Manual for QSim

An NMR simulation program for PCs

© Magnus Helgstrand and Peter Allard

Version 1.0

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1. Disclaimer ..................................................................................................................5 2. Introduction................................................................................................................6 3. QSim ..........................................................................................................................7

3.1. Installation...........................................................................................................7 3.2. File extensions ....................................................................................................7 3.3. QSim documents.................................................................................................7 3.4. View vs. Window................................................................................................7

4. Pulse sequence construction ......................................................................................9 4.1. Adding channels..................................................................................................9 4.2. Adding Pulse elements to a channel .................................................................10 4.3. Sim-markers......................................................................................................10 4.4. Pulse elements...................................................................................................11

4.4.1. Pulse...........................................................................................................11 4.4.2. Delay ..........................................................................................................12 4.4.3. Shaped pulse ..............................................................................................12 4.4.4. Decoupling sequence .................................................................................14 4.4.5. FID .............................................................................................................14

4.5. Variables ...........................................................................................................14 4.5.1. The Shape Variable....................................................................................16 4.5.2. The Decoupling variable............................................................................16

4.6. Watches.............................................................................................................17 5. Spin system construction .........................................................................................19

5.1. Spin properties ..................................................................................................19 5.2. Spin-spin interactions........................................................................................19

5.2.1. Scalar couplings .........................................................................................19 5.2.2. Residual dipolar couplings.........................................................................19

5.3. Relaxation .........................................................................................................20 5.3.1. Model of Dynamics....................................................................................20 5.3.2. Relaxation interference effects ..................................................................21

5.4. Chemical exchange ...........................................................................................22 6. Simulation ................................................................................................................23

6.1. Experimental parameters ..................................................................................23 6.2. Simulation parameters ......................................................................................24

6.2.1. Classical simulations..................................................................................24 6.2.2. Quantum mechanical simulations ..............................................................24 6.2.3. Gradients ....................................................................................................24 6.2.4. Average Liouvillians vs. Pulses.................................................................24

6.3. Advanced options..............................................................................................24 6.3.1. Using Watches ...........................................................................................24 6.3.2. Using custom start magnetization..............................................................25

6.4. Managing simulations.......................................................................................26 6.4.1. Simulation times ........................................................................................26

7. Processing ................................................................................................................27 7.1. Processing parameters.......................................................................................27

7.1.1. 1D...............................................................................................................27 7.1.2. 2D magnitude.............................................................................................27 7.1.3. 2D TPPI .....................................................................................................27 7.1.4. Stacked 1D.................................................................................................27 7.1.5. 2D States ....................................................................................................27

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7.1.6. 2D Echo-antiecho.......................................................................................27 7.1.7. Zero filling .................................................................................................27 7.1.8. Phasing.......................................................................................................28 7.1.9. Window functions......................................................................................28 7.1.10. Special processing....................................................................................28 7.1.11. Plotting parameters ..................................................................................28

7.2. The Spectrum window ......................................................................................29 8. Examples..................................................................................................................30

8.1. HSQC with sensitivity enhancement ................................................................30 8.2. TOCSY spectrum with TPPI ............................................................................31 8.3. Nondecoupled HSQC with chemical exchange................................................32 8.4. Classical simulation of a NOESY spectrum .....................................................33 8.5. Mixing sequence simulation .............................................................................34 8.6. QP-DD Cross correlation..................................................................................36 8.7. TROSY .............................................................................................................37 8.8. HNCO ...............................................................................................................38 8.9. Carbonyl 13C transverse relaxation ...................................................................39

9. Theoretical background ...........................................................................................41 9.1. Basic equations .................................................................................................41 9.2. Spin Hamiltonian ..............................................................................................42 9.3. Basis operators ..................................................................................................43 9.4. Chemical exchange ...........................................................................................43 9.5. Relaxation .........................................................................................................45 9.6. Model of dynamics ...........................................................................................47 9.7. Equilibrium density operator ............................................................................47 9.8. Pulsed field gradients ........................................................................................48 9.9. Simulating Spectra ............................................................................................48 9.10. Average Liouvillian ........................................................................................49 9.11. Signs and Phases .............................................................................................49

10. HME.......................................................................................................................50 10.1. The spin system class......................................................................................50 10.2. The spinoperator class.....................................................................................50 10.3. The superoperator class...................................................................................50

11. Contact information ...............................................................................................51 12. References..............................................................................................................52

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1. Disclaimer QSim is free to use for all users and is distributed “as is”. Although the authors have put a large effort into trying to make QSim as accurate as possible the authors take no responsibility for the correctness of any simulations performed with QSim.

The authors are in no way liable for any damage caused directly or indirectly by the use of QSim.

Although QSim is free to use, the program may not be redistributed without consent from the authors.

If errors are found in QSim please report them to the authors, see the contact information section.

A paper describing QSim will be published, please reference this paper when using QSim.

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2. Introduction This is the manual for QSim (version 1.0) running on PCs under Windows. QSim is a program for simulation of NMR pulse sequences.

QSim is intended as an accurate simulation program with a well known, easy to use graphical interface. QSim is intended for the advanced user wishing to simulate complicated interaction mechanisms and/or complicated pulse sequences. But, since QSim is easy to use, it is also intended for users with limited or no experience of NMR. QSim can then be used as a tool for learning how spins interact, what results pulse sequences produce and how FIDs are processed. QSim can for example be used to teach NMR, using predefined spin-systems and/or pulse sequences.

QSim performs simulations on any spin system with any number of spins. It performs calculations using quantum mechanics or optionally, for spin systems with spins with quantum numbers ½ or 1, classical mechanics. It handles first order equilibrium chemical kinetics. It calculates relaxation rates using chemical shift anisotropy, dipole-dipole and quadrupolar relaxation mechanisms in any combination, i.e. including all possible interference, cross-correlation, effects.

Relaxation and chemical exchange are always included in all parts of the simulation. QSim supports quantum mechanical simulations of partially oriented systems using residual dipolar and quadrupolar coupling constants. A method for calculating the effect of pulsed field gradients in the z-direction is also implemented.

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3. QSim

3.1. Installation QSim is distributed as a zip-file. To install QSim expand the zip-file in, for example, the “Program files” folder. QSim is then placed in a folder named “QSim”.

The QSim distribution contains the following elements:

• QSim\QSim.exe – the program • QSim\QSim.hlp – help file for QSim • QSim\Examples – a library of examples, described in the examples section in

this manual. • QSim\Doc – A folder with documentation, including this manual.

After uncompressing QSim the program can be run directly, without the need for any further installation steps.

3.2. File extensions QSim uses three file extensions for documents. The main documents for QSim have the extension “.qsd” and these files are associated to the program as soon as the program has been run once. The QSim documents then get their own icon and can be double clicked to start QSim.

QSim also uses the extension “.qss” for files containing exported spin systems and “.qsp” for exported pulse sequences. These file formats are binary and specific for QSim. These files are not associated to QSim in the way “.qsd” files are, i.e. it is not possible to directly double click on them to start QSim.

QSim also uses text-files, ASCII files, to store exported data from simulations. These files can be viewed by any text editor. The file extension for these file is “.txt”. Some of these text files are aimed at Matlab by giving data in a form readable to Matlab as a matrix.

3.3. QSim documents QSim documents by default contain the pulse sequence with variables, the spin system, simulation parameters, simulated FIDs and processed data. It is possible to exclude simulated FIDs and processed data by choosing “Document options…” from the “Tools” menu. By not saving simulated and processed data, files are getting considerably smaller, but still contain all data necessary to simulate the data. Examples distributed together with QSim are saved in this way to minimize the size of the distribution.

3.4. View vs. Window QSim uses the Multiple Document Interface (MDI) as implemented by Microsoft in Microsoft Foundation Classes (MFC). This is a user interface recognized by most Windows users and allows the application to have multiple open documents and multiple windows for each document.

There are two different menu entries to control what windows should be shown. By choosing “View” from the menu users can show and hide windows associated with a document. Checked windows are shown, others are not shown. The

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windows presented under the “Windows” menu are those that are checked under the “View” menu in any open document.

In practice, first show a window for a document selecting it under the “View” menu, the next time it should be shown use the “Window” menu to put it on top.

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4. Pulse sequence construction Pulse sequences in QSim are constructed using the mouse and a set of tools in a toolbox. Pulse sequences can be exported and imported in a program-specific format. To export a pulse sequence select “Export” from the “File” menu and then select “Pulse sequence”. To import a pulse sequence select: File-Import-Pulse sequence.

An Undo/Redo function is implemented in the Pulse sequence construction window. It is possible to undo in 20 steps. To undo, either choose “Undo” from the “Edit “ menu or press Ctrl+Z. To redo, choose “Redo” from the “Edit” menu or press Ctrl+Y

The pulse sequence can be copied to the clip-board as a Windows meta-file or printed by selecting “Copy to clipboard” from the “Edit” menu or by pressing Ctrl+C..

4.1. Adding channels The process of constructing a pulse sequence starts by adding one or more channels. To add a channel to a pulse sequence, open the “Sequence” menu and select “Add Channel…” A new channel is then added with the default nucleus 1H. To change the nucleus right-click to the left of the channel and select “Properties…” from the pop-up menu. The following dialog is then showed:

Figure 1. The channel properties dialog. The name field is used to name the channel and the isotope drop down list is used to select the isotope for the channel.

In the “Name:” field the name of the channel is set, the isotope for the channel is selected from the “Isotope:” drop-down list. Only isotopes listed can be chosen. The name of the channel is only used as a marker when viewing and printing the pulse-sequence. The isotope is, on the other hand, used in simulations and it is thus important to correctly set this value.

To add more channels for other isotopes, just repeat the process. It is important to only include one channel for a certain isotope. It is not permitted to include, for example, one channel for 13CO and one for 13C, since these channels should then be assigned the same isotope.

A channel (and all its content) can be deleted by right-clicking to the left of the channel and then selecting “Delete channel”. The channel is then immediately deleted.

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4.2. Adding Pulse elements to a channel To add a pulse element, such as a pulse, delay or mixing sequence, select one of the tools in the pulse element toolbar.

Figure 2. The pulse element toolbar. The tools are from the left: hard pulse, delay, shaped pulse, decoupling/mixing sequence, sim-marker and FID.

After a tool has been selected, add an element by clicking at the position where the element should be placed. Pulses, shaped pulses and decoupling/mixing sequences are colored red when added, indicating that no variables are defined for the elements (see further below).

To set the acquisition channel select the FID-tool and click to the right of the chosen detection channel. To change acquisition channel just use the FID-tool to select a new channel, and the FID-element will be moved to the new channel. To add a decoupling sequence for the acquisition period select the decoupling/mixing element tool and click to the right of the channel, in the same manner as for the FID.

To delete pulse elements from the pulse sequence, right click on the element (for delays just above the line) and choose “Delete element” from the pop-up menu.

Figure 3. A simple pulse sequence with a hard pulse, a delay and a FID. Variables are not defined, as indicated by the red color of the pulse and the FID, and the lack of variable name for the delay.

4.3. Sim-markers The sim-marker tool is used to indicate points that are simultaneous in time in multiple channels. Sim-markers are depicted as vertical lines crossing between

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channels. To add a sim-marker, choose the sim-marker tool from the toolbox and then click on the pulse elements in the channels that should be simultaneous. To end the sim-marker, double-click on the last element. After adding a sim-marker the program will adjust the pulse sequence accordingly. Below is an example of the appearance of a pulse sequence before and after adding a sim-marker.

Figure 4. To the left a pulse sequence before a sim-marker was added. To the right the result after adding a sim-marker. In this case the sim-marker was positioned in the center of the pulse elements.

Sim-markers can be placed at the beginning, in the center or at the end of pulse elements. The default setting is to set the sim-marker in the center, to change this right-click on the pulse element, select “Sim Marker”, “Set Position” and then choose position.

4.4. Pulse elements Pulse elements available are pulse, delay, shaped pulse and decoupling/mixing sequence. Properties for the pulse elements are set by right-clicking on the element and then choosing “Properties…” from the pop-up menu. The different elements will now be described.

4.4.1. Pulse Pulses are depicted by a filled rectangle. If all properties for a pulse have been defined the color is black, otherwise red. Pulses have four properties:

• Frequency • Phase • Length • Power

Each property must be assigned a variable. All variables, but the phase variable, are specific for the current channel.

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The graphical length of a pulse is assigned by the length variable. Note that this length is not automatically related to the actual length in time.

4.4.2. Delay Delays are depicted by a black horizontal line. Delays are only defined by a length. The name of the length variable associated to the delay is shown under the delay. The graphical length is set to the value defined in the length variable. If no variable is defined for the delay the delay is dynamic. The length of a dynamic variable is adjusted according to other elements and sim-markers. For the length of a dynamic delay to be calculated by the program, it must be surrounded by elements with defined lengths and/or sim-markers. See Figure 5 for examples on allowed and disallowed placements of dynamic delays.

4.4.2.1. Incremental delay Evolution periods, for spectra with higher dimensionalities, are defined by incremental delays, i.e. delays with incremental length variables (see below). Incremental delays must follow certain rules:

• Incremental delays must be surrounded by sim-markers on both sides. The

sim-markers must go through all channels (including the gradient channel if present). One sim-marker could for example be placed at the beginning of the incremental delay and another at the beginning of the element following the delay.

• No pulses, in any channel, are allowed during incremental delays. • Decoupling sequences are allowed during incremental delays. They must then

be dynamic and no longer than the evolution period. • If pulses should be present during evolution periods, the evolution period can

be divided into multiple periods. Each period then has to be assigned a incremental delay variable.

Examples of pulse sequences for 2D spectra are presented in the examples section.

4.4.3. Shaped pulse Shaped pulses are depicted by a Gaussian. If all properties for a shaped pulse have been defined the color is black, otherwise red. Shaped pulses have five properties:

• Frequency • Phase • Length • Power • Shape

Each property must be assigned a variable. All variables, but the phase and shape variables, are specific for the current channel.

The graphical length of a shaped pulse is assigned by the length variable. Note that this length is not automatically related to the actual length in time. The shape of shaped pulses is defined by the shape variable, and is defined by a number of amplitude and phase values (See below).

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1H

d1

15N

1H

d1

15N

1H

d1

15N

1H

d1

15N

AllowedNot allowed

Two dynamic delays nextto each other

Pulse with undefined timebetween two dynamic delays

Figure 5. Examples of disallowed placements of dynamic delays to the left and the solution to the right.

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4.4.4. Decoupling sequence Decoupling/mixing sequences are depicted by filled boxes. If all properties for a decoupling sequence have been defined the color is black, otherwise red. Decoupling/mixing sequences have five properties:

• Frequency • Phase • Length • Power • Decoupling sequence

Each property, except the length, must be assigned a variable. All variables, but the phase and decoupling variables, are specific for the current channel.

The graphical length is set to the value defined in the length variable. If no length variable is defined for the decoupling sequence, it is dynamic. The length of a dynamic decoupling sequence is adjusted according to other elements and sim-markers, in the same way as for dynamic delays. If a decoupling sequence is used under acquisition, a length variable does not have to be assigned. The pulses in the decoupling sequence are defined by the decoupling sequence variable, further discussed below.

4.4.5. FID The FID pulse element is used to define the acquisition channel and is depicted as an exponentially decaying sine wave. The only property of the FID is the receiver phase, and it is set by “right-clicking on the FID symbol. As long as the FID is not assigned a phase variable the color of the FID is red.

4.5. Variables Variables in simulations are used in a similar way as on an NMR-spectrometer. They have the following general properties:

• Variables are symbolic; they have a name and a value. • A single variable can be associated to multiple pulse elements. • Variables can be static or vary in different ways depending on transient

number. • Variables all have integer values. • If a variable is no longer used in a pulse sequence it is automatically deleted. • Variables are stored with the pulse sequence.

Variables are set in the Variables window. The variables window is opened by selecting “Variables” from the “View” menu. All variables are categorized and presented for each channel. To set the properties for a variable double-click on the name of the variable to get the variable properties dialog box.

Before a simulation can be started all variables must be set. The program remembers which variables are set and which are not set. A variable is defined as set if the variables properties dialog box has been opened for the variable, it is thus important to open the properties dialog box for all variables, also if the default value should be used.

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Figure 6. The variable properties dialog box. Shown are the properties for a delay variable.

In the variable properties dialog box the properties for the variables are set. The name of the variable can be changed. The graphical length is set for length variables; this field is inactive for variable for which the graphical length can not be set. The unit for the variable is set using a drop-down list. Only units available in the drop-down list can be used for the specific variable. The value of a variable is given as a static, cyclic, super cyclic or incremental value. Values must be given as integers; the resolution is given in Table 1.

Variable type Resolution Frequency 1 Hz Phase 1 º Length 1 ns Power 1 Hz

Table 1. Resolution of variables.

Static variables are constant throughout the entire simulation. Cyclic variables follow the transient number and are for example used for phase cycling. Super cyclic variables are cycled following the super cycle, as discussed in the simulation section. Super cyclic variables are typically used to obtain phase sensitivity in indirect dimensions using the States method1. Super cycles can be added to cycles, to be able to cycle a super cyclic variable. Cyclic and super cyclic variables are given as a list of numbers separated by any non numerical character. Incremental variables are incremented after a full super cycle and are, for example, used for evolution periods in indirect dimensions. The base for the increment can be set, by doing so the variable can be decremented by giving a negative increment. The order should be given a number larger than zero. Incremental variables with the same order are incremented synchronously. The number of increments is given among the simulation parameters. A cycle can be added to incremental variables, by doing so the TPPI method2 can be simulated.

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4.5.1. The Shape Variable Shape variables have special properties and a property dialog of their own. In the dialog box the shape of the pulse can be imported and exported. To import a shape for a shaped pulse click on “Import…” and select a file. The file should be a text file with two numbers on each row separated by for example a semicolon or a “tab” character. The first number on each row is the intensity in arbitrary numbers and the second is the phase in degrees. After the shape has been imported, the values for “Max. intensity” and “Rel. intensity” are given. The “Max. intensity” is the largest number found in the intensity column. The maximum intensity value is used to scale the shape, i.e. the power used for a particular point in the shape is given by the intensity in that point divided by the maximum intensity times the power variable.

The relative intensity compares the summed intensity of the shaped pulse to a hard pulse at the maximum intensity. This value can be used to calculate the length of, for example, a Gaussian pulse in relation to a hard pulse. The formula for calculating the relative intensity is

max

1

iN

ii

N

kk

R ⋅=∑=

in which iR is the relative intensity, ik is the intensity at point k, imax is the maximum intensity and N is the number of points in the shape.

Shape variables can also be given a short description for internal use. The shape variable properties dialog is shown in Figure 7.

Figure 7. The properties dialog for the shape variable.

4.5.2. The Decoupling variable Decoupling variables have special properties and a property dialog of their

own. In the dialog box the decoupling sequence can be imported and exported. To import a decoupling sequence click on “Import…” and select a file. The file should be

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a text file with three numbers on each row separated by, for example, a semicolon or a “tab” character. The first number on each row is the pulse angle in degrees, i.e. it is 90 for a 90-degree pulse. The second number is the phase in degrees and the third number is the relative power of the pulse. A 90-degree pulse with zero power is the same as a delay with the same length as a 90-degree pulse. The lengths of pulses are calculated using the power variable associated to the decoupling/mixing sequence. See Figure 8 for an example of a decoupling sequence variable properties dialog box.

Figure 8. The properties dialog for the decoupling variable

4.6. Watches Watches are used to probe the state of the spin-system at a certain position of the pulse sequence. The “watch” is associated to a certain pulse element, and can be placed at the end or at the beginning of the element. To add a watch, right-click on the pulse element, select “Watch” and then “Add”. Watches are depicted as small green circles in the pulse sequence. To move the watch right-click on the element to which it is associated and select “Watch”, “Position” and then select position.

Watches are not used in default simulations, so the flag to calculate the value for watches must be set among the simulation parameters. If watches are calculated at simulations, the simulation runs considerably slower.

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Figure 9. A pulse sequence with a watch place at the end of the 90-degree y-pulse. The watch will give the magnetization for the first point, at time zero, in the FID.

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5. Spin system construction Spin systems in QSim are constructed in the spin system window. To view the window select “View” and then “Spin system”. Spin systems can be exported and imported in a program-specific format. To export a spin system select “Export” from the “File” menu and then select “Spin system”. To import a spin system select: File-Import-Spin system.

5.1. Spin properties The first step in constructing a spin system is to give it a name and then give the number of spins that should be included in the system. After giving the number of spins, fields for entering properties for the spins are shown.

Set the isotope for the involved spins. If the nucleus given is recognized by the program the values for angular momentum and magnetogyric ratio is automatically set.

Isotropic chemical shifts are given in Hz. The rotating frame Hamiltonian used to simulate chemical shifts is given by

∑= izi IH πν2

in which υi is the chemical shift in Hz. If the spin system is partially oriented, there is an option to include residual

quadrupolar coupling constants for spins with angular momenta larger than ½. The residual quadrupolar coupling is then included in the simulations with the following rotating frame Hamiltonian:

⎟⎠⎞

⎜⎝⎛ −= 22

31 SSH zQπυ

In which υQ is the residual quadrupolar coupling constant in kHz as observed in spectra, without the factor 2 often used.

5.2. Spin-spin interactions

5.2.1. Scalar couplings The rotating frame Hamiltonian used to include scalar coupling is

( )∑ ++= lzkzlykylxkxkl IIIIIIJH 2π

for the homonuclear case and

∑= lzkzkl SIJH 2π

for the heteronuclear case. The scalar coupling constant between spin k and l is denoted Jkl and is given in Hz.

5.2.2. Residual dipolar couplings If the spin system is partially oriented, there is an option to include residual dipolar couplings between spins. The rotating frame Hamiltonian used to include residual dipolar couplings is

( )∑ ⎟⎠⎞

⎜⎝⎛ +−= lykylxkxlzkzkl IIIIIIDH

212π

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for the homonuclear case and

∑= lzkzkl SIDH 2π

for the heteronuclear case. The residual dipolar coupling constant between spin k and l is denoted Dkl, and is given in Hz as observed in spectra, without the factor 2 often used.

5.3. Relaxation To be able to calculate dipole-dipole relaxation rates the distances between the spins in the spin system must be known, it is thus necessary to give the distance in Å between the nuclei. If the distance is set to zero, it is interpreted as an infinite distance.

If distances are set to non-zero values the dipole-dipole relaxation mechanism is automatically included in simulations. Other relaxation mechanisms can be included by setting non-zero values. These are the Chemical Shift Anisotropy (CSA) and the quadrupolar relaxation mechanisms. For CSA the parallel and perpendicular components of the axially symmetric shielding tensor are given in ppm. For quadrupolar relaxation the quadrupolar coupling constant is given in kHz.

5.3.1. Model of Dynamics The model of dynamics used in simulations is derived using the Lipari-Szabo approach3, 4. The analytical spectral density function for this model is

( )( )

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

+

−+

+= 2

2

2

2

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152

i

i

m

m SSJ

ωττ

ωττ

ω

with

emi τττ111

+=

in which τm is the rotational correlation time, τe is the correlation time for internal motion and S2 the order parameter squared describing the balance between contributions to relaxation due to overall rotation and internal motion. In simulations in QSim all auto and cross-correlation relaxation mechanisms have their separate set of correlation times and order parameters. The longer correlation time is thus not assumed to be the isotropic rotational correlation time, but rather one of two correlation times approximating the dynamics of a particular vector or differential dynamics for a pair of vectors.

Diagonal elements in the matrices for S2, τm and τe are used to calculate quadrupolar and CSA relaxation rates, off diagonal elements are used to calculate dipole-dipole relaxation rates.

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Figure 10. The order parameters squared for a three-spin system. Diagonal elements are used to calculate quadrupolar and CSA relaxation rates. Off diagonal elements are used to calculate dipole-dipole relaxation rates.

5.3.2. Relaxation interference effects Relaxation interference effects can be included in simulations. The number of interference effects to include is given in the field “Nr of interference inter”. After a number is given, press “Enter” and boxes for filling in the interactions occur. Interference effects are given one by one (See Figure 11). The first field, in the column, is the first relaxation mechanism, use “CSA”, “DD” or “QP” for CSA, dipole-dipole and quadrupolar relaxation respectively. The second and third fields are used to define the nuclei associated to the first relaxation mechanism. Use only the second (named “First spin”) field for CSA and QP. The second together with the third fields are used for DD to define the two spins involved in the dipole-dipole relaxation. The fourth field (named “Second mech.) is used to define the second relaxation mechanism. Fields five and six are used in the same manner as fields two and three. Fields seven through nine are used for model free parameters. The last field is used to give the angle between the principal axes of the tensor and/or dipole-dipole vector(s).

Figure 11. Example of a relaxation interference effect between the CSA relaxation mechanism and the dipole-dipole relaxation mechanism.

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5.4. Chemical exchange First order chemical kinetics can be included in simulations. The number of states for the kinetics is given in the “Size of kinetics field”, just below the field for the number of spins in the spin system. After giving a number and pressing “Enter”, a kinetics matrix is presented. The kinetics matrix must be calculated by the user as shown in 9.4 (equation 22) in the theoretical backgrounds section and then given to QSim.

While adding chemical exchange to the spin system, the number of spins is automatically increased, as seen in Figure 12.

Figure 12. An example of a kinetics matrix for two-state chemical exchange. Note the extra spins added. In this case spin 1 is in exchange with spin 3 and spin 2 with spin 4. (This spin system is taken from the example in section 8.3)

If the size of the kinetics matrix is n, that is there are n states in chemical exchange, and the number of spins is s, then spin x is in chemical exchange with spins x+s, x+2*s, ..., x+(n-1)*s.

All states in chemical exchange have specific properties for all spin properties except, isotope, angular momentum and magnetogyric ratio. It is thus, for example, possible to have different scalar couplings in different states as well as having different dynamic properties. If relaxation interference effects are included, they must also be set for all spins in the spin system.

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6. Simulation Simulations in QSim are set up and started in the simulation window. To view the window, select “View” and then “Simulation”.

6.1. Experimental parameters Simulation parameters must be set prior to simulating the pulse sequence. The first parameter to set is the external magnetic field in Tesla. The number of transients for each phase cycle must be given. Transients are added together, and there is no way to access individual transients after the simulation.

The number of super cycles is the number of data sets that should be stored for each increment (see below). If, for example, 8 transients and 2 super cycles are used and the number of increments is set to 1, the number of FIDs simulated will be 2, each a sum of the 8 transients. If there are super cyclic variables the super cyclic variables will vary depending on the super cycle number (see example of States pulse sequence in the Examples section).

The number of points in the FID is the number of complex points that will be simulated for the FID. The dwell time between the points is given next. Remember that 1/dwell time equals the spectral width. Dwell times are given in seconds.

FID number Super Cycle Incremental order 1

Incremental order 2

1 1 1 1 2 2 1 1 3 3 1 1 4 4 1 1 5 1 2 1 6 2 2 1 7 3 2 1 8 4 2 1 9 1 1 2 10 2 1 2 11 3 1 2 12 4 1 2 13 1 2 2 14 2 2 2 15 3 2 2 16 4 2 2

Table 2. An example of the order of the FIDs for a simulation with a four step super cycle and two incremental orders with two increments each.

If there are incremental variables in the pulse sequence, the number of increments must be given. Double click on the incremental order and set the number of increments. If there is more than one incremental order present, the incremental order highest in the list will be incremented first and then the second, and so on. See Table 2, for an example on in which order FIDs are simulated and stored.

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6.2. Simulation parameters Simulation parameters determine how the simulation should be performed by the program. In most cases the default values could be used.

6.2.1. Classical simulations To perform classical simulations, as opposed to quantum mechanical simulations, click on this radio button. Classical simulations are performed using the extended Solomon and McConnel equations5, 6. Using classical simulations, large number of spins can be simulated. Classical mechanics simulations are typically used to simulated NOESY spectra for large spin systems.

6.2.2. Quantum mechanical simulations To perform quantum mechanical simulations, as opposed to classical simulations, click on this radio button. Quantum mechanical simulations are performed using the methodology presented in the Theoretical background section.

6.2.3. Gradients The number of slices the sample should be divided into is given in this field. The more slices the more accurate the result of the simulation, but the longer the simulation takes. If unexpected results are obtained when simulating pulse sequences with gradients, try to increase this number to see if it is an effect of the number of gradient slices.

The gradient slices are usually divided evenly through the sample (the sample length in simulations is 1 cm). This could potentially lead to “resonance effects”, and to avoid these effects the slices could be randomly distributed in the sample. Our experience is that a larger number of slices is necessary when using a random distribution as compared to an even distribution.

6.2.4. Average Liouvillians vs. Pulses Mixing and decoupling sequences can either be simulated using the actual mixing/decoupling pulse sequence or an average Liouvillian of the mixing/decoupling pulse sequence. Average Liouvillian simulations are faster than simulating the actual pulses, but have certain limitations as discussed in the Theoretical background section. If mixing/decoupling sequences are too long the average Liouvillian approach should not be used, the program will in these cases issue a warning.

6.3. Advanced options Advanced options are used to override some of the default simulation settings. It is for example possible to turn off the calculation of FIDs, which makes simulations in which Watches are the only interesting result a bit faster.

6.3.1. Using Watches Watches are used to probe the magnetization at certain points given in the pulse sequence. To be able to save results from Watches, at least one Watch must be included in the pulse sequence. To save Watches to file mark the checkbox and select the file to store watches to. Watches are stored for every transient, in every super cycle and increment, but watches for gradient slices are not stored separately. The magnitudes of all Cartesian product operators are stored at every Watch. Watches are numbered according to where they occur in the pulse sequence, the first Watch being

24

number one. The Watch-file is a semicolon separated text file, suitable to import into spread sheet software, such as Microsoft Excel. An example of a Watch file in Excel is given in Figure 12.

Figure 13. Excel document with an imported Watch file. E is the unity operator, and the spins are numbered 1, 2 and 3. Thus E*E*3x is the same as Ix for the third spin.

6.3.2. Using custom start magnetization The default start magnetization for a simulation is the equilibrium magnetization. Among the parameters in the advanced section the start magnetization can be altered in two ways, using a custom start magnetization and/or using the magnetization at the end of the previous transient.

The start magnetization can be set to any user given value for any of the Cartesian product operators. This is done by clicking the “Use custom start magnetization” check box and then clicking the “Set…” button. In the dialog box appearing the equilibrium values are given, to change them mark the operator for which the value should be changed, set the new value and then click “Change” to update.

As default the equilibrium magnetization is the starting value for each transient during simulation. This can be changed to instead using the resulting magnetization from the previous transient. This makes it possible to simulate inter scan delays. It also makes it possible to simulate the outcome of a mixing-sequence as a function of time (see 8.5).

25

6.4. Managing simulations Simulations are started by clicking the “Start Simulation” button. The pulse sequence is then compiled and the Hamiltonians, relaxation parameters and the Liouvillians constructed. If the pulse sequence can not be compiled, due to errors in the sequence, the simulation is aborted and the place where the error occurred will be marked with a red arrow in the pulse sequence.

After the simulation has started the progress of the simulation is shown in the “Simulation status” box. The simulation will run in the background with low priority in a special simulation process. This enables the user to use the computer for other tasks during simulations. It is also possible to run simultaneous simulations but they must be contained in different documents. Simultaneous simulations support multiple processors using modern operating systems.

If the simulation must be aborted, click the “Abort simulation” button. It may take some time to abort the simulation, depending on the complexity of the pulse sequence and the spin system. The program will notify the user when the simulation has aborted. Aborting one simulation will not affect other running simulations. Data from aborted simulations are not accessible.

When the simulation is completed the user will be notified by a pop-up dialog box. The data is not automatically stored, so it is recommended to store the data before continuing to processing.

6.4.1. Simulation times Simulations with complex pulse sequences and/or many spins may take long time. It is not recommended to use more than four to five spins in quantum mechanical simulations. Gradients also take long time to calculate and simulate, it is thus not recommended to have too many slices for gradients.

Using “Watches” will slow down simulations since some optimizations must be turned off for the program to be able to calculate the magnetization at these time-points. It is thus recommended to turn of the calculation of Watches, when they are not specifically required.

It is slightly more efficient to use average Liouvillians for decoupling/mixing than using pulse sequences, but the difference is not enormous due to optimization of pulse sequences before start of simulations. The most time is saved with average Liouvillians if decoupling is present during evolution periods, either or both in indirect and direct dimensions.

26

7. Processing If a dataset is available in a QSim document, the data can be processed. By default data, i.e. FIDs, are saved in the document, it is thus not necessary to perform the simulation again for a newly opened document. Processing in QSim is performed in the processing window. To view the window, select “View” and then “Processing”.

The unprocessed FIDs can be exported to file in a text format suitable for Matlab by choosing “Export FID…” from the “Processing” menu.

7.1. Processing parameters There are six different processing schemes available in QSim. They are:

• 1D • 2D magnitude • 2D TPPI • Stacked 1D • 2D States • 2D Echo-antiecho

7.1.1. 1D If at least one FID is present after a simulation, a 1D processing can be performed. The first FID in the dataset is transformed after application of window-functions and zero filling. Only the first FID can be processed and shown.

7.1.2. 2D magnitude This processing scheme is used if phase sensitivity is omitted in the indirect dimension.

7.1.3. 2D TPPI The TPPI processing scheme is used for simulations with the TPPI-method2 implemented for phase sensitivity in the indirect dimension.

7.1.4. Stacked 1D Stacked 1D processes all FIDs in the dataset and shows them stacked in a 2D-plot. The stacked 1D processing scheme is useful for 1D data in which, for example, a relaxation delay has been incremented.

7.1.5. 2D States The States processing scheme is used for simulations with the States-method1 implemented for phase sensitivity in the indirect dimension.

7.1.6. 2D Echo-antiecho The Echo-antiecho processing scheme is used for datasets in which sensitivity enhancement using the echo-antiecho method7 has been implemented.

7.1.7. Zero filling Zero filling is done if the number given is larger than the number of complex points acquired in the specific dimension. If the number is smaller than the number of points, the dataset is truncated.

27

7.1.8. Phasing Phasing parameters are given in degrees.

7.1.9. Window functions Window functions are given for both dimensions. Multiple window-functions can be used; they are then applied after each other.

7.1.9.1. Exponential window Multiplies the FID with the function , where the processing parameter LB (Hz) determines the line broadening.

tLBe ⋅⋅−π

7.1.9.2. Gaussian window

Multiplies the FID with the function , with 2btate −− LBa ⋅= π and

AQGBab⋅⋅

−=2

,

where AQ is the total acquisition time of a FID. LB and GB are processing parameters in Hz. LB should be entered as a negative value.

7.1.9.3. Sine bell window

Multiplies the FID with the function ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅− c

AQtcπsin , with

SSBc π= , where

SSB is the processing parameter and AQ is the total acquisition time of a FID. A sine wave is obtained for SSB = 1 and a cosine wave for SSB=2.

7.1.9.4. Squared sine bell window

Multiplies the FID with the function ( )2

sin ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅− c

AQtcπ , with

SSBc π= , where

SSB is the processing parameter and AQ is the total acquisition time of a FID. A squared sine wave is obtained for SSB = 1 and a squared cosine wave for SSB=2.

7.1.10. Special processing Special processing is primarily used for nuclei with a negative magnetogyric ratio8. See the theoretical background section.

7.1.11. Plotting parameters Spectra can either be plotted in 1D or 2D plots. 2D-spectra are contoured using parameters given in the Contouring section. Positive and negative parts of spectra can be included or omitted using the check boxes. The number of contours is given in the “Number of levels” field. The level multiplier gives the relative distance between two contours, contouring is always started from the point in the spectra with the largest absolute value. The first contour is drawn at the largest value divided by the level multiplier, the second contour at the last levels value divided by the level multiplier, and so on. If, for example, the largest value in the spectra is 1000 and the level multiplier is 10, the first contours will be at ±100, the next at ±10 and so on.

The size, in millimeters, of the spectrum is defined by the values in the “x-size” and “y-size” fields.

28

7.2. The Spectrum window After processing spectra are shown in the Spectrum window. To show the spectrum, select “View” and then “Spectrum”. The spectrum can be printed by selecting the “Print” command from the “File” menu. The spectrum can be exported in a text format, suitable for manipulations in Matlab, by selecting “Export Spectrum…” from the “Spectrum” menu.

To zoom in a spectrum make a box with the mouse by left clicking in the spectrum, right click within the box to get a pop-up menu and then select “expand” from the menu. To go back to show the full spectrum right click anywhere in the spectrum and choose “Full” from the menu.

29

8. Examples The QSim distribution contains some examples. In this section each example is briefly described. All examples are ready for simulation, just go to the simulations window and press start. All examples, except the example in 8.5, can be processed using stored parameters.

8.1. HSQC with sensitivity enhancement This example illustrates the use of gradients in simulations and processing using the sensitivity enhancement scheme. The pulse sequence used (Figure 13) and all parameter values are taken from the paper by Kay and co-workers7. The spin system simulated is a 1H and 15N spin system in a protein backbone.

To achieve phase sensitivity in the indirect dimension using the sensitivity enhancement scheme, the phi1 variable was set to a super cycle of 180º and -180º, simultaneously the amplitude of the last gradient was super cycled using the values 29.05 G/cm and -29.05 G/cm. 50 slices were simulated for the gradients.

Figure 14. Pulse sequence for sensitivity enhanced HSQC7.

After the simulation the data was processed using the “Echo-antiecho” processing scheme. The result is presented in Figure 14.

The HSQC with sensitivity enhancement example is found in the Examples section of the QSim distribution, the file name is se-hsqc.qsd. The document does not contain any simulated or processed data. The simulation time for the example is a couple of minutes.

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Figure 15. Spectrum simulated from the sensitivity enhanced HSQC pulse sequence.

8.2. TOCSY spectrum with TPPI This example illustrates the use of TPPI for phase sensitive detection in the indirect dimension. It also illustrates the use of average Liouvillians, instead of pulses for mixing sequences. The pulse sequence used is depicted in Figure 15. To obtain phase sensitive information in the indirect dimension the phi1 variable was incremented 90 degrees for each increment. In practice this means setting the incremental order of phi1 to the same value as for the time increment, t1. To phi1 a phase cycle was added by marking the “Add cyclic” box in the Variables properties dialog box (see Figure 6) and giving the phase cycle in the “Cyclic” field.

Figure 16. The pulse sequence for 2D TOCSY.

After simulation was done the spectrum was processed using the “2D TPPI” option to obtain the spectrum in Figure 16.

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Figure 17. Spectrum from the TOCSY with TPPI pulse sequence.

The TOCSY with TPPI example is found in the Examples section of the QSim distribution, the file name is tocsy_tppi.qsd. The document does not contain any simulated or processed data. The simulation time for the example is about ten minutes.

8.3. Nondecoupled HSQC with chemical exchange This example illustrates both cross correlation between CSA and dipole-dipole relaxation, but also chemical exchange. The pulse sequence used is an HSQC pulse sequence without decoupling. The States method was used to obtain phase sensitivity in the indirect dimension, by super cycling the phi2 variable in Figure 17. The spin system contains two spins, one 1H and one 15N. Two state chemical exchange is simulated and both the forward and backward reaction rates are 10 s-1. The magnetic field strength was set to 18.8 T.

Figure 18. The HSQC pulse sequence without decoupling.

32

The resulting spectrum shows crosspeaks due to chemical exchange in the slow regime. The cross correlation between CSA and dipole-dipole relaxation is visible as different line shapes for the four peaks. The lower right peak being the narrowest, this peak is the peak used in TROSY-type experiments.

Figure 19. The spectrum obtained from the nondecoupled HSQC pulse sequence, with chemical exchange present.

The nondecoupled HSQC example is found in the Examples section of the QSim distribution, the file name is undec_hsqc_chexch.qsd. The document does not contain any simulated or processed data. The simulation time for the example is about one minute.

8.4. Classical simulation of a NOESY spectrum This example shows how large spin systems can be simulated using classical mechanics and the Solomon equations. In this example arbitrary chemical shifts and distances were used. A 2D NOESY pulse sequence was simulated and the resulting spectrum is shown in Figure 19.

The NOESY example is found in the Examples section of the QSim distribution, the file name is noesy.qsd. The document does not contain any simulated or processed data. The simulation time for the example is about one minute.

33

-200 -100 0 100 200

-200

-100

0

100

200

Figure 20. Simulated NOESY spectrum using a spin system with 20 1H. The simulation was performed using classical mechanics.

8.5. Mixing sequence simulation This example illustrates how watches can be used to follow what happens to the magnetization during a mixing sequence. The simulated pulse sequence is illustrated in Figure 20. Note the watch at the beginning of the mixing sequence.

Figure 21. The pulse sequence used to follow the magnetization during a mixing sequence.

34

To be able to follow the magnetization during the mixing sequence the following advanced simulation parameters are set (as shown in Figure 21):

• Simulate FID is not selected • Save watches to file • Use custom magnetization. The custom magnetization is set to zero for

all product operators except E*E*3z, which is set to one. • Use output magnetization from previous transient.

The number of transients is set to a large value (in this case 100) and the number of super cycles is set to 1.

Figure 22. The simulation parameters for simulation of a Dipsi2rc mixing sequence.

The watch file is then imported into, for example, Microsoft Excel and the magnetization plotted, as shown in Figure 22. This example follows the Dipsi2rc sequence9. In this example the mixing sequence has been simulated as pulses, and it is thus an entire decoupling sequence between each point in the diagram.

The mixing sequence Cross correlation example is found in the Examples section of the QSim distribution, the file name is mixsim.qsd. The document does not contain any simulated or processed data. Before running the example, change the file to store watches to. The simulation time for the example is a couple of minutes.

35

0

1

0 50 100 150 200 250 300 350 400 450

time (ms)

Inte

nsity

E*E*3z E*2z*E 1z*E*E

Figure 23. Example of product operators and their magnitudes during a simulation. 1z*E*E is the same as Iz for the first spin, E*2z*E is Iz for the second spin and E*E*3z is Iz for the third spin.

8.6. QP-DD Cross correlation The effect of cross correlation between quadrupolar relaxation and dipole-dipole relaxation is illustrated in this example. The spin system is a 2H-13C moiety. The 2H has a quadrupolar constant of 170 kHz the scalar coupling constant is 20 Hz and the distance between the nuclei is 1.09 Å.

Figure 24. Spectrum from an experiment illustrating the effect or cross correlation between quadrupolar and dipole-dipole relaxation.

36

Note the shift of the center resonance in the triplet in Figure 23, which is a result of the cross correlation between the quadrupolar relaxation of 2H and the dipole-dipole relaxation. The effect has been described by Bax and co-workers10.

The QP-DD Cross correlation example is found in the Examples section of the QSim distribution, the file name is qpdd.qsd. The document does not contain any simulated or processed data. The simulation time for the spectrum in Figure 23 is a couple of seconds.

8.7. TROSY The TROSY example illustrates the cross correlation effect between CSA and dipole-dipole relaxation. The pulse sequence, Figure 24, is adapted from the one published by Pervushin and co-workers11.

Phase sensitive detection in the indirect dimension is performed using the States method1.

Figure 25. The pulse sequence for TROSY11.

The result after processing and plotting is shown in Figure 25. The x-axes show chemical shifts in Hz for 1H and the y-axes show chemical shifts in Hz for 13C.

The chemical shift for the 1H is 30 Hz and 10 Hz for 15N, since the coupling constant is -92 Hz, this agrees well with a peak at 36 Hz for 15N and 76 Hz for 1H. The simulation is performed at 14.1 T.

The TROSY example is found in the Examples section of the QSim distribution, the file name is trosy.qsd. The document does not contain any simulated or processed data. The simulation time for the spectrum in Figure 25 is a couple of seconds.

37

Figure 26. The result of the TROSY experiment. Chemical shifts for 1H is on the x-axes and chemical shifts for 15N is on the y-axes.

8.8. HNCO The HNCO spectrum is an example of a complex pulse sequence (Figure 26). The pulse sequence is taken from the book by Cavanagh12. The spin system simulated is a three spin system with 1H, 15N and 13C. Couplings and distances are set up to mimic the backbone of a protein.

Phase sensitive detection in the indirect dimension is performed using the States method1.

Figure 27. The HNCO pulse sequence in the HNCO example12.

38

The result after processing and plotting is shown in Figure 27. The x-axes show chemical shifts in Hz for 1H and the y-axes show chemical shifts in Hz for 13C.

Figure 28. A 2D plane from an HNCO spectrum. 1H on the x-axes and 13C on the y-axes.

The HNCO example is found in the Examples section of the QSim distribution, the file name is hnco.qsd. The document does not contain any simulated or processed data. The simulation time for the spectrum in Figure 27 is a couple of minutes.

8.9. Carbonyl 13C transverse relaxation This example shows the use of gradients, shaped pulses in a complex pulse sequence. The experiment is used to measure 13C transverse carbonyl relaxation, as described by Mulder and co-workers13. The pulse sequence is shown in Figure 28. The first shaped pulses in the 13C’ channel is an adiabatic pulse, the second a spin lock and the third an adiabatic pulse. The spin system simulated is a three spin system with 1H, 15N and 13C. Couplings and distances are set up to mimic the backbone of a protein.

The result of simulating the pulse sequence is shown in Figure 29. To simulate an entire relaxation set the spin lock length and the spin lock frequencies are varied separately.

The carbonyl 13C transverse relaxation example is found in the Examples section of the QSim distribution, the file name is COrelaxation.qsd. The document does not contain any simulated or processed data. The simulation time for the example is about half an hour.

39

Figure 29. Pulse sequence for measurement of measurement of 13C transverse relaxation rates. The pulse sequence is taken from a paper by Mulder and co-workers13.

Figure 30. The 1H-13C’ plane from the pulse sequence in Figure 28. 1H on the x-axes and 13C’ on the y-axes.

40

9. Theoretical background The quantum mechanical master equation describing the evolution of the

density operator including relaxation, the Liouville-von Neumann equation14-16, has the mathematical form of an inhomogeneous differential equation. The fact that it is inhomogeneous complicates its use for numerical simulations and analytical calculations. The standard approach to the problem is to ignore relaxation altogether. Without relaxation the Liouville-von Neumann equation is homogeneous. A more constructive approach is to ignore relaxation during RF pulses and to include relaxation during periods of free precession. The spin Hamiltonian commutes with the equilibrium density operator in the absence of an RF field, and a simple substitution of the density operator makes the master equation homogeneous during delays. None of these methods is an optimal solution to the problem. It has been shown, however, that the Liouville-von Neumann equation can be rewritten in a homogeneous form without introducing any approximations17-21. All simulations in QSim are performed in superspace (Liouville space) using the homogenous form of the master equation or the corresponding homogeneous classical mechanical system of equations6, 22.

9.1. Basic equations The common inhomogeneous form of the quantum mechanical master

equation, the Liouville-von Neumann equation for the density operator σ, can be written as14-16

[ ] ( 0ˆ, σσσσ −Γ−−= Hi

dtd ) (1)

or

( ) 0ˆ ˆ ˆˆ ˆ ˆd iH

dtσ σ σ= − +Γ +Γ ⋅ (2)

with

[ˆ ,H H ]σ σ= . (3)

where H is the spin Hamiltonian, Γ is the relaxation superoperator, ˆH is the Liouville superoperator without relaxation and σ0 is the density operator at equilibrium.

The Liouville-von Neumann equation can easily be rewritten in a homogeneous form when the unity operator is included in the set of basis operators according to

( ) ( )0ˆ ˆˆ ˆ ˆˆ ˆˆ ˆ ˆ

improved totd diH iH Ldt dt

ˆσ σ σ σ σ= − +Γ +Γ → = − +Γ = − σ (4)

where is the total homogeneous Liouvillian. The way to do it is to include the vector corresponding to the relaxation superoperator multiplied with the equilibrium

density operator, , as a column in the matrix at the position corresponding to the unity operator, column 1. The first element in the vector describing the density operator should finally be set to 1.

ˆtotL

0ˆσΓ ˆ

totL

41

9.2. Spin Hamiltonian The multiple rotating frame spin Hamiltonian is assumed and used in all

calculations. The Zeeman Hamiltonian, H, is given by15 i iz

iH = Ω∑ I

F

(5)

with 02 i i Rπν ω= Ω = −ω

)

(6) where ν is the chemical shift offset frequency in s-1, Ω is the chemical shift offset frequency in rad/s, ω0 is the resonance frequency in rad/s and ωRF is the frequency of the RF field in rad/s. The rotating frame spin Hamiltonian is

(2kl kx lx ky ly kz lzH J I I I I I Iπ= + +∑ , (7) for homonuclear scalar interactions and

2kl kz lzH J I Sπ=∑ , (8) for heteronuclear scalar interactions, where Jkl is the scalar coupling constant between spin k and l in Hz. The spin Hamiltonian in the rotating frame for the applied RF field is

( x x y yH Iω ω= +∑ )I (9)

where ωx and ωy are the RF magnetic field components along the x and y axes in rad/s with

( )( )

1

1

cos ,

sin ,x

y

B

B

ω γ φ

ω γ φ

= −

= − (10)

where γB1 is the magnetogyric ratio times the strength of the RF field and φ is the phase of the RF field.

In partially oriented systems, i.e. non isotropic liquids, a small fraction of the quadrupolar and dipolar interactions remain active due to incomplete averaging23, 24. The spin Hamiltonian, H, for spin S>½ with nuclear quadrupolar coupling and an axially symmetric electric field gradient tensor is

2 13Q zH Sπν ⎛= −⎜

⎝ ⎠2S ⎞⎟ , (11)

where νQ is the quadrupolar splitting. The spin Hamiltonian for residual dipolar coupling in nonisotropic liquids is

(122kl kz lz kx lx ky lyH D I I I I I Iπ ⎛= − +⎜

⎝ ⎠∑ )⎞⎟ , (12)

for homonuclear interactions, and 2kl kz lzH D I Sπ=∑ , (13)

for heteronuclear interactions, where Dkl is the residual dipolar coupling between spin k and l in Hz. Both residual coupling terms, νQ and Dkl, are the coupling observed in a spectrum, without the factor 2 often used.

The coherent part of the Liouville-von Neumann equation can be calculated from the spin Hamiltonian matrix element by element according to15

[ ] ( )

†

†

ˆ ,ˆ r s sr sr s r s srs

r r r r r r r r

Tr B HB B HB H BB H B B HB B HH

B B B B B B Tr B B

−−= = = = ,(14)

where B is a complete set of basis operators and † stands for the adjoint.

42

9.3. Basis operators We have used a complete set of Hermitian operators as basis operators for

quantum mechanical calculations25. For spin quantum number S=½ the Hermitian operator basis set corresponds to the ordinary Cartesian product operators. The main advantage with using an Hermitian basis operators is that all relevant superoperators are represented as real matrices and all relevant spinoperators as real vectors25. With any other set of basis operators the superoperators (matrices) become complex. The calculations will in theory be faster using real algebra, even though efficiency of implementation is obviously also very important.

The Hermitian basis is based on linear combinations of irreducible tensor operators25. The matrix representations of the irreducible tensor operators can be calculated using Wigner 3-J symbols, according to26

( )( ) MpSMSMpqMSkS

kSTS

SM

S

SMp

MSSqk ,,11212, ⎥

⎦

⎤⎢⎣

⎡−

−++= ∑ ∑−= −=

− , (15)

where S is the spin quantum number, 0 ≤ k ≤ 2S and -k ≤ q ≤ k in steps of 1. T is normalized so that for any spin quantum number S. Linear combinations of the irreducible tensor operators can be formed

ET S =0,025

( )( )(, ,

1 31

2qS S

k qx k q k q

S SC T −

+= + ),

ST− , (16)

( )( )(, ,

1 31

2qS S

k qy k q k q

i S SC T −

+= − ),

ST− , (17)

( ) Sk

Szk TSSC 0,, 31+= , (18)

where and 0 ≤ q ≤ k. These operators form a complete set of basis operators which are orthogonal to each other and Hermitian25. The basis is normalized so that independently of the spin quantum number and . This implies

yS

yxS

x SCSC == ,1,1 , zS SC =0,1

( )( )†,

1 2 13r s r s r s

S S SB B Tr B B δ

+ += = , (19)

where Br and Bs are the basis operators. It is easy to construct physically relevant operators, such as Hamiltonian operators, density operators and detection operators using these Hermitian basis operators.

9.4. Chemical exchange Chemical kinetics are included in the simulations by QSim as the

homogeneous version of the stochastic Liouvillian equation6 in the case of quantum mechanics or the homogeneous McConnell equations6 in the case of classical mechanics. The chemical exchange processes are described with a square matrix of (pseudo) first order rate constants12, 15. As an example we set up a simple first-order chemical exchange reaction with three components, A, according to15

43

[ ] [ ]

[ ] [ ]

[ ] [ ]113

313

332

232

221

121

Ak

kA

Ak

kA

Ak

kA

(20)

where k are the exchange rate constants for the forward and reverse reactions. The differential equation system for the chemical exchange process is easily set up according to the chemical reaction rate law

[ ] ( )[ ] [ ] [ ]

[ ] [ ] ( )[ ] [ ]

[ ] [ ] [ ] ( )[ ]

1 12 13 1 21 2 31

2 12 1 21 23 2 32 3

3 13 1 23 2 31 32 3

d A k k A k A k Adtd A k A k k A k Adtd A k A k A k k Adt

= − + + +

= − + +

= + − +

3

(21)

which can be written in matrix form as [ ][ ][ ]

( )( )

( )

[ ][ ][ ]

1 12 13 21 31

2 12 21 23 32

3 13 23 31 32

1

2

3

A k k k k Ad A k k k kdt

AA k k k k

⎡ ⎤ ⎡ ⎤ ⎡ ⎤− +⎢ ⎥ ⎢ ⎥ ⎢= − +⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥− +⎣ ⎦ ⎣ ⎦ ⎣ ⎦A

⎥⎥ . (22)

This equation can be generalized to many coupled first-order reactions according to

AKAdtd

= , (23)

where A is a vector of concentrations and with the elements of the kinetic matrix K defined as15

∑≠

−=

≠=

jrjrjj

rjjr

kK

jrkK

.

, (24)

It is also possible to handle higher order chemical reactions by defining pseudo first-order rate constants15. These are calculated by dividing the reaction rates with the concentration of the reactant molecule. In the case of non-equilibrium reactions these pseudo first-order rate constants are time dependent making the kinetic matrix K time dependent. However, if the system is in chemical equilibrium the kinetic matrix becomes time independent and equivalence with true first-order kinetics is obtained15.

A product space between chemical configuration space and magnetization, superspace, is required in order to account for the flow of magnetization during the chemical exchange17, 27, 28,

[ ]IS SI

IS SI

k k compositeMagnetization

k k space−⎡ ⎤ ⎡ ⎤

⊗ =⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦ (25)

All information about chemical kinetics necessary for NMR simulations are included in the kinetic matrix and it is thus used as input to QSim. The null-space vector of the kinetic matrix defines the equilibrium concentrations. The equilibrium concentrations are used when calculating the density operator at equilibrium, σ0.

44

The chemical exchange should be a Markovian random process17 and the sudden jump approximation is assumed, which implies that the magnetization does not change orientation during the chemical exchange17.

9.5. Relaxation The relaxation superoperator is calculated using operator algebra as described

in Abragam14. Interaction operators for dipole-dipole, axially symmetric chemical shift anisotropy and axially symmetric quadrupolar relaxation are included. All possible permutations of 1 to 4 nuclear spins and 1 or 2 interaction operators can be used. All possible cross-correlation, interference, interactions between different relaxation mechanisms can thus be calculated.

The elements of the relaxation matrix, including dynamic frequency shifts, in the basis of any complete set of operators is calculated according to

( ) ( )( )†

, ,

, ,118

m nq r q snm nm

rs n mn m q r r

A B A BJ iL Tr

B Bω ω ω ω

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎣ ⎦ ⎣ ⎦Γ = − ⎨ ⎬⎪ ⎪⎩ ⎭

∑ , (26)

where † denotes the adjoint and in which An is an interaction operator for relaxation mechanism n. The subscript q defines the different interaction operators with their respective frequencies. The factor ωn is the strength of relaxation mechanism n. J(ω) and L(ω) are the spectral density at the angular frequency ω. The indices n and m run over the different relaxation mechanisms involved. If n equals m the auto-correlation relaxation rate of a particular mechanism is calculated whereas if n is not equal to m the cross-correlation, interference, relaxation rate between two mechanisms is calculated. The indices r and s denote the two operators in the basis set between which the relaxation rate is calculated. If r equals s the auto-relaxation rate of a particular basis operator is calculated whereas if r is not equal to s the cross-relaxation rate between two basis operators is calculated.

The interaction operators for homonuclear dipole-dipole relaxations and their respective frequencies are

( ) ( )

( ) ( )

( )

0

1

2

12 ,2

6 ,2

6 , 2 ,2

ISz z

ISz z IS

ISIS

A S I I S I S

A I S I S

A S I

ω

ω ω

ω ω

− + + −

± ± ±

± ± ±

= − + =

= + = ±

= = ±

m

0 ,

, (27)

and the interaction operators for heteronuclear dipole-dipole relaxation are

45

( )

( )

( )

( )

( )

( )

0

1, 1

1, 1

0, 1

1,0

1, 1

2 , 0 ,1 , ,21 , ,26 , ,

26 , ,

26 , ,

2

ISz z

ISS I

ISI S

ISz S

ISz I

ISI S

A S I

A I S

A I S

A I S

A I S

A S I

ω

ω ω ω

ω ω ω

ω ω

ω ω

ω ω ω

− + − +

+ − + −

± ±

± ±

± ± ± ±

= =

= − = −

= − = −

= = ±

= = ±

= = ±

m

m

±

(28)

the interaction operators for chemical shift anisotropy relaxation are ( )

( )

0

1

2 , 0 ,

6 , .2

Sz

SS

A S

A S

ω

ω ω± ±

= =

= =m ± (29)

and finally, the interaction operators for quadrupole relaxation are ( )

( ) ( )

( )

2 20

1

22

3 , 0 ,

6 ,2

6 , 2 .2

Sz

Sz z

SS

A S S

A S S S S

A S

ω

ω ω

ω ω

± ± ±

± ±

= − =

= + = ±

= = ±

m ,S (30)

The secular approximation is applied and commutators are only calculated for pair of interaction operators with the same frequency.

The strength of the dipole-dipole relaxation mechanism is

033

4I S

dISr

µ γ γωπ

⎛ ⎞⎛ ⎞= ⎜⎜ ⎟⎝ ⎠⎝ ⎠

h⎟

S

(31)

where µ0 is the permeability of vacuum; rIS is the distance between spins I and S. The strength of chemical shift anisotropy relaxation is

( )|| 0c Bω σ σ γ⊥= − − , (32)

where σ|| and σ⊥ are the shielding constants for the parallel and perpendicular directions an axially symmetric shielding tensor, respectively and B0 is the static magnetic field strength of the magnet. The strength of quadrupole relaxation mechanism is given by

( )2 3

4 2 1Q S Sπ χω =

−, (33)

where χ is the nuclear quadrupole coupling constant and S is the spin quantum number. The relaxation rates are calculated assuming axially symmetric electric field gradient tensors. The direct influence on relaxation of an RF field has not been taken into account. This effect has been shown to be very small in practical applications29, 30. The indirect influence of a RF field, which is important when chemical exchange processes are taking place, is taken care of using the stochastic Liouvillian for

46

quantum mechanics and the homogenous McConnell equations for classical mechanics.

9.6. Model of dynamics The relaxation rates and dynamic frequency shifts are dependent on the

dynamics of the nucleus studied. A dynamic model is used in order to describe the motional properties of vectors in molecules. The dynamic model is then used to calculate an analytical complex spectral density function, G(ω). The relaxation rates are functions of the real part of the spectral density function, J(ω), at certain angular frequencies. We have in the simulations by QSim used the model for dynamics derived using the Lipari-Szabo approach3, 4. The analytical spectral density function for this model is

( )( )

( )( ) ⎥⎦

⎤⎢⎣

⎡

+−

++

= 2

2

2

2

11

152

i

i

m

m SSJωτ

τωττω

(34)

with

emi τττ111

+= (35)

where τm is the rotational correlation time, τe is the correlation time of internal motions, and S2 the order parameter squared that describes the balance between contributions to relaxation due to overall rotation and internal motion. The assumptions behind the Lipari-Szabo model is that that the two motions are statistically independent and the overall rotation is isotropic. The imaginary contribution to the complex spectral density function, L(ω), is defined as the Hilbert transform of J(ω) and is the cause of a dynamic frequency shift for nuclei with S≥131. The imaginary contribution is

( )( )

( )( )

2 22 2

2

125 1 1

im

m i

SSL 2

ωτωτωωτ ωτ

⎡ ⎤−⎢ ⎥= +⎢ ⎥+ +⎣ ⎦

. (36)

All auto and cross-correlation relaxation mechanisms have their separate set of correlation times and order parameters in QSim. The longer correlation time is thus not assumed to be the isotropic rotational correlation time, but rather one of two correlation times approximating the dynamics of a particular vector (auto-correlation) or differential dynamics for pair of vectors (cross-correlation). There are thus two different kind of spectral density functions to be considered, corresponding to auto-correlation, and cross-correlation, respectively. If isotropic rotational diffusion of a rigid body is assumed, there is a simple relation between the different spectral densities

( ) ( ) ( ) ( )( 21 3cos 12

auto crossJ J Jω ω ω ϕ= = )− , (37)

where ϕ is the angle between the unique axes of the two interactions involved. This equation is not rigorously correct in the presence of internal motion, i.e. if S2 < 1, but it should be a good approximation if ϕ is small32.

9.7. Equilibrium density operator The renormalized equilibrium density operator used in QSim is

0 ,i eq i izi

K c Iσ γ= ∑ , (38)

47

where γ is the magnetogyric ratio, ceq is the normalized equilibrium concentration, Iz is the spinoperator for longitudinal magnetization, the sum i is over all spins in the spin system. The constant K is set so that the absolute value of largest element in the vector describing the equilibrium density operator is 1. The normalized equilibrium concentration is calculated from the null-space of the kinetic matrix as discussed previously, or set to unity in case of no chemical exchange. Standard approximations have been applied.

9.8. Pulsed field gradients We have implemented the PFG in the following way33, 34. The effect of a

pulsed field gradient is a z rotation identical to the effect of the chemical shift. When the PFG is applied a term proportional to the PFG field strength is added to all chemical shift terms according to

( )0 ,I I B zγΩ − = (39) for classical mechanics, where B0(z) is the strength of the applied PFG as a function of the physical height of the sample tube, or

( )0ˆ

Ispins

iL B z Iγ− = ∑ ˆz , (40)

for quantum mechanics, where ˆzI is the commutator superoperator of the

corresponding spinoperator. Only a single value of z can be used at a time in the numerical calculations, corresponding to a single plane in the sample tube. The calculations are therefore repeated with linearly spaced values of z and the normalized results are added. In this way integration over the height of the sample is performed.

9.9. Simulating Spectra The solution to the homogeneous master equation

( ) ( )ˆtot

d t Ldtσ = − tσ

)1σ

, (41)

is

( ) (1ˆexp tott t t L t t tσ ⎡ ⎤= + ∆ = − ∆ =⎢ ⎥⎣ ⎦

, (42)

where and σ are a matrix and vector, respectively. The important calculation step when performing simulations in QSim is thus the exponent of a real matrix. The observable magnetization can be calculated using

ˆtotL

†Obs Obs σ= . (43) where observable x and y-magnetization correspond to the operators Obs = Sx or Sy, respectively. A complete free induction decay can be calculated recursively with

( ) †1

ˆx x tot dwell kFID k S exp L t σ −

⎡ ⎤= −⎢ ⎥⎣ ⎦ (44)

( ) †1

ˆy y tot dwell kFID k S exp L t σ −

⎡ ⎤= −⎢ ⎥⎣ ⎦ (45)

where

1ˆ

k tot dwellexp L tσ kσ −⎡ ⎤= −⎢ ⎥⎣ ⎦

. (46)

48

Calculating a free induction decay in QSim is simply a question of repeatedly multiplying a vector with a previously calculated matrix and then taking the dot product of this vector with a detection vector.

9.10. Average Liouvillian Average Hamiltonian theory has been used in the development of composite pulse sequences in which a specific Hamiltonian is required during a certain time interval15. The homogeneous master equation makes it possible to formulate an average Hamiltonian theory in combination with an average relaxation superoperator, which is

named average Liouvillian theory18, 19. The average relaxation Liouvillian over a

discrete number of steps, n, with their respective Liouvillians, , and time periods, ∆t

ˆeffL

ˆnL

n, is defined by

2 2 1 1ˆ ˆ ˆˆ ˆ ˆexp exp ......exp expeff tot n n

ˆL t L t L t L⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡− = − ∆ − ∆ − ∆⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣t ⎤⎥⎦

, (47)

where is the total time for the pulse sequence. The average relaxation

Liouvillian is calculated according to

∑=

∆=n

iitot tt

1

2 2 1 11ˆ ˆ ˆˆ ˆ ˆln exp ......exp expeff n ntot

ˆL L t L t L tt

⎡ ⎤ ⎡ ⎤ ⎡= − − ∆ − ∆ − ∆⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣⎤⎥⎦

. (48)

It should be noted that the effective Liouvillian approach has certain limitations. In particular, the equality

1ˆ ln expeff effˆL L t

t⎡= − − ∆⎢⎣ ⎦∆

⎤⎥ (49)

is only true when ∆t→0. Interactions, such as the chemical shift or scalar coupling, that evolve more than π radians during ∆t, will be folded and resonance lines might appear at a unexpected positions. The total time over which an average Liouvillian is calculated should be kept short compared to the inverse of the largest interaction of interest in order to avoid problems.

9.11. Signs and Phases The signs of radio frequency phases and frequencies is a source of confusion

in NMR8. In particular, the sign of the magnetogyric ratio and how that should be handled is a problem. The problems appear in pulses, when acquiring data and when processing data. The strength of RF pulses is input in QSim as -γB1 which is the magnetogyric ratio times the strength of the RF field. All pulses behave as expected if this value is input as a positive value, i.e. a 90 degree pulse with x-phase rotates Iz to -Iy while a 90 degree pulse with y-phase rotates Iz to Ix. If -γB1 is input as a negative value for nuclei with negative magnetogyric ratios, the effect of current spectrometer hardware is emulated. Detection of magnetization in QSim is also independent of the sign of the magnetogyric ratio. Detecting Ix with receiver phase x or detecting Iy with receiver phase y both always gives an positive in-phase spectrum. Finally, during processing in QSim a flag is available that both take the complex conjugate of the FID before Fourier transformation and then plots the spectrum with the frequency increasing from right to left. This is the recommended way to present spectrum of nuclei with negative magnetogyric ratios because this ensures that less-shielded spins appear on the left of the spectrum8.

49

10. HME The NMR simulation software package HME is a stand alone part of QSim

and is written in C++. 35C++ is an object oriented programming language with classes, inheritance and operator overloading35. The HME software package defines 3 classes of importance. These classes are the spin system class, spin_sys, the spinoperator class, spin_op, and the superoperator class, super_op.

10.1. The spin system class The spin system class, spin_sys, is a container for all information about a spin

system. This includes number of spins, spin quantum numbers, chemical shifts and possibly a kinetic matrix describing chemical kinetics. All information about relaxation is also stored in this class.

10.2. The spinoperator class The spinoperator class, spin_op, describes spinoperators in superspace.

Typical spinoperators in NMR simulations are the Hamiltonian operator, the density operators at equilibrium and the density operator as a function of the pulse sequence. Spinoperators in super space are vectors of real numbers when the basis operators are Hermitian25. The spinoperator class inherits, in an object oriented fashion, many of its traits from a class of real vectors. A spinoperator object also contains a pointer to the spin system it belongs to. This pointer makes it possible to check that no algebra is performed between spinoperators belonging to different spin systems. The problem of multiplication of spinoperators in super space is also solved using the pointer to the spin system object. Operator multiplication is not defined in superspace, but using the pointer it is possible to temporarily go to Hilbert space and do the multiplication using complex matrices, before returning to superspace and real vectors. This is of course done without user intervention.

10.3. The superoperator class The final class of importance in HME is the superoperator, super_op, class.

Superoperators in NMR simulations are typically the relaxation superoperator, the Liouvillian superoperator and superoperator propagators. Physically relevant superoperators in superspace are matrices of real numbers if the basis operators are Hermitian25. The superoperator class thus inherits most if its traits from a real matrix. A superoperator object also contains a pointer to the spin system it belongs to in the same way as the spinoperator object does. The superoperator class does not strictly need this pointer for anything but error control. One such error check is to notice if the spin system has changed since the superoperator was created and warn the user if attempting to use an obsolete superoperator. Algebraic operations between spinoperators and superoperators belonging to different spin systems are not allowed.

50

11. Contact information QSim, with examples, and HME can be downloaded from www.bpc.lu.se/QSim. Up to date information on bugs and an FAQ can also be found there. If you find bugs and/or errors in simulations please report them to Magnus Helgstrand or Peter Allard.

Full addresses to the authors:

Magnus Helgstrand Dept. of Biophysical Chemistry Lund University Box 124 SE-221 00 Lund Sweden

magnus.helgstrand@bpc.lu.se

Peter Allard KTH, AlbaNova University Center Structural Biochemistry Department of Biotechnology SE-106 91 Stockholm Sweden allard@kth.se

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