### Bachelor Research Project

## Increasing the efficiency of

## molecular spectroscopy simulations by means of noise reduction

Molecular spectroscopy experiments are used to study properties of materials on short timescales and small distances. Simulations of these experiments are made for

a better interpretation. Often, many samples are needed in the calculations, and this can take a lot of time. To decrease this time, a smart noise reduction method has been implemented and analysed in MATLAB. First, the method is tested for a

linear Gaussian response, then a linear non-Gaussian response. For both of them, the method seems succesful since the number of samples needed for converged spectra can be decreased by at least one order of magnitude for the linear Gaussian

response. We will also cover 2D responses, which is the main purpose of this project, the extension towards 2D has been succesfull for the Gaussian response,

and for the non-Gaussian response it looks promising as well.

### Author:

### J. F. Kruiger

### Supervisors:

### Dr. T. L. C. Jansen Dr. M. S. Pshenichnikov

### Theory of Condensed Matter group

### June 29, 2013

### Contents

1 Introduction 2

2 Implementation for linear Gaussian response 3

2.1 Generating the linear response . . . 3

2.2 Noise reduction . . . 4

2.3 Defining α(t) . . . 5

2.4 Results . . . 6

2.5 N -dependence of spectrum errors . . . 6

2.6 Conclusion . . . 7

3 Application: Carbon monoxide on a platinum surface 7 4 2D spectroscopy: Gaussian response 9 4.1 Uncorrelated Gaussian response . . . 10

4.2 Correlated Gaussian response . . . 15 5 Application: Carbon monoxide on a platinum surface (2D) 18

6 Conclusion 20

7 Acknowledgements 20

### 1 Introduction

Spectroscopy is a very convenient way to study materials on a short timescale (up to
10^{−15}s) and small distances (submolecular). Molecules may respond differently when
different kinds of light interact with the molecule. The molecule absorbs the light that
is close to one of its resonant frequencies. These resonant frequencies are interesting
to look at, because they tell us something about the composition and properties of
the molecule. Spectroscopy is also a great tool for identifying what elements are in a
material, because the electronic structure of atoms determines at which wavelengths
the atom absorbs and emits. These wavelengths are called spectral lines. One of
many applications of this tool is determining the composition of stars[1].

A powerful and relatively new tool is 2D spectroscopy, which allows for dynamics
of molecules to be followed down to the femtosecond (10^{−15}s) timescale.[2] The new
dimension in 2D spectroscopy is the fact that the system is first ‘pumped’ with a

‘pump’ laser pulse, and some time after that the absorbtion will be measured with a second ‘probe’ laser pulse. Both the pump and probe pulses must be evaluated over a certain range of frequencies to produce a 2D spectrum. By first pumping with a laser, and later probing it, something can be said about the memory of the system.

The 2D spectra allow for sensitive measurements of the coupling between states, and provide information of the population transfer between states as well as structural information.

2D infrared spectroscopy (2DIR) has become possible by ultrafast lasers, which can send pulses on the scale of femtoseconds. Ultrafast lasers have previously been utilised for the observation of chemical reactions on the small timescale on which they occur. A. H. Zewail earned the Nobel prize in Chemistry for his studies of the transition states of chemical reactions using femtosecond spectroscopy[3, 4].

2D spectroscopy has succesfully measured processes like hydrogen bond breaking in water[5], and this allows sensitive measurements of protein folding[6] and solution conforming of DNA[7]. Protein folding is of particular importance, since it’s used for the treatment of diseases like Alzheimer’s and Parkinson’s[8]. Other useful things can be achieved with 2D spectroscopy as well. For example, population transfer in the green sulphur bacteria has been measured because of its great photosynthetic properties and its efficiency of directing light excitations[9]

When one measures 2D spectra, they are often difficult to interpret because of overlapping peaks. Therefore, simulations are needed for a better understanding of the measurements. These simulations often need to take an ensemble average of a vast amount of molecules because otherwise noise will play a bigger role. This can be expensive and computationally lengthy. When performing experiments, many molecules are used, so sampling is not a problem there.

The noise usually comes from the simulated environment the molecules are in, which is often a liquid. When a molecule is in a slightly different environment, the electromagnatic interaction of the molecule will also differ slightly. When using too few molecules, the response function is non-coherent, and that is why you need to average over many molecules, so the response function becomes coherent. There are different strategies for reducing the computational costs. Two will be covered in this report, and both work by altering the response-function, which is a function of time that describes the interaction of the molecule with an electromagnetic field. When using a finite ensemble, there will always be noise in the response-function, originating from decoherence of the single molecule responses, and both strategies will reduce the noise in their own way. When the noise is accurately reduced, the number of samples in the ensemble can be reduced as well. And this will help bringing down the computational costs.

The first method, we here denote it the “lifetime-method”[2, ch. 9], where one simply multiplies the original response with an exponent to surpress the noise, i.e.:

R_{1}= R_{0}e^{−}^{2τ}^{t}

Where τ is the lifetime, which is generally on the order of picoseconds^{1}. The
lifetime is also a physical quantity of the system. It is a measure of how long the
system’s response will be active after the system has been pulsed.

The lifetime method broadens the peaks in the spectrum, because it not only surpresses the noise, but also part of the information we want to keep.

This strategy cleary has its deficiencies, and therefore, we have developed and analysed a new method in this report, which we will call the α-method.

In this report, responses (and spectra) altered with the lifetime-method will be called R1, responses altered with the α-method will be called R2, and responses that have not been altered with will be called R0.

The new method will first be tested for linear Gaussian responses, along with this, the method will be explained. After that, the method will be tested for a linear non- Gaussian response. 2D spectra will also be covered, both Gaussian and non-Gaussian.

### 2 Implementation for linear Gaussian response

### 2.1 Generating the linear response

To simulate the problem and test the strategy, the first task was generating linear response-functions. A simple linear response function is a sum of cosines and sines:

R0(t) = 1 N

N

X

j=1

cos(ωjtc) + i sin(ωjtc)

Where the ωj’s are wavenumbers in cm^{−1}, taken from a Gaussian distribution
around a wavenumber ω_{0}with some standard deviation σ.^{2} t is the time in ps and c
is the speed of light in cm/ ps. N is the number of realisations (molecules) that have
been used. A higher N corresponds to a situation with less noise. The wavenumbers
are taken from a Gaussian distribution, because all molecules will be in a slightly
different environment, resulting in a somewhat different response, caused by for ex-
ample a different velocity which causes a Doppler shift in the photon’s wavenumber.

Variations like this are often well described by using a Gaussian distribution. In a real system, the wavenumber that a molecule can absorb, can also change over time. In this model, the wavenumber is static for every molecule, which is an approximation, but in Section 3, a system with non-static wavenumbers will be treated.

In Figure 1, it can be seen that noise decreases when more realisations are eval-
uated. Decreasing the noise in this way seems reasonable, but is computationally
expensive. Statistically, the noise magnitude will be about ^{√}^{1}

N.

1τ = 1 ps has been used.

2In this section, a mean of 1000 cm^{−1}and a standard deviation of 50 cm^{−1}have been used.

0 1 2 3 4 5 0

0.2 0.4 0.6 0.8 1

Time (ps)

|Response| (AU)

N = 100 N = 500 N = 5000

Figure 1: Absolute values of generated response-functions, with different N .

### 2.2 Noise reduction

To enhance the spectrum for smaller ensembles, we want to reduce the noise. We do this as follows:

1. Fit^{3} |R_{0}| to the sum of a Gaussian function and a noise term, i.e.:

Rf it= e^{−}(τ^{t})^{2}

| {z }

Rsignal

+ ν

1 − e^{−}^{τ}^{t}

| {z }

R_{noise}

where τ and ν are the fit parameters. This way, we can extract parts of the fit that can be used for the correction. See Figure 2 for an illustration of a linear fit.

2. Define a function of time, which will be called α(t), and it can take values between 0 and 1. α(t) = 0 corresponds to fully trusting the original response, and α(t) = 1 corresponds to fully using Rsignal. How α(t) is defined will be explained later. Essentially, α(t) is a measure of how much we can trust the original response at a certain time t.

3. Construct the corrected response-function in the following way:

R2(t) = R0(t)

α(t)R_{signal}(t)

|R0(t)| + (1 − α(t))

Where Rsignal is the Gaussian term of the fit. It can be seen that the new
response function fully utilises the fitted Gaussian distribution when α(t) = 1
and it trusts the original data, R_{0}, when α(t) = 0. This way, α(t) decides what
fraction of the original response, R_{0}, is used for the corrected response, R_{2}. This
is exactly what α was intended for.

3This has been done with MATLAB’s built-in function fminsearch().

0 2 4 6 8 0

0.2 0.4 0.6 0.8 1

Time (ps)

|Response| (AU)

|R_{0}|
Rsignal

Rnoise

Figure 2: The fit terms illustrated for N = 500.

### 2.3 Defining α(t)

Since the value of all responses will be 1 at t = 0, the value of R_{0}can be fully trusted
at that time, i.e. α(0) = 0. However, at some point, the noise will begin to dominate
in R_{0}. To prevent this, we will set α(t) to start growing linearly at the time where
the R_{signal}term is equal to the noise term, and to stop growing linearly when R_{signal}
is equal to half of the noise. These conditions can of course be chosen differently. A
number of different conditions were evaluated, but this one worked reasonably well
overall. See Figure 3 for an illustration of the behaviour of α(t).

0 1 2 3

0 0.2 0.4 0.6 0.8 1

Time (ps)

(AU)

|R0|

|R_{2}|
R_{signal}
Rnoise

α(t)

Figure 3: α(t): |R_{2}| goes from |R_{0}| to the fit as α(t) increases. This is for N = 200.

### 2.4 Results

The aim of this method is to decrease the noise in the spectrum. To show that this has been achieved, Figure 4 demonstrates the impact of this new method, and the other method, to the spectrum. The new method (R2) does not significantly broaden the spectrum like the lifetime method (R1) does, while accurately reducing the noise.

The exact distribution has also been plotted. This has been obtained using a simple normal distribution, using the same mean and standard deviations that were used when creating the original response. It can be seen that the new method is more in agreement with the exact distribution than the lifetime method.

800 900 1000 1100 1200
ω (cm^{−1})

Intensity (AU)

R_{0}
R1

R2

Rexact

Figure 4: Spectra for N = 200. The exact spectrum is the Gaussian distribution from which the original spectrum was taken. All spectra have been scaled to the exact spectrum, for a better comparison.

### 2.5 N -dependence of spectrum errors

It is of course interesting to look at how this method behaves with different values of the number of realised molecules, N . In the new, corrected spectra, the sum of squared errors (SSE),P

D(Rexact− R1,2)^{2}, with the spectrum of the original normal
distribution (which has been used to generate the response) is a great measure of the
impact of the method. It has been compared with the lifetime method. The SSE
has been calculated by comparing the spectrum with the exact spectrum. The errors
are summed over the domain D = [ω0− 5σ, ω0+ 5σ].

0 5000 10000 15000 20000
10^{−8}

10^{−6}
10^{−4}
10^{−2}

N

SSE (AU)

Original response Lifetime−method Alpha−method

Figure 5: Sum of squared errors (SSE) for different N and different methods.

### 2.6 Conclusion

The new method works good for the Gaussian response. It can be seen that when the best other method uses 20,000 samples, the new α-method uses only about 1,000 samples for the same error. This is a significant improvement, and this could be really useful for simulating new linear spectra.

It is however not completely fair what happens in this section. The response function that was generated used a Gaussian distribution, and we fitted a function to it that made use of that fact. For this method to be a more useful method, it also has to work for response functions that are generated in a different, more advanced way.

### 3 Application: Carbon monoxide on a platinum surface

dr. T.L.C. Jansen has provided some of his recent simulation data for this section.

It consists of simulated response functions for a single monolayer of carbon monoxide
molecules on a platinum surface. This is a well studied system[10, 11] and it can be
seen as a benchmark for spectroscopy. In practice, the system is used as a catalyst
to turn toxic carbon monoxide into the less toxic carbon dioxide. The data is created
with the SFG (Sum Frequency Generation) technique^{4}[12].

In this system, the data is not Gaussian, and therefore the previous method would result in incorrect spectra. To solve this, another term has been added to the fit- function: an exponential, which would (seperately) result in a Lorentzian distribution in the spectrum. Now, the fit function is the following:

Rf it= Ae^{−}

t τg

2

+ (1 − A)e^{−}^{τl}^{t}

| {z }

R_{signal}

+ ν

1 − e^{−}^{τg}^{t}

| {z }

R_{noise}

Where A and (1 − A) are just there to make sure the fit is 1 at t = 0. The fit
parameters are now τ_{G}, τ_{L}, A and ν, with the condition that 0 ≤ A ≤ 1.

4The average site frequency is 2073 cm^{−1}which fluctuates with a standard deviation of 20 cm^{−1}.
The nearest neighbour coupling is 2.7 cm^{−1}. The correlation time is set to 1 ps, and the anharmonic-
ity is set to be 22 cm^{−1}.

Here, the α(t) part works similar. Only now, Rsignalconsists of both a Gaussian and a Lorentzian term. The results for the response and the spectra are shown in Figures 6 and 7, respectively. Do note that N now corresponds to the number of realisations of 100 CO molecules as a monolayer.

0 500 1000 1500 2000 0

0.2 0.4 0.6 0.8 1

Time (fs)

Response

|R0|

|R2|

|Rbest|
R_{signal}

Figure 6: Responses for carbon monoxide. N = 10 has been corrected, and compared
with R_{best}which used N = 5000. It can be seen that |R_{2}| leaves |R_{0}| just as the noise
begins to dominate in |R0|.

1950 2000 2050 2100 2150 2200
ω (cm^{−1})

Intensity (AU)

R_{0}
R1

R2

R_{best}

Figure 7: Spectra for carbon monoxide (N = 10).

The method also does what it should do when this non-Gaussian response is used.

The N = 10 response is generally in agreement with the N = 5000 response, which seems to indicate an efficiency increment of a factor 500. However, this is likely not the case, since the simulation of, say N = 2500 might also have sufficed for Rbest. This part of the analysis was more qualitatively, since not all values of N could be easily evaluated in the simulations.

The method does work for the non-Gaussian response as well, which is fortunate, and brings us hope.

### 4 2D spectroscopy: Gaussian response

The topic of this section is the extension towards 2D spectroscopy. This is useful, because 2D spectroscopy simulations in particular are computationally expensive, and thus would benefit from a noise reduction method.

In 2D spectra, there are two frequency axes, one for the pump pulse, and one for the probe pulse. So for the method to work, we need to extend the fit functions to suitable two-dimensional functions. We also need to find a new way to define α, since the linear interpolation technique will no longer work properly when a second dimension is added.

When measuring 2D spectra, the spectrum that is observed during the experiment is the sum of two parts: The rephasing part and the non-rephasing part[2, sec. 2.8].

The rephasing response is a function of the difference between t1 and t3, while the non-rephasing response is a function of the sum of t1and t3.

It was found that a suitable way of constructing and correcting a 2D spectrum is the following:

1. Construct two seperate response function: Rephasing and non-rephasing.

2. Apply the noise reduction methods to both response functions seperately.

3. Take the Fourier transforms of the responses to retrieve a rephasing and non- rephasing spectrum.

4. Add the two spectra to retrieve the final spectrum.

### 4.1 Uncorrelated Gaussian response

Just like the linear Gaussian responses, the 2D responses are constructed by adding sines and cosines. However, they have been replaced by exponentials using Euler’s formula to make it more concise:

R_{0,RP}(t_{1}, t_{3}) = 1
N

N

X

j=1

e^{i(ω}^{j,3}^{t}^{3}^{−iω}^{j,1}^{t}^{1}^{)c}

R0,N RP(t1, t3) = 1 N

N

X

j=1

e^{i(ω}^{j,3}^{t}^{3}^{+ω}^{j,1}^{t}^{1}^{)c}

Where ωj,i are wavenumbers taken from a Gaussian distribution around ω0,1 and ω0,3 with standard deviations σ1and σ3. R0,RP(t1, t3) and R0,N RP(t1, t3) denote the rephasing and non-rephasing responses, respectively. Variations in the wavenumbers over time are neglected in this model, so they are static.

Just like with the linear response functions, the noise will be more present when less molecules are realised, i.e. when N is lower. This has been illustrated in Figure 8, where the 2D rephasing response functions and 2D spectra for different N have been plotted. Note that the the exact spectrum should be circular, since the standard deviations are equal in both dimensions.

0 2 4 0

1 2 3 4 5

t1 (ps) t 3 (ps)

N = 100

0 2 4

0 1 2 3 4 5

t_{1} (ps)
t 3 (ps)

N = 1000

0 2 4

0 1 2 3 4 5

t1 (ps) t 3 (ps)

N = 5000

800 1000 1200 700

800 900 1000 1100 1200 1300

ω_{1} (cm^{−1})
ω 3 (cm−1 )

N = 100

800 1000 1200 700

800 900 1000 1100 1200 1300

ω_{1} (cm^{−1})
ω 3 (cm−1 )

N = 1000

800 1000 1200 700

800 900 1000 1100 1200 1300

ω_{1} (cm^{−1})
ω 3 (cm−1 )

N = 5000

Figure 8: Generated responses (rephasing) and full spectra for ω0,1 = ω0,3 =
1000 cm^{−1} and σ1= σ3= 100 cm^{−1}.

The fit function for the absolute value of the 2D uncorrelated Gaussian response would be:

Rf it= e^{−}

_{t1}

τ1

2

−_{t3}

τ3

2

| {z }

Rsignal

+ N (1 − e^{−}^{t1}^{τ1}^{−}^{t3}^{τ3}

| {z }

R_{noise}

)

The definition of α(t1, t3) has changed slightly. For the 2D α-method, both Rnoise

and Rsignal will be evaluated at every point in the time domain^{5}:

When _{R}^{R}_{signal}^{noise} < a, α(t1, t3) will be 0.

When _{R}^{R}_{signal}^{noise} > b, α(t1, t3) will be 1.

When a ≤ _{R}^{R}_{signal}^{noise} ≤ b, α(t1, t_{3}) will be

Rnoise Rsignal−a

b−a .

This way, again, α(t_{1}, t_{3}) will be zero when the response dominates, one when the
noise dominates, and a value in between in the transition period.

When the fit is made, and α(t1, t3) is initialised, we can construct the corrected response function, R2(t1, t3):

R2(t1, t3) = R0(t1, t3)

α(t1, t3)Rsignal(t1, t3)

|R0(t_{1}, t_{3})| + (1 − α(t1, t3))

The results of this method, along with the lifetime method, are shown in Figures 9 and 10.

5The values of a and b are currently 0.5 and 1, respectively, but again, they could be chosen differently, as long as a < b.

500 1000 1500 400

600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{0} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{1} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{2} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{exact} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{0} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{1} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{2} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{exact} spectrum

Figure 9: Spectra for N = 500, ω0,1 = ω0,3 = 1000 cm^{−1}, σ1 = 50 cm^{−1} and σ3 =
250 cm^{−1}. The R0 spectrum is a lighter blue where it is zero, because there are also
negative values.

500 1000 1500 400

600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{0} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{1} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{2} spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{exact} spectrum

Figure 10: Spectra for N = 5000, ω0,1 = ω0,3 = 1000 cm^{−1}, σ1 = 50 cm^{−1} and
σ3= 250 cm^{−1}. The R0 spectrum is a lighter blue where it is zero, because there are
also negative values.

To the eye, the spectra corrected with the new α-method seem to be corrected quite well. Fortunately, the calculated error with respect to the exact spectrum also tells us that the α-method behaves as we have expected. The N -dependence of the sum of squared errors has been plotted in Figure 11. The domain on which the errors are summed over is the domain where the exact spectrum is greater than or equal to half its maximum.

0 2000 4000 6000 8000 10000
10^{−10}

10^{−9}
10^{−8}
10^{−7}

N

SSE (AU)

R_{0} error
R_{1} error
R2 error

Figure 11: SSE for different N with ω_{0,1}= ω_{0,3} = 1000 cm^{−1}, σ_{1}= σ_{3}= 100 cm^{−1}.
In Figure 11, it can be seen that for the 2D Gaussian response, the lifetime spec-
trum when using 10,000 samples has the same error as the α-method with only 1,000
samples. This indicates that the method also works efficiently for 2D Gaussian re-
sponses.

### 4.2 Correlated Gaussian response

When the Gaussian response is correlated, it means that the t3 part of the response not only follows ω0,3, but also ω0,1. As a consequence of this, the response function will spread out over the t1= t3 diagonal. The magnitude of this spreading is determined by the correlation level. The spectrum will also change shape. It will be spread out over the diagonal ω1= ω3. This can be seen in Figure 12.

The response function for the correlated Gaussian response can be constructed like this:

R_{0,RP}(t_{1}, t_{3}) = 1
N

N

X

j=1

e^{−iω}^{j,1}^{t}^{1}^{c+i}(^{aω}j,1+√

1−a^{2}ω_{j,3}t_{3})^{c}

R0,N RP(t1, t3) = 1 N

N

X

j=1

e^{iω}^{j,1}^{t}^{1}^{c+i}(^{aω}j,1+√

1−a^{2}ω_{j,3}t_{3})^{c}

Where a is a measure of correlation. If we construct the response in this way, we end up with a uncorrelated response when a = 0 and a fully correlated response when a = 1.

The initialisation for α(t1, t3) remains the same as in Section 4.1, likewise the construction of the corrected response, R2(t1, t3).

0 0.5 1 0

0.2 0.4 0.6 0.8 1

t1 (ps) t 3 (ps)

R0, a = 0.8

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R0 spectrum, a = 0.8

Figure 12: Response (rephasing) and full spectrum for N = 1000 and a = 0.8.

The distribution of the spectrum is around ω0,1 = ω0,3 = 1000 cm^{−1} with standard
deviations σ1= σ3= 200 cm^{−1}.

It was thought that the fit function would need to be modified to correct for the correlation, but this was not the case. In Figure 13, the result is plotted for N = 1000 and a = 0.8, and the spectrum is corrected quite well.

0 0.5 1 0

0.2 0.4 0.6 0.8 1

t1 (ps) t 3 (ps)

R0

0 0.5 1

0 0.2 0.4 0.6 0.8 1

t_{1} (ps)
t 3 (ps)

R1

0 0.5 1

0 0.2 0.4 0.6 0.8 1

t1 (ps) t 3 (ps)

R2

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R0 spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R1 spectrum

500 1000 1500

400 600 800 1000 1200 1400 1600

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R2 spectrum

Figure 13: Responses (rephasing) and full spectra for N = 1000 and a = 0.8. The same distribution is used as in Figure 12.

### 5 Application: Carbon monoxide on a platinum surface (2D)

2DSFG[13, 14, 15] (2D Sum Frequency Generation) is a very new technique for generating 2D responses. T. L. C. Jansen provided some data, made with this new technique for this section[12]. Again, the simulated system is a monolayer of carbon monoxide on a platinum surface. The waiting time between the pump and probe pulses has been set to 0 in the simulation.

Because this is not a Gaussian response, we will alter the fit function to the following:

R_{f it}= Ae^{−}

_{t1}

τg,1

2

− _{t3}

τg,3

2

+ (1 − A)e^{−}^{τl,1}^{t1} ^{−}^{τl,3}^{t3}

| {z }

Rsignal

+ N (1 − e^{−}^{τg,1}^{t1} ^{−}^{τg,3}^{t3}

| {z }

Rnoise

)

This is similar to the linear fit for this system. Again, a Lorentzian term is added
to R_{signal}. The rest of the method works the same as in Section 4.1.

In Figures 14 and 15, the spectra are plotted for N = 10 and N = 500, respectively, to see the methods performing on the generated responses. A high sampled N = 5000 spectrum is also plotted as a reference. Note that also here, N corresponds to the number of realisations of 100 carbon monoxide molecules.

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R0 spectrum

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R1 spectrum

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R2 spectrum

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

Rbest spectrum

Figure 14: 2D spectra for carbon monoxide. N = 10 is used for the corrections, while
R_{best} uses N = 5000.

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{0} spectrum

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R_{1} spectrum

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

R2 spectrum

1500 2000 2500 1500

2000 2500

ω_{1} (cm^{−1})
ω 3 (cm−1 )

Rbest spectrum

Figure 15: 2D spectra for carbon monoxide. N = 500 is used for the corrections, while Rbest uses N = 5000.

If one looks at the negative (blue) region of the spectrum in Figure 14, it looks like the lifetime method did a better job of correcting the response. However, in the positive (red) region, it looks like the α-method did a better job. It looks like the lifetime method is better in the negative region, because by chance, the original spectrum is too narrow in that region. In Figure 15, that same region is less narrow.

When we look at the total sum of squared errors for N = 10, the new method has a smaller error than the lifetime method by a factor of 6.08, and a smaller error than the original response by a factor of 1.35. Relative errors with other N are listed in Table 1.

N ^{SSE(R}_{SSE(R}^{0}^{)}

2)

SSE(R_{1})
SSE(R2)

10 1.35 6.08

100 1.03 18.87

500 1.03 290.23 1000 0.96 949.78

Table 1: Relative SSE’s for different N ’s, summed over the entire frequency domain.

Errors are calculated by comaring with Rbest, which used N = 5000.

It seems the method also works for non-Gaussian 2D responses. However, the value 0.96 at N = 1000 in Table 1 seems strange. This could be because the corrected

response is actually better than Rbest, which could indeed result in a value lower than 1. It is also peculiar that the errors in R1are higher than in R0, and that the error in R1 also rises relative to the other errors when N increases. This could be caused by the broadening of the spectrum and the fact that we sum over the entire frequency domain.

### 6 Conclusion

The new method of noise reduction seems like a succesful method. If it indeed could reduce the number of samples by a factor of 10, as for the 2D Gaussian response, it would be a succesful new tool for spectroscopy simulations. If, for example, a simulation would have previously taken 10 days, a corrected simulation with the same error could be made within only 1 day. Nonetheless, it is questionable wether the new method would actually result in such fast simulations, because the results in Section 5 are debatable.

A proposal for further research would be to do more experiments with the new method on simulations like in Section 5, and, if possible, more quantitatively. The fit functions could also be altered, and the definition of α could be revisited.

### 7 Acknowledgements

I want to thank T. L. C. Jansen and C. P. van der Vegte for supporting me during the process of the project and for assisting me with the report and the presentation, and I would like to thank M. S. Pshenichnikov for attending my presentation and reviewing the final report. Also, I want to thank the Theory of Condensed Matter Physics group for welcoming me to the group.

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