## Friday, January 31, 2014

## Sunday, January 26, 2014

### Roundtable discussion on good teaching at the university

I am invited to a roundtable discussion on good teaching at the university, at the Ministry of Science, Innovation and Higher Education. Topics of discussion are given below, and they want emphasis on personal experience, and this post is to help me figure out what I want to say. Any comments from you are most welcome.

Most undergraduate teaching is "just in case" as in "here it is just in case you need it later". This focusses on simple, unrealistic problems unrelated to real life or research problems. Instead use the "just in time" approach, i.e. pose a complex problem and introduce underlying concepts as you need it.

Also, introduce IT tools in problem solving to attack more complex problems.

Textbooks are not designed for this kind of teaching. Replace textbooks by Google, wikipedia, lecture notes and video lectures.

Finally, all bachelor students at my College need to complete a (15 ECTS) bachelors project. In addition my institute also offers two optional mini-bachelor projects (7.5 ECTS).

I use the flipped classroom/peer instruction method for all my courses. During the lecture period students are actively discussing and voting. The have to take online "reading" quizzes before the lecture periods, to ensure they are prepared. I provide homework answers online using the free site PeerWise, so that they can get "help" when working on the projects at home. For some courses I use project-based reports instead of exams.

In my experience Danish students are good team-players and collaborators. It's easy to get them to discuss with each other. They are weak in traditional math skills.

I have started giving students a range of problems to choose from - with some very difficult ones aimed at the stronger students. Sometimes, a homework problem consists of writing a homework problem (on PeerWise), here the strong students can really shine if they choose. The same is true for the in-class clicker question discussions.

For weaker students, immediate and thorough feedback on problems is important. Sometimes I make videos where I carefully explain how to solve an assigned problem. Also, video lectures can be paused or watched repeatedly. I encourage the use of computers for computing/tedious math using, for example, MAPLE and Wolfram|Alpha.

On-line quizzes, provide immediate feedback and gather data.

Video lectures freeing up "lecture" time and supplementing/replacing the textbook.

Removing tedious math or other components, making more interesting assignments possible.

Removing tedious memorization components with Google.

Dynamic visualization and simulations (virtual experiments).

Many teachers won't allow computers during the exam because the are afraid students will email each other answers. Students won't embrace tools they cannot use during the exam. I allow computers during exams and have not seen this, but I am not sure what to do about this fear.

I think teaching needs to be held to the same general kinds of standards as research: productivity, innovation, and dissemination. For example, US NSF CAREER Award proposals must contain a strong educational component. Should this be introduced in Denmark?

Another idea is a limited number of teaching bonuses that one can apply for, based on documented productivity, innovation, and dissemination - perhaps in the form of a short proposal. Good student evaluations are not enough. This is different from an award or prize, which you usually get only once. For example, I will never get another teacher of the year (Årets Harald) award from KU.

I think collegial supervision is key. Lead by example.

Also, most introductory courses at my department are co-taught. This allows more experiences teachers help the less experiences.

I do this almost any way I can think of: I share what I do on blogs, social media (Twitter, Google+, Facebook) and the departmental newsletter, I organize seminars about it, invite people to observe me in the classroom, give talks to other departments and at meetings, submit write-ups to various outlets such as the magazine for high school teachers (helped by our departmental communication specialist) and write scholarly articles. Also I make as much of the material available on the web (e.g. YouTube and blogs) under the CC-BY license.

It would be nice if alternative teaching methods were even considered.

Somewhat related:

To experiment with teaching you have to break a lot of rules because you can't ask for permission each time. For example, I recently changed the curriculum of my part of the course. I should have asked permission to do this

Also, any sweeping changes or radical experiments are discouraged by the always looming accreditation.

This work is licensed under a Creative Commons Attribution 4.0

**1 What does "good teaching" mean to you?****1.1 What current developments and initiatives are especially interesting for strengthening the quality of education?***Flipped classroom*- knowledge acquisition is done outside the classroom, problem solving is done inside the classroom.*Blended and online learning*- Video-lectures are used instead or in addition to textbook reading. Electronic (formative) assessment e.g. video questions or online reading quizzes with immediate feedback. This can be combined with*gamification*, where points and badges are given as motivation.*Formative*means the questions are mostly or entirely unrelated to the course grade.*Peer instruction*- In-class "clicker" questions with student-student discussion. One model is to lecture and then assess. Another is the flipped classroom approach, where the "lecture" period is used entirely for clicker questions. I use the free site Socrative.com for in class voting.*Project-based learning*- multi-week projects rather than short homework questions. I find this hard to implement for large courses, because grading the resulting reports is time-consuming.*The "students as natural learners" or "get out of the way" approach*- while I have no experience with this approach I find it fascinating and alluring. The general idea is that humans, and especially young children, are natural learners and if you simply let them follow their interests with minimal supervision they learn what they need to in general and become deep learners in topics that interest them. Examples/proponents of this approach include the Sudbury Valley School, Sugata Mitra and Ken Robinson. Related to this is Googles 20% time rule.**1.2 How do you ensure general and research relevance in teaching?**Most undergraduate teaching is "just in case" as in "here it is just in case you need it later". This focusses on simple, unrealistic problems unrelated to real life or research problems. Instead use the "just in time" approach, i.e. pose a complex problem and introduce underlying concepts as you need it.

Also, introduce IT tools in problem solving to attack more complex problems.

Textbooks are not designed for this kind of teaching. Replace textbooks by Google, wikipedia, lecture notes and video lectures.

*Why do we teach as though the internet doesn't exist?*Finally, all bachelor students at my College need to complete a (15 ECTS) bachelors project. In addition my institute also offers two optional mini-bachelor projects (7.5 ECTS).

**1.3 How do you design your course to that the students participate actively both inside and outside the classroom?**I use the flipped classroom/peer instruction method for all my courses. During the lecture period students are actively discussing and voting. The have to take online "reading" quizzes before the lecture periods, to ensure they are prepared. I provide homework answers online using the free site PeerWise, so that they can get "help" when working on the projects at home. For some courses I use project-based reports instead of exams.

**1.4 What strengths and backgrounds do the students bring with then? How do you differentiate your teaching to serve students with different academic backgrounds?**In my experience Danish students are good team-players and collaborators. It's easy to get them to discuss with each other. They are weak in traditional math skills.

I have started giving students a range of problems to choose from - with some very difficult ones aimed at the stronger students. Sometimes, a homework problem consists of writing a homework problem (on PeerWise), here the strong students can really shine if they choose. The same is true for the in-class clicker question discussions.

For weaker students, immediate and thorough feedback on problems is important. Sometimes I make videos where I carefully explain how to solve an assigned problem. Also, video lectures can be paused or watched repeatedly. I encourage the use of computers for computing/tedious math using, for example, MAPLE and Wolfram|Alpha.

**1.5 What new possibilities results from the use of IT in teaching?**On-line quizzes, provide immediate feedback and gather data.

Video lectures freeing up "lecture" time and supplementing/replacing the textbook.

Removing tedious math or other components, making more interesting assignments possible.

Removing tedious memorization components with Google.

Dynamic visualization and simulations (virtual experiments).

*However*, if you use IT, then the infra structure needs to be there*and be dependable*. Electrical outlets, WiFi, AV, etc. This also applies to the exams.Many teachers won't allow computers during the exam because the are afraid students will email each other answers. Students won't embrace tools they cannot use during the exam. I allow computers during exams and have not seen this, but I am not sure what to do about this fear.

**2 How do you create good conditions for teaching?****2.1 Does "teaching" need to be more recognized? How?**I think teaching needs to be held to the same general kinds of standards as research: productivity, innovation, and dissemination. For example, US NSF CAREER Award proposals must contain a strong educational component. Should this be introduced in Denmark?

Another idea is a limited number of teaching bonuses that one can apply for, based on documented productivity, innovation, and dissemination - perhaps in the form of a short proposal. Good student evaluations are not enough. This is different from an award or prize, which you usually get only once. For example, I will never get another teacher of the year (Årets Harald) award from KU.

**2.2 Can training, collegial supervision, etc help strengthen teaching quality? How?**I think collegial supervision is key. Lead by example.

Also, most introductory courses at my department are co-taught. This allows more experiences teachers help the less experiences.

**2.3 Should more be done to disseminate knowledge on good teaching? How?**I do this almost any way I can think of: I share what I do on blogs, social media (Twitter, Google+, Facebook) and the departmental newsletter, I organize seminars about it, invite people to observe me in the classroom, give talks to other departments and at meetings, submit write-ups to various outlets such as the magazine for high school teachers (helped by our departmental communication specialist) and write scholarly articles. Also I make as much of the material available on the web (e.g. YouTube and blogs) under the CC-BY license.

**2.4 Is a better connection between learning objective, teaching methods and exam types needed? How?**It would be nice if alternative teaching methods were even considered.

Somewhat related:

To experiment with teaching you have to break a lot of rules because you can't ask for permission each time. For example, I recently changed the curriculum of my part of the course. I should have asked permission to do this

*one year*before! Not gonna happen.Also, any sweeping changes or radical experiments are discouraged by the always looming accreditation.

This work is licensed under a Creative Commons Attribution 4.0

## Friday, January 24, 2014

### Quantum Biochemistry: the rise of semiempirical methods

**Quantum Biochemistry: the rise of semiempirical methods**from

**molmodbasics**

An updated version of a talk I gave last year. To be presented at the Barcelona Supercomputer Center in a week.

## Wednesday, January 15, 2014

### EFMO-PCM: A roadmap. Version 2

Having read the EFP-PCM interface more carefully again I need to make a major update of a previous post. Since Blogger doesn't have version numbers, here comes a separete post.

One of the things still missing in the EFMO method is an interface to PCM. Here I attempt to sketch the method, based on the FMO-PCM and EFP-PCM interfaces.

The EFMO-PCM electrostatic interaction free energy between solute and solvent plus the polarization energy is

$\mathbf{q}$ are the apparent surface charges (ASCs), which for large systems are obtained by solving this equation iteratively

$\mathbf{V}$ is the electrostatic potential from the static multipoles and induced dipoles, respectively:

The simplest approximation to $\mathbf{V}$ is

$$\mathbf{V} =\sum_I^N \mathbf{V}_{I} = \sum_I^N (\mathbf{V}^{\text{mul}}_{I} +\mathbf{V}^{\mu}_{I})$$

The potential at tesserae $j$ due to induced dipoles on fragment $I$ is given by:

$$\mathbf{V}_I^\mu (j)=\sum_{i \in I} (\mathbf{R}^T)_{ji}\boldsymbol{\mu_{i}}$$

The induced dipoles are obtained iteratively:

$$\boldsymbol{\mu}_i=\boldsymbol{\alpha}_i\left(\mathbf{F}^{\text{mul}}_i+\mathbf{F}^q_i-\sum_{i\neq j}\mathbf{D}_{ij}\boldsymbol{\mu}_j\right)$$

where $\boldsymbol{\alpha}_i$ is the dipole polarizability tensor at site $i$ and $\mathbf{F}^{\text{mul}}_i$ and $\mathbf{F}^q_i$ are the electrostatic fields due to all static multipoles and ASCs felt at point $i$.

Procedure:

1. Compute EFMO gas phase energy

2. Use gas phase static multipoles and $\boldsymbol{\mu}$ to construct $\mathbf{V}$

3. Solve $\mathbf{Cq=-V}$

$$G_{sol}^x=\mathbf{V}^x \mathbf{q}^T+\frac{1}{2}\mathbf{q}^T\mathbf{C}^x\mathbf{q}$$

$$G_{pol}^x=-\mathbf{F}^{mul,x} \boldsymbol{\mu}^T+\boldsymbol{\mu}^T\mathbf{D}^x\boldsymbol{\mu}$$

The most tricky part is $\mathbf{V}^{\mu,x} \mathbf{q}^T$:

$$ \sum_n^N V_n^{\mu,x} q_n = \sum_i^I V_i^{\mu,B,x_i} q_i+ \sum_j^J V_j^{\mu,A,x_A} q_j$$

$$V_i^{\mu ,B,{x_i}} = \sum\limits_{m \ne A}^{} {{{\left( {\frac{{ - ({{\bf{r}}_i} - {{\bf{r}}_m})}}{{{{\left| {{{\bf{r}}_i} - {{\bf{r}}_m}} \right|}^3}}}} \right)}^{{x_i}}}} \boldsymbol{\mu}_m^B$$

$$V_j^{\mu,A,{x_A}} = {\left( {\frac{{ - ({{\bf{r}}_j} - {{\bf{r}}_A})}}{{{{\left| {{{\bf{r}}_j} - {{\bf{r}}_A}} \right|}^3}}}} \right)^{{x_A}}}\boldsymbol{\mu}^A$$

Here $n$ sum over the all the tesserae ($N$). The tesserae set $I$ belong to the sphere centered on atom $A$ - the atoms whose coordinate ($x_A = x$) we are taking the derivative wrt. $x_i$ refers to the coordinate of a ASC associated with a $I$ tessera. $J$ are all other tessera belonging to atoms collectively referred to as $B$. Notice that the gradient involving tessera and induced dipoles belonging to the same atom is zero, $V_i^{\mu,A,x_A}=0$

The tricky thing in general with the EFMO gradient involving induced dipoles is that they are not centered on atoms. In text S1 of this paper we describe how we deal with this for bond-dipoles, $$ V_j^{\mu,A,{x_A}}=f(R)V_j^{\mu,A,{x_{LMO-A}}}$$but we don't talk about how to do it for lone pairs. Perhaps we simply set $x_A = x_{LMO-A}$? Anyway, it should only be an issue for $V_j^{\mu,A,{x_A}}$.

For more accurate results one can approximate $\mathbf{V}$ as
$$\mathbf{V}=\sum_I^N \mathbf{V}_{I}+\sum^N_{I}\sum^N_{J<I} (\mathbf{V}_{IJ}-\mathbf{V}_{I}-\mathbf{V}_{J})$$

This essentially means that the gas phase static multipoles and $\alpha$'s are corrected in step 2. E.g. for static monopoles ($q$)'s:

$$V_I^q(i)=\frac{q_i^{I}}{|r-r_i|}$$

and

$$\begin{align*}

V(i)&=V_I^q(i)+\sum^N_{J<I} (V_{IJ}^q(i)-V_I^q(i))\\

& = [q^{I}_i+\sum^N_{J<I}(q^{IJ}_i-q^{I}_i)]\frac{1}{|r-r_i|} \\

\end{align*}$$

All other steps are the same.

I found the EFP-PCM paper a big mouthful and had to revert to an old trick that I'll share with you here. Mainly what makes the paper complex is its length so that definitions and equations are pages apart. For such papers I usually fire up PowerPoint and Snapndrag (for screenshots) and grab what I think are the essential pieces, rearrange some, and add "missing equations" when needed. You can see the final result here (the first two pages are the main results and the rest are "derivations").

I highly recommend this approach for complex papers or making sense of a collection of papers on a specific topic.

This work is licensed under a Creative Commons Attribution 4.0

One of the things still missing in the EFMO method is an interface to PCM. Here I attempt to sketch the method, based on the FMO-PCM and EFP-PCM interfaces.

The EFMO-PCM electrostatic interaction free energy between solute and solvent plus the polarization energy is

$G_{pol+sol}=-\frac{1}{2}\mathbf{F}^{mul} \boldsymbol{\mu}^T+\frac{1}{2}\mathbf{V}^{mul} \mathbf{q}^T$

As Hui and Mark wrote:

"The interaction between the induced dipoles and charges is implicitly described with the matrices $\mathbf{R}$ and $\mathbf{R}^T$, just as the interaction between the EFP induced dipoles is implicitly described with the off-diagonal elements of matrix $\mathbf{D}$, and the interaction between the PCM induced charges is implicitly described with the off-diagonal elements of matrix $\mathbf{C}$"

"The interaction between the induced dipoles and charges is implicitly described with the matrices $\mathbf{R}$ and $\mathbf{R}^T$, just as the interaction between the EFP induced dipoles is implicitly described with the off-diagonal elements of matrix $\mathbf{D}$, and the interaction between the PCM induced charges is implicitly described with the off-diagonal elements of matrix $\mathbf{C}$"

$\mathbf{Cq}=-\mathbf{V}$

$\mathbf{V}$ is the electrostatic potential from the static multipoles and induced dipoles, respectively:

$\mathbf{V=V^{\text{mul}}+V^{\mu}}$

The simplest approximation to $\mathbf{V}$ is

$$\mathbf{V} =\sum_I^N \mathbf{V}_{I} = \sum_I^N (\mathbf{V}^{\text{mul}}_{I} +\mathbf{V}^{\mu}_{I})$$

The potential at tesserae $j$ due to induced dipoles on fragment $I$ is given by:

$$\mathbf{V}_I^\mu (j)=\sum_{i \in I} (\mathbf{R}^T)_{ji}\boldsymbol{\mu_{i}}$$

The induced dipoles are obtained iteratively:

$$\boldsymbol{\mu}_i=\boldsymbol{\alpha}_i\left(\mathbf{F}^{\text{mul}}_i+\mathbf{F}^q_i-\sum_{i\neq j}\mathbf{D}_{ij}\boldsymbol{\mu}_j\right)$$

where $\boldsymbol{\alpha}_i$ is the dipole polarizability tensor at site $i$ and $\mathbf{F}^{\text{mul}}_i$ and $\mathbf{F}^q_i$ are the electrostatic fields due to all static multipoles and ASCs felt at point $i$.

Procedure:

1. Compute EFMO gas phase energy

2. Use gas phase static multipoles and $\boldsymbol{\mu}$ to construct $\mathbf{V}$

3. Solve $\mathbf{Cq=-V}$

4. Use $\mathbf{q}$ and $\boldsymbol{\mu}_i=\boldsymbol{\alpha}_i\left(\mathbf{F}^{\text{mul}}_i+\mathbf{F}^q_i-\sum_{i\neq j}\mathbf{D}_{ij}\boldsymbol{\mu}_j\right)$ to find new $\boldsymbol{\mu}$

5. Repeat steps 2-4 until self consistency

6. Compute $G_{pol+solv}$

**Gradient**

**$$G_{pol+sol}^x=G_{sol}^x+G_{pol}^x$$**

$$G_{sol}^x=\mathbf{V}^x \mathbf{q}^T+\frac{1}{2}\mathbf{q}^T\mathbf{C}^x\mathbf{q}$$

$$G_{pol}^x=-\mathbf{F}^{mul,x} \boldsymbol{\mu}^T+\boldsymbol{\mu}^T\mathbf{D}^x\boldsymbol{\mu}$$

The most tricky part is $\mathbf{V}^{\mu,x} \mathbf{q}^T$:

$$ \sum_n^N V_n^{\mu,x} q_n = \sum_i^I V_i^{\mu,B,x_i} q_i+ \sum_j^J V_j^{\mu,A,x_A} q_j$$

$$V_i^{\mu ,B,{x_i}} = \sum\limits_{m \ne A}^{} {{{\left( {\frac{{ - ({{\bf{r}}_i} - {{\bf{r}}_m})}}{{{{\left| {{{\bf{r}}_i} - {{\bf{r}}_m}} \right|}^3}}}} \right)}^{{x_i}}}} \boldsymbol{\mu}_m^B$$

$$V_j^{\mu,A,{x_A}} = {\left( {\frac{{ - ({{\bf{r}}_j} - {{\bf{r}}_A})}}{{{{\left| {{{\bf{r}}_j} - {{\bf{r}}_A}} \right|}^3}}}} \right)^{{x_A}}}\boldsymbol{\mu}^A$$

Here $n$ sum over the all the tesserae ($N$). The tesserae set $I$ belong to the sphere centered on atom $A$ - the atoms whose coordinate ($x_A = x$) we are taking the derivative wrt. $x_i$ refers to the coordinate of a ASC associated with a $I$ tessera. $J$ are all other tessera belonging to atoms collectively referred to as $B$. Notice that the gradient involving tessera and induced dipoles belonging to the same atom is zero, $V_i^{\mu,A,x_A}=0$

The tricky thing in general with the EFMO gradient involving induced dipoles is that they are not centered on atoms. In text S1 of this paper we describe how we deal with this for bond-dipoles, $$ V_j^{\mu,A,{x_A}}=f(R)V_j^{\mu,A,{x_{LMO-A}}}$$but we don't talk about how to do it for lone pairs. Perhaps we simply set $x_A = x_{LMO-A}$? Anyway, it should only be an issue for $V_j^{\mu,A,{x_A}}$.

**Miscellaneous**

For more accurate results one can approximate $\mathbf{V}$ as

This essentially means that the gas phase static multipoles and $\alpha$'s are corrected in step 2. E.g. for static monopoles ($q$)'s:

$$V_I^q(i)=\frac{q_i^{I}}{|r-r_i|}$$

and

$$\begin{align*}

V(i)&=V_I^q(i)+\sum^N_{J<I} (V_{IJ}^q(i)-V_I^q(i))\\

& = [q^{I}_i+\sum^N_{J<I}(q^{IJ}_i-q^{I}_i)]\frac{1}{|r-r_i|} \\

\end{align*}$$

All other steps are the same.

**A tip on dealing with complex papers**I found the EFP-PCM paper a big mouthful and had to revert to an old trick that I'll share with you here. Mainly what makes the paper complex is its length so that definitions and equations are pages apart. For such papers I usually fire up PowerPoint and Snapndrag (for screenshots) and grab what I think are the essential pieces, rearrange some, and add "missing equations" when needed. You can see the final result here (the first two pages are the main results and the rest are "derivations").

I highly recommend this approach for complex papers or making sense of a collection of papers on a specific topic.

This work is licensed under a Creative Commons Attribution 4.0

## Tuesday, January 7, 2014

### PhD fellowship at the Danish Technical University

3-year PhD fellowships are available at the Technical University of Denmark and you are invited to compete for one of these working in the group of Klaus B. Møller and Niels Engholm Henriksen at the Department of Chemistry, Technical University of Denmark.

The activities in our group are centered around the theoretical description of ultrafast chemical reactions and other dynamical processes in molecular systems – in particular, how snapshots of atomic positions are obtained using ultrashort (femtosecond) pulses of laser light and X-rays and you will work on the theory and simulation of such experiments.

You should have – or be in the final stages of obtaining – a M.Sc. degree in theoretical/computational chemistry or physics with outstanding grades.

The scholarships for the PhD degree are subject to academic approval and the salary and appointment terms are consistent with the national rules for PhD students.

If you are interested, contact Klaus B. Møller (klaus.moller@kemi.dtu.dk) or Niels Engholm Henriksen (neh@kemi.dtu.dk)

before January 8, 2014 at 23.59

Please include a phone number, a transcript of your grades and a date for your obtaining a master’s degree if you have not obtained it yet.

Literature:

K.B. Møller, N.E. Henriksen, “Time-resolved X-ray diffraction: The dynamics of the chemical bond”, Structure and Bonding, 142, 185:212 (2012)

J.H. Lee et al., “Filming the birth of molecules and accompanying solvent rearrangement”, Journal of the American Chemical Society, 135, 3255:3261 (2013)

## Thursday, January 2, 2014

### EFMO-PCM: A road map

One of the things still missing in the EFMO method is an interface to PCM. Here I attempt to sketch the method, based on the FMO-PCM and EFP-PCM interfaces.

The PCM electrostatic interaction free energy between solute and solvent is

For EFMO $\mathbf{V}$ is the electrostatic potential from the static multipoles and induced dipoles

The simplest approximation to $\mathbf{V}$ is

$$\begin{align*}

\mathbf{V}& =\sum_I^N \mathbf{V}_{I}\\

& = \sum_I^N (\mathbf{V}^{\text{mul}}_{I}+\mathbf{V}^{\mu}_{I})

\end{align*}$$

The potential at tesserae $j$ due to induced dipoles on fragment $I$ is given by:

$$\mathbf{V}_I^\mu (j)=\sum_{i \in I} (\mathbf{R}^T)_{ji}\boldsymbol{\mu_{i}}$$

The induced dipoles are obtained iteratively:

$$\boldsymbol{\mu}_i=\boldsymbol{\alpha}_i\left(\mathbf{F}^{\text{mul}}_i+\mathbf{F}^q_i-\sum_{i\neq j}\mathbf{D}_{ij}\boldsymbol{\mu}_j\right)$$

where $\boldsymbol{\alpha}_i$ is the dipole polarizability tensor at site $i$ and $\mathbf{F}^{\text{mul}}_i$ and $\mathbf{F}^q_i$ are the electrostatic fields due to all static multipoles and ASCs felt at point $i$.

Procedure:

1. Compute EFMO gas phase energy

2. Use gas phase static multipoles and $\boldsymbol{\mu}$ to construct $\mathbf{V}$

3. Solve $\mathbf{Cq=-V}$

This essentially means that the gas phase static multipoles and $\alpha$'s are corrected in step 2. E.g. for static monopoles ($q$)'s:

$$V_I^q(i)=\frac{q_i^{I}}{|r-r_i|}$$

and

$$\begin{align*}

V(i)&=V_I^q(i)+\sum^N_{J<I} (V_{IJ}^q(i)-V_I^q(i))\\

& = [q^{I}_i+\sum^N_{J<I}(q^{IJ}_i-q^{I}_i)]\frac{1}{|r-r_i|} \\

\end{align*}$$

All other steps are the same.

This work is licensed under a Creative Commons Attribution 4.0

The PCM electrostatic interaction free energy between solute and solvent is

$G_s=\frac{1}{2}\mathbf{V}^T \mathbf{q}$

$\mathbf{q}$ are the apparent surface charges (ASCs), which for large systems are obtained by solving this equation iteratively

$\mathbf{Cq=-V}$

For EFMO $\mathbf{V}$ is the electrostatic potential from the static multipoles and induced dipoles

$\mathbf{V=V^{\text{mul}}+V^{\mu}}$

The simplest approximation to $\mathbf{V}$ is

$$\begin{align*}

\mathbf{V}& =\sum_I^N \mathbf{V}_{I}\\

& = \sum_I^N (\mathbf{V}^{\text{mul}}_{I}+\mathbf{V}^{\mu}_{I})

\end{align*}$$

The potential at tesserae $j$ due to induced dipoles on fragment $I$ is given by:

$$\mathbf{V}_I^\mu (j)=\sum_{i \in I} (\mathbf{R}^T)_{ji}\boldsymbol{\mu_{i}}$$

The induced dipoles are obtained iteratively:

$$\boldsymbol{\mu}_i=\boldsymbol{\alpha}_i\left(\mathbf{F}^{\text{mul}}_i+\mathbf{F}^q_i-\sum_{i\neq j}\mathbf{D}_{ij}\boldsymbol{\mu}_j\right)$$

where $\boldsymbol{\alpha}_i$ is the dipole polarizability tensor at site $i$ and $\mathbf{F}^{\text{mul}}_i$ and $\mathbf{F}^q_i$ are the electrostatic fields due to all static multipoles and ASCs felt at point $i$.

Procedure:

1. Compute EFMO gas phase energy

2. Use gas phase static multipoles and $\boldsymbol{\mu}$ to construct $\mathbf{V}$

3. Solve $\mathbf{Cq=-V}$

4. Use $\mathbf{q}$ and $\boldsymbol{\mu}_i=\boldsymbol{\alpha}_i\left(\mathbf{F}^{\text{mul}}_i+\mathbf{F}^q_i-\sum_{i\neq j}\mathbf{D}_{ij}\boldsymbol{\mu}_j\right)$ to find new $\boldsymbol{\mu}$

5. Repeat steps 2-4 until self consistency

6. Compute $G_s$

For more accurate results one can approximate $\mathbf{V}$ as

$$\mathbf{V}=\sum_I^N \mathbf{V}_{I}+\sum^N_{I}\sum^N_{J<I} (\mathbf{V}_{IJ}-\mathbf{V}_{I}-\mathbf{V}_{J})$$This essentially means that the gas phase static multipoles and $\alpha$'s are corrected in step 2. E.g. for static monopoles ($q$)'s:

$$V_I^q(i)=\frac{q_i^{I}}{|r-r_i|}$$

and

$$\begin{align*}

V(i)&=V_I^q(i)+\sum^N_{J<I} (V_{IJ}^q(i)-V_I^q(i))\\

& = [q^{I}_i+\sum^N_{J<I}(q^{IJ}_i-q^{I}_i)]\frac{1}{|r-r_i|} \\

\end{align*}$$

All other steps are the same.

This work is licensed under a Creative Commons Attribution 4.0

Subscribe to:
Posts (Atom)