspondence to the intuitive chemical picture ● All-electron description possible ● Can be tuned for each application (and even each atom) ● No implicit periodicity ● Non-orthogonal ● Depend on the atomic positions (Pulay forces) ● Numerical problems due to over-completeness ● Basis set superposition error (BSSE) ● Systematic improvement is less straightforward

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 69 Plane waves (PW)

● Plane wave function for a cell of volume : 1 = Ω 𝑖𝑖𝒈𝒈⋅𝒓𝒓 ● Expansion of the periodic𝜙𝜙 functions𝒈𝒈 𝑒𝑒 1 Ω 1 = ( ) = ( ) 𝑖𝑖𝒈𝒈⋅𝒓𝒓 𝑖𝑖𝒌𝒌 𝑖𝑖𝒌𝒌 𝑖𝑖 𝑖𝑖𝒌𝒌 in plane𝑢𝑢 waves𝒓𝒓 results� in 𝑐𝑐the Kohn𝒈𝒈 𝜙𝜙-Sham𝒈𝒈 orbitals� 𝑐𝑐 𝒈𝒈 𝑒𝑒 Ω 𝒈𝒈 Ω 𝒈𝒈 1 = = 𝑖𝑖𝒌𝒌⋅𝒓𝒓 −𝑖𝑖 𝒈𝒈+𝒌𝒌 ⋅𝒓𝒓 𝑖𝑖𝒌𝒌 𝑖𝑖𝒌𝒌 𝑖𝑖𝒌𝒌 and the𝜓𝜓 electron𝒓𝒓 density𝑒𝑒 𝑢𝑢 𝒓𝒓 � 𝑐𝑐 𝒈𝒈 𝑒𝑒 Ω 𝒈𝒈 1 = = ( ) , ∗ ′ −𝑖𝑖 𝒈𝒈+𝒌𝒌 ⋅𝒓𝒓 −𝑖𝑖𝒈𝒈⋅𝒓𝒓 𝑖𝑖 𝑖𝑖𝒌𝒌 𝑖𝑖𝒌𝒌 𝜌𝜌� 𝒓𝒓 � �𝑓𝑓 𝒌𝒌 𝑑𝑑𝒌𝒌 �′ 𝑐𝑐 𝒈𝒈 𝑐𝑐 𝒈𝒈 𝑒𝑒 � 𝜌𝜌� 𝒈𝒈 𝑒𝑒 𝑖𝑖 𝒈𝒈 Matthias Krack (PSI) Ω ThUL School𝒈𝒈 2014𝒈𝒈 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 70 Plane waves: Pros and Cons

● Orthonormal ● Independent of the atomic positions (no Pulay forces) ● No basis set superposition error (BSSE) ● Systematic improvement simply by increasing the cut-off ● Implicit periodicity ● No selective tuning possible ● Large number of basis functions is needed ● Pseudopotentials / frozen-core approximation required ● Chemical information not directly accessible

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 71 Hybrid basis sets

● Multiple representations of the electronic charge density are concurrently available: . : density matrix (atom-centred Gaussian-type orbital basis set)

. 𝐏𝐏( ): real-space density (auxiliary plane wave basis set) . 𝜌𝜌� ̃(𝒓𝒓g): g-space density (auxiliary plane wave basis set) ● Efficient𝜌𝜌 solution of the Coulomb problem is enabled:

= −1 collocate FFT FFT integrate 𝜌𝜌� 𝒈𝒈 𝐏𝐏 𝜌𝜌� 𝒓𝒓 𝜌𝜌� 𝒈𝒈 → 𝑉𝑉H( 𝒈𝒈 ) 2 𝑉𝑉H 𝒓𝒓 𝐕𝐕H 𝑔𝑔 ● Hartree potential build scales 𝑂𝑂(quasi)𝑛𝑛 log 𝑛𝑛 linearly ● Attempts to combine the best of two worlds

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 72 Electronic structure codes

● Selection of popular codes suited for condensed phase systems:

GPL or academic license Commercial license ABINIT (PW, PAW) ADF (STO) BigDFT (Wavelets) CASTEP (PW) CP2K (Hybrid GPW, GAPW) CRYSTAL (GTO) CPMD (PW) DMol3 (NAO) DACAPO (PW) FHI-aims (NAO) FLEUR (FP-LAPW) Gaussian (GTO) GPAW (real-space grids) ONETEP (PW) OpenMP (PW) VASP (PW, PAW) PWSCF (PW) Wien2k (FP-LAPW) SIESTA (NAO) ELK (FP-LAPW)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 73 Electronic structure versus force field methods

Electronic structure methods Force field methods

Motion of nuclei and electrons Motion of atoms

No a priori knowledge of the pre-defined fitted interaction potentials interatomic interactions (empirical potentials)

Dynamic (re)bonding processes Bonding (topology) pre-defined

Predictive Only limited predictive

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 74 (MM)

● Static description using classical (empirical) force field methods

● Structure optimisation by a minimisation based on energy gradients

● Cell optimisation based on a minimisation of the stress tensor elements

● Reaction path optimisation, e.g. using the nudged elastic band (NEB) method

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 75 (MD)

● The force acting on an atom is obtained by

=𝑖𝑖 = 𝜕𝜕𝑉𝑉 𝒓𝒓 𝒇𝒇𝑖𝑖 𝒓𝒓 − 𝑚𝑚𝑖𝑖𝒂𝒂𝑖𝑖 ● provides the atomic accelerations by𝜕𝜕𝒓𝒓 𝑖𝑖Newton’s law ● A small time step is used to numerically integrate the equations of motion (EOM) ● Symplectic, time-reversible integrator of Verlet (Velocity Verlet scheme) ● Various thermodynamical ensembles are enabled by applying thermostats and barostats to the system:

• Microcanonical ensemble (NVE) • Canonical ensemble (NVT) • Isobaric-isothermal ensemble (NpT)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 76 Classical force fields

● Potential for a system consisting of atoms ( ) = ( , , … , ) 𝑛𝑛 ● The potential energy may𝑉𝑉 𝒓𝒓be divided𝑉𝑉 𝒓𝒓1 𝒓𝒓into2 𝒓𝒓𝑛𝑛

( ) = 𝑛𝑛 ( ) + 𝑛𝑛 𝑛𝑛 ( , ) + 𝑛𝑛 𝑛𝑛 𝑛𝑛 ( , , ) +

𝑉𝑉 𝒓𝒓 � 𝑣𝑣1 𝒓𝒓𝑖𝑖 � � 𝑣𝑣2 𝒓𝒓𝑖𝑖 𝒓𝒓𝑗𝑗 � � � 𝑣𝑣3 𝒓𝒓𝑖𝑖 𝒓𝒓𝑗𝑗 𝒓𝒓𝑘𝑘 ⋯ ● Four-body𝑖𝑖 and higher𝑖𝑖 terms𝑗𝑗>𝑖𝑖 are small and𝑖𝑖 can𝑗𝑗>𝑖𝑖 𝑘𝑘be>𝑗𝑗 >neglected𝑖𝑖 ● Implicit inclusion of average three-body effects into an effective pair potential

( ) 𝑛𝑛 ( ) + 𝑛𝑛 𝑛𝑛 ( ) eff 𝑉𝑉 𝒓𝒓 ≈ � 𝑣𝑣1 𝒓𝒓𝑖𝑖 � � 𝑣𝑣2 𝑟𝑟𝑖𝑖𝑖𝑖 𝑖𝑖 = 𝑖𝑖 𝑗𝑗 >𝑖𝑖

Matthias Krack (PSI) ThUL School 2014 𝑟𝑟in𝑖𝑖 Actinide𝑗𝑗 Chemistry𝒓𝒓𝑖𝑖 − (June𝒓𝒓 2𝑗𝑗-6, 2014, KIT, Karlsruhe) 77 Classical force fields

● The effective pair potential can be decomposed in a long- and a short- range part

= + eff 𝑣𝑣2 𝑟𝑟𝑖𝑖𝑖𝑖 = 𝑣𝑣0lr 𝑟𝑟 𝑖𝑖 𝑖𝑖 for 𝑣𝑣sr >𝑟𝑟𝑖𝑖𝑖𝑖 𝐿𝐿 sr 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ● The long-range part is𝑣𝑣 given𝑟𝑟 by the Coulomb𝑟𝑟 interaction2 of the ionic particles (atomic units)

( ) = 𝑞𝑞𝑖𝑖𝑞𝑞𝑗𝑗 𝑣𝑣lr 𝑟𝑟𝑖𝑖𝑖𝑖 ● The short-range is a sum of a (Pauli) repulsion,𝑟𝑟𝑖𝑖𝑖𝑖 a dispersion, and a covalent term ( ) = ( ) + ( ) + ( )

𝑣𝑣sr 𝑟𝑟𝑖𝑖𝑖𝑖 𝑣𝑣repulsive 𝑟𝑟𝑖𝑖𝑖𝑖 𝑣𝑣dispersion 𝑟𝑟𝑖𝑖𝑖𝑖 𝑣𝑣covalent 𝑟𝑟𝑖𝑖𝑖𝑖 Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 78 Classical force fields (material specific)

● Buckingham-type repulsive term

( ) = −𝐵𝐵𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖 repulsive 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ● Simple (C6) dispersion𝑣𝑣 term 𝑟𝑟 𝐴𝐴 𝑒𝑒

( ) = 𝐶𝐶𝑖𝑖𝑖𝑖 𝑣𝑣dispersion 𝑟𝑟𝑖𝑖𝑖𝑖 − 6 ● Morse-type covalent interaction term (rather𝑟𝑟 𝑖𝑖small)𝑖𝑖

( ) = [( ( ) 1) 1] 0 −𝛽𝛽𝑖𝑖𝑖𝑖 𝑟𝑟𝑖𝑖𝑖𝑖−𝑟𝑟𝑖𝑖𝑖𝑖 2 𝑣𝑣covalent 𝑟𝑟𝑖𝑖𝑖𝑖 𝐷𝐷𝑖𝑖𝑖𝑖 𝑒𝑒 − −

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 79 Classical force fields (material specific)

● Buckingham four-range potential for nonbonded interactions:

, 5 degree polynomial−𝐵𝐵𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖, < 𝐴𝐴𝑖𝑖𝑖𝑖𝑒𝑒 𝑟𝑟𝑖𝑖𝑖𝑖 ≤ 𝑟𝑟1 = 3th degree polynomial, < 𝑟𝑟1 𝑟𝑟𝑖𝑖𝑖𝑖 ≤ 𝑟𝑟min rd 𝑣𝑣 𝑟𝑟𝑖𝑖𝑖𝑖 , 𝑟𝑟min >𝑟𝑟𝑖𝑖𝑖𝑖 ≤ 𝑟𝑟2 𝐶𝐶𝑖𝑖𝑖𝑖 − 6 𝑟𝑟𝑖𝑖𝑖𝑖 𝑟𝑟2 𝑟𝑟𝑖𝑖𝑖𝑖

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 80 Classical force fields (O-O interaction potential)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 81 Classical force fields (cascade simulation)

● Short interatomic distances in cascade simulations ● Extra repulsive potential for the cascade simulations, e.g. the potential of Ziegler, Biersack, and Littmark (ZBL) = 0.1818 . + 0.5099 . + 0.2802 . + 0.02817−3 2.𝑥𝑥 −0 9423𝑥𝑥 −0 4029𝑥𝑥 ZBL 𝑣𝑣 𝑥𝑥 𝑒𝑒 −0 2016𝑥𝑥 𝑒𝑒 𝑒𝑒 ( . . ) = 𝑒𝑒 with .0 23 0 23 𝑟𝑟𝑖𝑖𝑖𝑖 𝑍𝑍𝑖𝑖 +𝑍𝑍𝑗𝑗 ● Fit of an universal screening𝑥𝑥 0 8854 function𝑎𝑎0

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 82 Classical force fields (O-O interaction potential)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 83 Empirical potentials

Rigid-ion models: ● Fixed (often formal) ion charge ionic

● Simple classical Buckingham-type→ pair potentials for the short-range nonbonded interactions ● Ewald method for the description of Coulomb interactions (FFT)

Core-shell model: ● Method of Dick and Overhauser (1958) ● Split in core and shell particles linked by a (usually) harmonic spring ● Consideration of polarisability effects

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 84 Core-shell model

● The massive point represents the nucleus and its inner electron shells ● The lighter shell represents the valence electron shell, core and shell interact through an harmonic potential ● The short-range interatomic potential acts between the shells only ● The electrostatic interaction between a core and its own shell is explicitly excluded ● By contrast to the rigid-ion model, the possible polarisation of an atom is included

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 85 Relaxed (massless or static) core-shell model

● Shells follow instantaneously their respective cores ● Tight optimization of the shells to their zero-force positions is required in each MD time step ● Significant computational overhead due to the iterative optimization ● Small energy drift even for tight optimisation due to residual forces

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 86 Adiabatic (dynamical) core-shell model

● Shells are also treated as particles, but with a very small mass compared to the mass of the corresponding core ● Shells are treated as fictitious dynamical variables analogous to the Car-Parrinello method explicit dynamics also for the shells

● Cores and shells are propagated⇒ concurrently ● MD time step limited by core-shell motion ● Smaller MD time steps required to ensure adiabaticity ● An explicit thermostat for the shell particles can be applied

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 87 Uranium dioxide (UO2, urania, fluorite structure)

This image cannot currently be displayed.

5.48 Å

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 88 3OxygenUnperturbed×3×3 sublattice unit oxygencells inof sublatticethepristine presence UO 2 (300of an K) O vacancy

Ucore

Ushell

Ocore

Oshell

The core-shell distance is displayed x10

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 90 Choice of the empirical potentials

Rigid-ion and core-shell potentials from the literature has been assessed concerning: ● Structural and mechanical properties ● Defect formation and migration energies

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 91 Structural and mechanical properties of UO2 (rigid-ion)

Property E a c c c B coh 0 11 12 44 0 dB /dP A (eV) (Å) (GPa) (GPa) (GPa) (GPa) 0 Potential Arima1 -103.0 5.4532 479.5 100.7 95.8 227.0 3.6 0.506 Arima2 -50.9 5.4545 435.4 113.7 106.3 220.9 5.3 0.661 Basak -43.2 5.4547 408.4 59.3 59.6 175.7 4.4 0.341 Grimes -105.6 5.4619 523.0 145.5 147.0 271.4 3.9 0.779 Karakasidis -100.6 5.4657 367.9 86.8 70.5 180.5 3.1 0.502 Lewis1 -103.6 5.3893 426.1 120.5 119.7 222.3 3.5 0.783 Morelon -65.9 5.4475 216.2 78.5 78.6 124.4 3.3 1.141 Sindzingre -100.7 5.4488 369.2 85.4 65.7 180.0 3.1 0.463 Tharmalingam -103.0 5.4104 472.2 95.3 84.5 220.9 3.5 0.449 Walker -104.1 5.3284 470.3 101.6 89.7 224.5 3.5 0.487 Yakub -37.0 5.4440 345.3 70.1 66.8 161.8 4.6 0.485 Yamada -45.6 5.4667 419.0 57.2 54.5 177.8 4.3 0.301 Experiment -106.7 5.4731 389.3 118.7 59.7 208.9 4.7 0.441

• K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Krack, Mater. Res. Soc. Symp. 1383, mrsf11-1383-a03-11 (2012)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 92 Structural and mechanical properties of UO2 (core-shell)

Property E a c c c B coh 0 11 12 44 0 dB /dP A (eV) (Å) (GPa) (GPa) (GPa) (GPa) 0 Potential Busker -104.5 5.4683 531.9 120.5 119.1 257.7 3.9 0.579 Catlow1 -103.1 5.4506 419.2 124.6 66.5 222.8 3.5 0.451 Catlow2 -94.5 5.5211 434.0 99.2 57.6 210.8 3.7 0.344 Grimes -105.6 5.4619 523.0 145.5 89.4 271.3 3.9 0.474 Jackson1 -103.1 5.4479 427.8 125.0 67.1 225.9 3.9 0.443 Jackson2 -103.4 5.4516 396.8 131.1 73.8 219.7 3.4 0.556 Lewis2 -103.6 5.3900 425.9 120.4 88.7 222.3 3.5 0.581 Lewis3 -103.5 5.3815 426.0 118.3 50.1 220.9 3.5 0.326 Meis -100.8 5.4688 387.6 116.1 59.9 206.6 4.1 0.442 Read -101.9 5.4693 390.5 115.5 58.3 207.2 3.5 0.424 Experiment -106.7 5.4731 389.3 118.7 59.7 208.9 4.7 0.441

• K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Krack, Mater. Res. Soc. Symp. 1383, mrsf11-1383-a03-11 (2012)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 93 Frenkel pair formation energies (rigid-ion)

Potential [O] (eV) [U] (eV) ∞ ∞ Arima1 𝑬𝑬𝒇𝒇 6.7 𝑬𝑬𝒇𝒇 24.0 Arima2 7.9 22.8 Basak 5.8 16.7 Grimes 9.0 32.8 Karakasidis 4.7 19.1 Lewis1 6.7 25.1 Morelon 3.8 15.4 Sindzingre 4.3 18.5 Tharmalingam 5.9 22.0 Walker 5.8 22.1 Yakub 5.6 15.8 Yamada 5.8 18.2 DFT+U 5.3 15.3 Experiment 3.0-4.0; 4.6 ± 0.5 9.5 • K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Bertolus et al., F-BRIDGE deliverable D221 (2010) • M. Krack, Mater. Res. Soc. Symp. 1383, mrsf11-1383-a03-11 (2012) and unpublished data

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 94 Frenkel pair formation energies (core-shell)

Potential [O] (eV) [U] (eV) ∞ ∞ Busker 𝑬𝑬𝒇𝒇 6.3 𝑬𝑬𝒇𝒇 22.0 Catlow1 5.0 18.3 Catlow2 5.0 16.2 Grimes 6.8 23.7 Jackson1 5.1 18.5 Jackson2 4.8 18.7 Lewis2 5.3 20.0 Lewis3 5.1 19.0 Meis 4.5 17.9 Read 4.5 17.1 DFT+U 5.3 15.3 Experiment 3.0-4.0; 4.6 ± 0.5 9.5

• K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Bertolus et al., F-BRIDGE deliverable D221 (2010) • M. Krack, Mater. Res. Soc. Symp. 1383, mrsf11-1383-a03-11 (2012) and unpublished data

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 95 Schottky defect formation energies (rigid-ion)

Potential (eV) (eV) ∞ 𝐛𝐛 Arima1 𝑬𝑬𝒇𝒇10.2 𝑬𝑬𝒇𝒇4.5 Basak 10.8 5.7 Morelon 8.0 3.9 Sindzingre 6.3 2.5 Tiwary 7.1 − Walker 8.3 3.5 Yamada 13.5 8.2 DFT+U 9.4 2.5 Experiment − −

• K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Bertolus et al., F-BRIDGE deliverable D221 (2010)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 96 Melting temperature of bulk UO2 (rigid-ion)

Property

Tm (K) Potential Basak 3540 Morelon 3500 Sindzingre 3150 Walker 3435 Yamada 4155 Experiment 3140

• K. Govers et al., J. Nucl. Mater. 376, 66 (2008) • M. Bertolus et al., F-BRIDGE deliverable D221 (2010)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 97 Defect migration energies (rigid-ion)

Potential E[IO] (eV) E[VO] (eV) E[IU] (eV) E[VU] (eV) Arima1 0.52 0.24 4.39 5.53 Arima2 1.73 0.57 4.27 5.32 Basak 1.04 0.27 3.54 4.09 Karakasidis 0.07 0.04 2.72 3.69 Lewis1 (0.69) 0.46 4.01 6.19 Morelon 0.69 0.34 2.35 3.88 Sindzingre 0.00 0.00 2.53 3.24 Tharmalingam 0.09 0.11 3.92 4.48 Walker 0.09 0.13 3.72 4.58 Yamada 0.89 0.27 3.74 4.30 DFT+U 0.9 0.7 7.9 3.7 Experiment 0.8-1.0 0.5-0.6 ≈2.0 ≈2.4

• K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Bertolus et al., F-BRIDGE deliverable D221 (2010) • M. Krack, Mater. Res. Soc. Symp. 1383, mrsf11-1383-a03-11 (2012)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 99 Defect migration energies (core-shell)

Potential E[IO] (eV) E[VO] (eV) E[IU] (eV) E[VU] (eV) Busker 0.70 0.33 5.84 6.62 Catlow1 not conv. 0.54 4.44 5.18 Catlow2 0.21 0.32 4.29 4.67 Grimes 1.02 0.72 5.84 6.74 Jackson1 0.61 0.55 4.46 5.23 Jackson2 0.73 0.62 4.51 not conv. Lewis2 not conv. 0.44 4.49 5.48 Lewis3 not conv. 0.47 3.64 4.51 Meis 0.62 0.54 3.55 4.58 Read 0.34 0.46 3.85 4.69 DFT+U 0.9 0.7 7.9 3.7 Experiment 0.8-1.0 0.5-0.6 ≈2.0 ≈2.4

• K. Govers et al., J. Nucl. Mater. 366, 161 (2007) • M. Bertolus et al., F-BRIDGE deliverable D221 (2010) • M. Krack, Mater. Res. Soc. Symp. 1383, mrsf11-1383-a03-11 (2012)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 100 Choice of the simulation method

● Study of collision (displacement) cascades requires . Large simulation cells, ideally µm scale . Description of the dynamics at the atomic level . Simulation time scale of at least 100 ps  computationally efficient . Simple and efficient but also sufficiently accurate description of the structural and

mechanical properties for pristine and defective UO2

. Consideration of stoichiometric effects  UO2±x  Ten thousands to millions of atoms  Atomic forces (ideally analytic and computationally cheap)  Millions of molecular dynamics (MD) time steps  Classical force field methods using empirical potentials  Proper description of polarisability and different oxidation states

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 102 Displacement cascades in pristine UO2

● Simulation box: . ~14 nm edge length (187500 atoms) 10 keV . ~22 nm edge length (768000 atoms) 20 keV → ● Simulation setup: → . NVE ensemble, T = 700K . Thermostat applied to the sides of the box (thermal region) . Short-range repulsive potential of Ziegler, Biersack, Littmark (ZBL) . Primary Knock-on Atom (PKA): Uranium . PKA energy: 10 keV and 20 keV . 1st part: 0.00 ps < t < 0.75 ps rigid-ion model (Morelon et al.) . 2nd part: t > 0.75 ps core-shell model (Meis and Chartier) → . CP2K code (http://www.cp2k.org) →

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 103 10 keV cascade simulation in UO2

1st movie part: All-atom sample equilibrated (700 K, 1 bar) 2nd movie part: Evolution of defects only

Uint

Oint

Uvac

Ovac

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 104 10 keV cascade simulation in UO2

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 105 Molecular dynamics codes

● Selection of popular codes suited for condensed phase systems:

GPL or academic license (some are not free) AMBER CHARMM CP2K DL_POLY GROMACS GROMOS GULP LAMMPS MOLDY NAMD

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 106 What method or code should I use?

● System type (e.g. condensed phase or molecular system) ● Target properties ● System size ● Requested accuracy ● Available computational resources

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 107 CP2K Program Package (www.cp2k.org)

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 108 Thank you for your attention!

Questions: [email protected]

Matthias Krack (PSI) ThUL School 2014 in Actinide Chemistry (June 2-6, 2014, KIT, Karlsruhe) 109