hb407@cam.ac.uk

Holomorphic Hartree Fock Theory:Beyond the Coulson–Fischer point

Hugh G. A. Burton1, Alex J. W. Thom1

1Department of Chemistry, University of Cambridge, U.K.;

There are many solutions to the SCF equations

SCF solutions are stationary points of the Hartree–Fock energy with respectto occupied-virtual orbital rotations:

• Many local maxima, minima and saddle points exist.• Local stationary points correspond loosely to excited electronic states.• SCF solutions are quasidiabatic states.

Stationary points may disappear as geometry changes:

• States coalesce and vanish at instability thresholds.• O�en related to emergence of symmetry broken solutions.• Leads to discontinuous energies when used as a basis for NOCI.

States of H2 vanish as geometry changes

✓↵�1.5 �1.0 �0.5 0.0 0.5 1.0 1.5

✓�

�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Energy

/Eh

�1.0

�0.9

�0.8

�0.7

�0.6

�0.5

�0.4

�0.3

�0.2

✓↵�1.5 �1.0 �0.5 0.0 0.5 1.0 1.5

✓�

�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Energy

/Eh

�1.0

�0.5

0.0

0.5

UHF energy as a function of ✓↵ and ✓�. 8 stationary points exist at 4.00 Å (le�) but only 4 at 0.75 Å (right).

The UHF orbitals for H2 (STO-3G) can berepresented by two angles ✓↵ and ✓�.

↵ = cos(✓↵)�g + sin(✓↵)�u � = cos(✓�)�g + sin(✓�)�u.

Holomorphic Hartree–Fock prevents states from disappearing

SCF orbitals are a linear combination of basis or-bitals,

i =X

↵

c↵i�↵.

The Hartree–Fock energy depends on the coe�i-cients c↵i and their complex conjugates c⇤↵i

E[ ] =h |H| ih | i .

Hartree–Fock states can disappear because the de-pendence of E on c⇤i means its gradient is not well–defined for complex coe�icients.

Remove the complex conjugate to define the Holo-morphic Hartree–Fock energy

eE[ ] =h ⇤|H| ih ⇤| i .

eE is analytic for c↵i and where UHF solutions disap-pear, the h-UHF coe�icients become complex.

E[ hUHF] is real and the h-UHF states can be usedas a basis for NOCI.

h-UHF solutions for H2 (STO–3G). Where UHF states disappear, theirh-UHF counterparts continue with complex coe�icients. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Bond Length / A

�1.2

�1.0

�0.8

�0.6

�0.4

�0.2

0.0

0.2

Energy

/Eh

�2u

�2g

�1g�

1u

H+ � H�

H. � H.

RHF

h-RHF

UHF

h-UHF

Holomorphic states provide a basis for NOCI

SCF states are notmutually orthogonal sowemust solve the generalised eigen-value equation

Hv = ESv,

where Hwx = hw |H|x i and Swx = hw |x i.

• Energies are size-consistent and scaling is similar to SCF.

• NOCI states provide excited statetrial wavefunctions for DMC.

DMC energies of H2 at 2.5 Å using NOCI trial wave-

functions in 6-31G. Lowest cc-pV5Z FCI energies are

�1.00552 Eh, �0.99783 Eh and �0.71396 Eh.

T ENOCI ENOCI-DMC Abs. Error/ Eh / Eh / mEh

1⌃+g �0.9967 �1.00532(9) 0.2

3⌃+u �0.9902 �0.99775(5) 0.1

1⌃+u �0.6078 �0.7070(4) 6.9

Is F2 charge-shi� bound?

Valence bond theory results suggest bonding in F2 arises from fluctuations ofthe electron pair density between cov and F+F� / F�F+ structures, creatinga “charge-shi�” bond.

NOCI using only ground state RHF and diradical h-UHF states can also yielda bound wavefunction.

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Bond Length / A

�198.7

�198.6

�198.5

�198.4

Energy

/Eh

RHF

UHF

h-UHF

NOCI

NOCI energies of F2 (6-31G) using basis of ground

RHF state and diradical h-UHF states.

Covalent bonding achieved withoutionic states.

|�NOCIi =cRHF| RHFi+cF"F#| F"F#i + cF#F"| F#F"i

• Suggests static correlation giveskey contribution to bonding.

NOCI mirrors Pauling descriptionwhere bonding arises from electronsinterchanging between two AOs.

The number of holomorphic states is constant

Algebraic geometry shows the number of h-RHF states for two electron sys-tems with n basis functions is

1

2(3n � 1) .

• Must exist for all geometries and all atomic charges• Provides a rigorous upper bound on number of real RHF solutions.• States can be complex across all geometries.

HZ in STO-3G provides a model for symmetric and asymmetric diatomics.

R/A

0.51.01.52.02.53.03.54.0

✓

�1.5�1.0�0.50.0 0.5 1.0 1.5

Energy

/E

h �1.2�1.0�0.8�0.6�0.4�0.20.0

R/A

0.51.01.52.02.53.03.54.0

✓

�1.5�1.0�0.50.0 0.5 1.0 1.5

Energy

/E

h �1.2�1.0�0.8�0.6�0.4�0.20.0

1.0 2.0 3.0 4.0

Bond Length / A

-3.0

-2.0

-1.0

0.0

1.0

Energy

/Eh

QZ = 1.00 a.u.

1.0 2.0 3.0 4.0

Bond Length / A

QZ = 1.15 a.u.

RHF

h-RHF

1.0 2.0 3.0 4.0

Bond Length / A

QZ = 2.00 a.u.

Above: h-RHF states of HZ (STO-3G) as nuclear charge QZ is varied. At

QZ = 2.00 a.u., an h-RHF pair exists that are complex everywhere.

Le�: Conventional Hartree–Fock energy of HZ (STO-3G) as a function of

occupied-virtual rotation angle ✓ for QZ = 1.00 a.u. (top) and

QZ = 1.15 a.u. (bo�om). Symmetry controls the type of coalescence point.

Acknowledgements and References

HGAB thanks the Cambridge and Common-wealth Trust for a PhD studentship andAJWT thanks the Royal Society for a Univer-sity Research Fellowship.

[1] A. J. W. Thom and M. Head-Gordon, J. Chem.Phys. 131, 1 (2009).

[2] H. G. Hiscock and A. J. W. Thom, J. Chem. TheoryComput. 10, 4795 (2014), arXiv:1406.2457 .

[3] H. G. A. Burton and A. J. W. Thom, J. Chem. The-ory Comput. 12, 167 (2016), arXiv:1509.01179 .