Multiscale Modelling of X-ray Spectra

1

Royal Institute of Technology School of Biotechnology

Theoretical Chemistry and Biology

Hans Agren

Theoretical Chemistry and Biology Royal Institute of Technology

Stockholm, Sweden agren@theochem.kth.se

Multiscale Modelling of

X-ray Spectra

Drug  Design

Computational    Life  Science  and  Health

First  signs  of   diseases  at  the   molecular  level RATIONAL

SYSTEM BIOLOGY

INFORMATICS

GOOGLE  :    MULTISCALE   MODELS  IN  MATERIALS  SCIENCE  

Two philosophies:

1) Provide data to build next scale model

2) Merge algorithms

Multiphysics  modeling  of    propagation  of   strong  X-­ray  pulses

Cross  section Transmission

Conversion Wave  equation

(Maxwell’s  equations)

Nonlinear   polarization Transition

moments and  energies

Density  matrix

equation Relaxation

times

Result:  DNA  N1s  NEXAFS

5

I:  π*(-­N=) II:  π*(-­NH-­)

III:  σ*

ØStacking effect of pairs has very little influence Ø-N= and -NH- nitrogens: noticeable energy range

ØGood agreement

[Hua et al, J. Phys Chem. B 2010, 114, 7016]

N1s  NEXAFS  spectra  of  solvated  peptide

Blocked alanine

Supermolecular model  (1 snapshot)

Hua  et  al,  PCCP,   2012,  14,  9666

Supermolecular-­‐continuum   model Supermolecular   model  (600  snapshots)

Linear  Scaling  

Quantum  Mechanics

QM

MM QM PCM

MM QM

3-­level  Multiscale  Modelling

QM  =  Quantum  mechanics MM  =  Molecular  Mechanics PCM  =  Polarizable  Continuum

*  Focus  on  the  interesting  part

*  QM  for  the  important  physics/chemistry    

*  MM  for  the  large  part

*  Discrete  nature  of  molecules  is    preserved

QMMM philosophy

• H _QM/MM =  H _el +H _vdw +H _pol

• H

= Electrostatic  interaction

• H

= van  der Waals  interaction  (dispersion  +  short  range  repulsion)

•  H

= Polarization   Interaction

Warshel et al. J. Mol. Biol. 103 (1976) 227

QM

è

MM

€

H →H _QM + H _MM + H _{QM / MM}

Enzymechromic shift nile  red  in  betalactoglobulin  and  water

The  QM/MM  model

Potential Induction

Effective  Interaction €

ϕ

(r) = T

q − T

µ

+1/ 3T

θ

+ ...

€

µ

[ ρ ] = ( α

− T

)

( ^F

^{[ ρ ] + F}

)

€

h

[ ρ ] = − h

s

∑ ^{− ( µ}

^{[ ρ ])}

s

∑

^t

pq s T

s MM ind

QM

pq

t

h

[ ^δρ ] ^{= −} ∑ ( ^µ [ ^δρ ])

€

µ

^[ δρ ] = ( α

− T

)

^F

[ δρ ^]

( )

Contribution  to  linear

Response  Function

Two  important  aspects  

Integration  with  Dynamics Decompoition  of  MM  interaction

Original LoProp:   L.  Gagliardi,   R.  Lindh,   G.  Karlström,   J.  Chem.   121,   4494,   (2004)

New    LoProp   :  I.  Harczuk,   O.  Vahtras,   H.  Ågren,   Phys.   Chem.   Chem.   Phys.   17,   12  7800   (2015)

; A,B – atomic sites

MM  Parameters

”Generalized  LoProp  technique”

• Charges  -­ multipoles

• Frequency  dependent  polarizabilities  and   hyperpolarizabilities

• Capacitances

• Dispersion  coefficients  (localized)

Original

New    LoProp   :  I.  Harczuk,   O.  Vahtras,   H.  Ågren,  Phys.   Chem.  Chem.   Phys.   17,  12  7800   (2015)

Basic Methods for X-ray Spectra

1. STATIC EXCHANGE APPROXIMATION(STEX) - single channel, single up excitations

2. TAMM-DANCOFF APPROXIMATION (TDA) - multi channel, single up excitations

3. RANDOM PHASE APPROXIMATION (RPA) - multi channel, single up and down excitations

STEX h{p}: TDA {hp}: RPA {hp}+{ph}: SOPPA {hhpp}+ {pphh}.

( TRANSITION POTENTIAL and EQUIVALENT CORE models are important approximations of STEX)

=

=

• The  cross  section  for  linear  absorption  of   radiation  by  a  randomly  oriented  molecule   sample  is  

• denotes  the  trace  of  the  complex  electric   dipole  polarizability  tensor.  

( ) 4 Im ( ) c

σ ω = πω α ω

α

X-ray Polarization Propagator

Response  theory

Outer  projection  of  resolvent:  sum-­over-­state Linear  response  

function

Innner  projection  of  resolvent:  solving  linear  

systems  of  equations

HF,  CI,  MCSCF,  DFT

Variational Perturbational

CC,  MP,  ADC

Quantum Mechanics-Molecular Mechanics (QMMM)

Relativistic

4-­component   Dirac

Multiscale

Time-­Dependent  (Response)  Properties

The  QM/MM  interaction  energy

Induced

charges Induced

dipoles Permanent

charges Induced

dipoles

QM/CMM: Theoretical Foundations

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput.,  2014,  10  (3),  pp  989–1003.      

Induced charges and dipoles are determined by solving charge equilibration

equation

T interaction matrices

Need to (Gaussian) distribute charges and dipoles !

Between distributed charges

Between distributed charges and dipoles

Between distributed dipoles

QM/CMM: Theoretical Foundations

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput.,  2014,  10  (3),  pp  989–1003.      

H ˆ

= 1

2 q

( ϕ

+ ϕ

+ T

∑

^q

^{) − 1} ₂ ^p

^(E

^{+ E}

^{+ T}

∑

^q

⁾

∑

∑

+ q

( ϕ

+ ϕ

⁾

∑

^{+ H}

Interaction Hamiltonian between QM region and heterogeneous MM region

Includes:

- contributions from induced distributed charges and distributed dipoles in the MM region

- contributions from permanent charges in the MM region - induced charges and dipoles are determined by solving charge

equilibration equation within capacitance-polarization model

Kohn-­Sham  theory  for  QMMM

Lagrangian for the energy functional

QM energy functional

from varying and

Linear  Reponse  – TDDFT/MM

Ehrenfest Principle

Linear response Function ...in matrix form

contributions to E

Solve to 1st order.

P. Salek, O. Vahtras, T. Helgaker, and H. Agren, J. Chem. Phys. 117, 9630 (2002).

δ

and

New relay equation

Perturbed QMMM Kohn-Sham operator

term in

describes the interaction between QM and MM due to the first order perturbed density of the QM region

Differentiation with respect to that perturbed density gives

TheoChemBio

Fibril-­probe

pH  probe

QMMM-­Virtual  Laboratory  for  Molecular  Probes

pH-­probe

Intrinsic Biomarkers  

and  

GFPs

QM/CMM: Applications

Absorption in UV/Vis region - thymidine on gold surface

Z. Rinkevicius et. al., J. Chem. Theory Comput., 2014, 10 (3), pp 989–1003.

Electronic circular dichroism

- aminohelicene on gold

X. Li et. al., J. Phys. Chem. C, 2014, 118 (11), pp 5833–5840.

Two-photon absorption - 4-nitro-4′-amino-trans- stilbene on gold and silver

surface

X. Li et. al., J. Chem. Theory Comput., 2014, 10 (12), pp 5630–5639.

• On Au(111), no solvent

• On Au(111), in aprotic

trichlorobenzene solution

• On Au(111), in protic octanoic acid solution

QMCMM for Nanoparticle Hybrids

PNA on gold nanoparticles: one-photon absorption

X.  Li  et.  al.,  J.  Chem.  Theory  Comput., 2016, Articles  ASAP.

(a)

(b)

(c)

(d)

NP887

NP3007 NP1505

NP1985

CPP-QM/CMM: from UV/Vis to X-ray spectroscopy

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput., 2016, 12 (6),  pp  2661–2667 UV/Vis Spectra Of PNA on Au(111)

surface

Carbon K-edge Spectra Of PNA on Au(111) surface

Linear  absorption  cross-­section  of  para-­nitroaniline  

adsorbed  on  Au(111)  surface  in  UV/Vis  region  than  in  X-­ray  

region

plastocyanin active site

histidine

vacuum

polarisable embedding

Cu

Energy shifts of 0.2–0.6 eV due to embedding

CPP/CAM-B3LYP(100%)

NEXAFS with PE-CPP

Pedersen et al, J. Chem. Theory Comput. 2014, 10, 1164

QMMM    calculations  of  XPS  shifts  in  ethanol-­water

T. Loytynoja, J. Niskanen, K. Jankala, O. Vahtras, Z. Rinkevicius, H. Agren, J. Chem. Phys. 118(46):13217-25 (2014)

Experiment A. Naves de Brito, O. Björneholm 2016

Non-­linear  X-­ray  spectroscopy  with  the  Gamma   tensor

RIXS

CARS hyper Raman

2-Photon Photocuring

Low Energy Cure

Photodynamic Therapy

Noninvasive Cancer Treatment

3 D Optical Data Storage

1000 CDs in 1 cm

2&3-Photon Pumped Upconverted Lasing

Blue Light From a Plastic Laser

MPA

Macromolecules

Multiphoton Excitation: Applications

2-Photon Nanofabrication

Couplers, Gratings Sensor Platforms

2-Photon Fluorescence Microscopy

Bio  Detection        Bio-­imaging

Detector Lens

Flow Cell

Membrane Bacteria

Flourophor Nano-particles Diode-Laser

Vaia, AFRL Normal light

High intensity light Optical Control

Sensor

QMMM

Zilvinas Rinkevicius (KTH) Xin Li (KTH)

Jaime-Axel Sandberg (KTH) Kurt Mikkelsen (Copenhagen U) Jacob Kongsted (Odense U)

Decomposition

Ignat Harczuk (KTH) Olav Vahtras (KTH)

X-ray Faris Gelmukhanov (KTH) Victor Kimberg (KTH)

Patrick Norman (KTH)

Thanks to

CPP-QM/CMM: from UV/Vis to X-ray spectroscopy

Linear absorption cross-section

Complex polarizability in CPP- QM/CMM

Ψ

[ ˆ Ω,[ ˆ κ

^{, ˆH}

+ ˆH

] + ˆH

+ ˆH

] Ψ

+( ω +i γ ⁾ Ψ

[ ˆ Ω, ˆ κ

^] Ψ

= Ψ

[ ˆ Ω, ˆV

] Ψ

Embedding Hamiltonian in QM/CMM formalism

- for metallic part of the MM region is described by capacitance-

polarization model

- for non-metallic part of the MM is described by non-polarizable force

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput., 2016, 12 (6),  pp  2661–2667

PNA on gold nanoparticles: two-photon absorption

X.  Li  et.  al.,  J.  Chem.  Theory  Comput., 2016, Articles  ASAP.

(a)

(b)

(c)

(d)

NP887

NP3007 NP1505

NP1987

QM/CMM: Theoretical Foundations

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput.,  2014,  10  (3),  pp  989–1003.      

Assumptions in partitioning of the heterogeneous system

- the QM region is selected from the non-metallic part of the system - the QM and MM regions must be selected in such way that strong

chemical bonds between the QM region and the metallic part of

the MM region are avoided

• On Au(111), no solvent

• On Au(111), in aprotic

trichlorobenzene solution

• On Au(111), in protic octanoic acid solution

HYBRID DFT/CMM METHOD

Metal  surface

• Almost-­free  motions  of  electrons – Image  charge  effects

• Physisorption  or  Chemisorption

• The  QM/CMM  approach

– Heterogeneous  environment   for   multiscale  QM/MM  properties

17:32 40

Quantum  Mechanics  

Capacitance  Molecular  Mechanics

(Rinkevicius)

Features  of  X-­ray  spectroscopy Good  features:

• Element  specific

• Chemical  specific  (chemical  shifts)

• Maps  local  electronic  structure But:

Chemistry  of  initial  or  final  state  ?

Core  holes  embedded  in  continua

Random Phase Approximation - RPA

TDA:

STEX:

TDDFT:parameterization

ˆ ( )^t

0 t = e

KS determinant at time t

unperturbed KS determinant

Time evolution operator (parameterized)

ˆ( )

( )

t t a a

κ = ∑ κ

Time  dependent  variational  theory  – response  functions

• The  Ehrenfest  equation  expanded  to  first  order   (frequency  domain)

• -­ A  linear  system  of  equations  for  the  first-­order   parameters  

• The  first-­order  parameters  determine  the  linear   response  function

† (0)

0 [ a a

,[ κ

, H ] + H

] + ωκ

0 = 0

, 0 [ , ] 0

A V

A

ω

= κ

The  QM/MM  model

QM/CMM: Theoretical Foundations

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput.,  2014,  10  (3),  pp  989–1003.      

H ˆ

= 1

2 q

( ϕ

+ ϕ

+ T

∑

^q

^{) − 1} ₂ ^p

^(E

^{+ E}

^{+ T}

∑

^q

⁾

∑

∑

+ q

( ϕ

+ ϕ

⁾

∑

^{+ H}

Interaction Hamiltonian between QM region and heterogeneous MM region

Induced charges and dipoles are determined by solving charge equilibration

equation

Hybrid QM/MM: Response theory

H ˆ

(t) + ˆV(t) 0 = i d

dt 0 ,

0 [ ˆQ,exp[ ˆ κ(t)]( ˆH _KS (t) + ˆV(t)−i d

dt )exp[ − ˆ κ(t)]] 0 ,

- Ehrenfest’s principle based response theory

- QM/MM or QM/CMM coupling in case of linear response function

0 [ ˆq,[ ˆ κ ^ω , ˆH _KS ⁰ ] + H _KS ^ω ] 0 + ω 0 [ ˆq, ˆ κ ^ω − ˆV ^ω ] 0 = 0,

H

= f

ˆE

= φ

^q

^ˆT

∑

^{(r) − p}

^ˆT

^(r)

∑

^φ

^ˆE

• Time-­dependent   expectation   value

Response  theory

• Linear   response   function

• Molecular   properties   via  response   functions

• Electronic   polarizability  tensor

• Magnetic   shielding   tensor

Response  theory

• Linear   response   function

• QM/MM  contribution   to  the  E matrix

The  QM/MM  linear  response

Induced polarizable embedding energy

QM/CMM: Theoretical Foundations

Z.  Rinkevicius  et.  al.,  J.  Chem.  Theory  Comput.,  2014,  10  (3),  pp  989–1003.      

Capacitance-polarization model:

Parametrization:

- A matrix is defined via atom like empirical polarizabilities, which are explicitly dependent on frequency and size of the metallic

part of MM region

- C matrix is defined via distributed capacitances, which are explicitly dependent on frequency and size of the metallic part of

MM region

!!!

A −M 0

−M

C 1 0 1 0

⎛

⎝

⎜ ⎜

⎜

⎞

⎠

⎟ ⎟

⎟

p

q

λ

⎛

⎝

⎜ ⎜

⎜

⎞

⎠

⎟ ⎟

⎟ =

E V q

⎛

⎝

⎜ ⎜

⎜

⎞

⎠

⎟ ⎟

⎟

.

T interaction matrices

Often need to (Gaussian) distribute charges and dipoles !

Between distributed charges

Between distributed charges and dipoles

Between distributed dipoles

QM/CMM: Applications

Absorption in UV/Vis region

- thymidine on gold surface

Z. Rinkevicius et. al., J. Chem.

Theory Comput., 2014, 10 (3), pp 989–1003.

Electronic circular dichroism

- aminohelicene on gold surface

X. Li et. al., J. Phys. Chem. C, 2014, 118 (11), pp 5833–5840.

Two-photon absorption

- 4-nitro-4′-amino-trans- stilbene on gold and silver

surface

X. Li et. al., J. Chem. Theory Comput., 2014, 10 (12), pp 5630–5639.

The  QM/MM  interaction  energy

Induced

charges Induced

dipoles Permanent

charges

T interaction matrices

Often need to (Gaussian) distribute charges and dipoles !

Charge interaction opertors

Dipole interaction operator

TD-DFT/CMM: Molecules on metal nanoparticles

X.  Li  et.  al.,  J.  Chem.  Theory  Comput., 2016, Articles  ASAP.

Cutting  and  capping

[1]  J.  Chem.  Phys.,  119,  pp3599-­3605,  (2003)

Zhang, D. W.; Zhang, J. Z. H. Molecular fractionation with conjugate caps for full quantum mechanical calculation of protein–molecule interaction energy. The Journal of Chemical

Physics 2003, 119, 3599,

Linear  Reponse  – TDDFT/MM

Ehrenfest Principle

Linear response Function from 1st order Ehrenfest Eq.

...in spectral form

, 0 [ , ] 0

A V

A

ω

= κ

Solve to 1st order.

...in matrix form

P. Salek, O. Vahtras, T. Helgaker, and H. Agren,J. Chem. Phys. 117, 9630 (2002).

δ

All  Spectroscopies

Towards  exact  QM/MM  solution

Insulin monomer

Explicitely  Correlated  Electrostatic  potential (MP2)

Courtesy Branislav   Jansik

QM/MM  approach:  QM  and  MM  regions   interaction

region MM  

QM  region electrostatic  int.

polarization   int.

Van  der   Waals  int.

• Electrostatic interaction: permanent multipoles in MM region interacting with QM region

• Polarization interaction: induced dipoles in MM region interacting with QM region

• Van der Waals interaction: empirical Lennard-Jones

potential

2-­level  multiscale  modelling

K.  Mikkelsen,   H.  Agren,   H.J.  Aa.  Jensen,   and  T.   Helgaker,   J.  Chem.   Phys.    89,   3086   (1988)

• The  solvent   is  homogeneous   structureless   medium

• Electrostatic   and   polarization effects are accounted for

• Implicit  averaging   over   solute-­solvent   phase   space

• Computationally   cheap   – one  calculation   for  molecular   properties

• Definition  of  molecular   cavity  is  not  unique

• Neglects   specific   intermolecular   interactions

H.  Agren,   C.  Medina-­Llanos,   and  K.  Mikkelsen,    Chem.   Phys.  115,   43   (1987).

QM PCM

QM